mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.AffineSpace.Independent

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
ι : Type u_4
inst✝ : Nontrivial k
p : ι → P
ha : AffineIndependent k p
s₁ s₂ : Set ι
hs₁₂ : affineSpan k (p '' s₁) = affineSpan k (p '' s₂)
hne : s₁ ≠ s₂
i : ι
⊢ i ∈ s₁ ↔ i ∈ s₂

Tactic:
simp_rw [← ha.mem_affineSpan_iff, hs₁₂]

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

Goal state before:
pp: R : Type u_1
E : Type u_3
ι : Type u_5
inst✝² : Field R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
i : ι
w : ι → R
z : ι → E
hw : w i ≠ 0
⊢ (w i)⁻¹ • w i • z i = z i

Tactic:
match_scalars

Goal state after:
R : Type u_1
E : Type u_3
ι : Type u_5
inst✝² : Field R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
i : ι
w : ι → R
z : ι → E
hw : w i ≠ 0
⊢ (w i)⁻¹ * (w i * 1) = 1
Module: Mathlib.LinearAlgebra.AffineSpace.Independent

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
ι : Type u_4
p : ι → P
h : AffineIndependent k p
w w₁ w₂ : ι → k
s : Finset ι
hw : ∑ i ∈ s, w i = 0
hw₁ : ∑ i ∈ s, w₁ i = 1
hw₂ : ∑ i ∈ s, w₂ i = 1
r : k
hr : (s.weightedVSub p) (r • (w₁ - w₂) - w) = 0
i : ι
hi : i ∈ s
⊢ ∑ j ∈ s, (r • (w₁ - w₂) - w) j = 0

Tactic:
simp_rw [Pi.sub_apply, Pi.smul_apply, Pi.sub_apply, smul_sub, Finset.sum_sub_distrib, ←
        Finset.smul_sum, hw, hw₁, hw₂, sub_self]

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

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
inst✝¹⁰ : Ring 𝕜
inst✝⁹ : LinearOrder 𝕜
inst✝⁸ : IsStrictOrderedRing 𝕜
inst✝⁷ : DenselyOrdered 𝕜
inst✝⁶ : TopologicalSpace 𝕜
inst✝⁵ : OrderTopology 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : TopologicalSpace E
inst✝² : ContinuousAdd E
inst✝¹ : Module 𝕜 E
inst✝ : ContinuousSMul 𝕜 E
x y : E
⊢ [x -[𝕜] y] ⊆ closure (openSegment 𝕜 x y)

Tactic:
rw [segment_eq_image, openSegment_eq_image, ← closure_Ioo (zero_ne_one' 𝕜)]

Goal state after:
𝕜 : Type u_2
E : Type u_3
inst✝¹⁰ : Ring 𝕜
inst✝⁹ : LinearOrder 𝕜
inst✝⁸ : IsStrictOrderedRing 𝕜
inst✝⁷ : DenselyOrdered 𝕜
inst✝⁶ : TopologicalSpace 𝕜
inst✝⁵ : OrderTopology 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : TopologicalSpace E
inst✝² : ContinuousAdd E
inst✝¹ : Module 𝕜 E
inst✝ : ContinuousSMul 𝕜 E
x y : E
⊢ (fun θ ↦ (1 - θ) • x + θ • y) '' closure (Ioo 0 1) ⊆ closure ((fun θ ↦ (1 - θ) • x + θ • y) '' Ioo 0 1)
Module: Mathlib.Analysis.Convex.Topology

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
inst✝¹³ : Ring 𝕜
inst✝¹² : LinearOrder 𝕜
inst✝¹¹ : IsStrictOrderedRing 𝕜
inst✝¹⁰ : DenselyOrdered 𝕜
inst✝⁹ : PseudoMetricSpace 𝕜
inst✝⁸ : OrderTopology 𝕜
inst✝⁷ : ProperSpace 𝕜
inst✝⁶ : CompactIccSpace 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : TopologicalSpace E
inst✝³ : T2Space E
inst✝² : ContinuousAdd E
inst✝¹ : Module 𝕜 E
inst✝ : ContinuousSMul 𝕜 E
x y : E
⊢ closure (openSegment 𝕜 x y) = [x -[𝕜] y]

Tactic:
rw [segment_eq_image, openSegment_eq_image, ← closure_Ioo (zero_ne_one' 𝕜)]

Goal state after:
𝕜 : Type u_2
E : Type u_3
inst✝¹³ : Ring 𝕜
inst✝¹² : LinearOrder 𝕜
inst✝¹¹ : IsStrictOrderedRing 𝕜
inst✝¹⁰ : DenselyOrdered 𝕜
inst✝⁹ : PseudoMetricSpace 𝕜
inst✝⁸ : OrderTopology 𝕜
inst✝⁷ : ProperSpace 𝕜
inst✝⁶ : CompactIccSpace 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : TopologicalSpace E
inst✝³ : T2Space E
inst✝² : ContinuousAdd E
inst✝¹ : Module 𝕜 E
inst✝ : ContinuousSMul 𝕜 E
x y : E
⊢ closure ((fun θ ↦ (1 - θ) • x + θ • y) '' Ioo 0 1) = (fun θ ↦ (1 - θ) • x + θ • y) '' closure (Ioo 0 1)
Module: Mathlib.Topology.AlexandrovDiscrete

Goal state before:
pp: ι : Sort u_1
α : Type u_3
inst✝¹ : TopologicalSpace α
inst✝ : AlexandrovDiscrete α
f : ι → Set α
⊢ (closure[inst✝¹] (⋃ i, f i))ᶜ = (⋃ i, closure[inst✝¹] (f i))ᶜ

Tactic:
by
    simpa only [← interior_compl, compl_iUnion] using interior_iInter fun i ↦ (f i)ᶜ

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

Goal state before:
pp: case mpr.inr
𝕜 : Type u_4
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
s : Set 𝕜
hs : Convex 𝕜 s
x : 𝕜
hx : x ∈ interior s
h : s.Nontrivial
⊢ s.Nontrivial

Tactic:
· exact h

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Independent

Goal state before:
pp: case inr.refine_2
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
p : P
hp : p ∈ s
b : Set V
hb₁ : b ⊆ ⇑(Equiv.vaddConst p).symm '' s
hb₂ : Submodule.span k b = Submodule.span k (⇑(Equiv.vaddConst p).symm '' s)
hb₃ : AffineIndependent k fun p_1 ↦ ↑p_1
hb₀ : ∀ v ∈ b, v ≠ 0
⊢ affineSpan k ({p} ∪ ⇑(Equiv.vaddConst p) '' b) = affineSpan k s

Tactic:
rw [Equiv.coe_vaddConst_symm, ← vectorSpan_eq_span_vsub_set_right k hp] at hb₂
    apply AffineSubspace.ext_of_direction_eq
    · have : Submodule.span k b = Submodule.span k (insert 0 b) := by simp
      simp only [direction_affineSpan, ← hb₂, Equiv.coe_vaddConst, Set.singleton_union,
        vectorSpan_eq_span_vsub_set_right k (Set.mem_insert p _), this]
      congr
      change (Equiv.vaddConst p).symm '' insert p (Equiv.vaddConst p '' b) = _
      rw [Set.image_insert_eq, ← Set.image_comp]
      simp
    · use p
      simp only [Equiv.coe_vaddConst, Set.singleton_union, Set.mem_inter_iff]
      exact ⟨mem_affineSpan k (Set.mem_insert p _), mem_affineSpan k hp⟩

Goal state after:
Module: Mathlib.Topology.Connected.LocallyPathConnected

Goal state before:
pp: case inl.refine_1.hsF
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
x✝ y z : X
ι : Type u_3
F : Set X
inst✝¹ : LocallyPathConnectedSpace X
inst✝ : LocallyPathConnectedSpace Y
u : Set (X ⊕ Y)
hu : IsOpen[instTopologicalSpaceSum] u
x : X
hxu : inl x ∈ u
⊢ inl '' pathComponentIn (inl ⁻¹' u) x ⊆ u

Tactic:
exact (image_mono pathComponentIn_subset).trans (u.image_preimage_subset _)

Goal state after:
Module: Mathlib.Topology.Connected.LocallyPathConnected

Goal state before:
pp: case inr.refine_1.hsF
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
x y✝ z : X
ι : Type u_3
F : Set X
inst✝¹ : LocallyPathConnectedSpace X
inst✝ : LocallyPathConnectedSpace Y
u : Set (X ⊕ Y)
hu : IsOpen[instTopologicalSpaceSum] u
y : Y
hxu : inr y ∈ u
⊢ inr '' pathComponentIn (inr ⁻¹' u) y ⊆ u

Tactic:
exact (image_mono pathComponentIn_subset).trans (u.image_preimage_subset _)

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

Goal state before:
pp: 𝕜 : Type u_4
V : Type u_5
P : Type u_6
inst✝¹⁴ : Field 𝕜
inst✝¹³ : LinearOrder 𝕜
inst✝¹² : IsStrictOrderedRing 𝕜
inst✝¹¹ : TopologicalSpace 𝕜
inst✝¹⁰ : OrderClosedTopology 𝕜
inst✝⁹ : CompactIccSpace 𝕜
inst✝⁸ : ContinuousAdd 𝕜
inst✝⁷ : AddCommGroup V
inst✝⁶ : TopologicalSpace V
inst✝⁵ : IsTopologicalAddGroup V
inst✝⁴ : Module 𝕜 V
inst✝³ : ContinuousSMul 𝕜 V
inst✝² : AffineSpace V P
inst✝¹ : TopologicalSpace P
inst✝ : IsTopologicalAddTorsor P
n : ℕ
s : Simplex 𝕜 P n
this : IsCompact (⇑↑(AffineEquiv.vaddConst 𝕜 (s.points 0)).symm '' s.closedInterior)
⊢ IsCompact s.closedInterior

Tactic:
apply (Homeomorph.vaddConst (s.points 0)).symm.isCompact_image.mp

Goal state after:
𝕜 : Type u_4
V : Type u_5
P : Type u_6
inst✝¹⁴ : Field 𝕜
inst✝¹³ : LinearOrder 𝕜
inst✝¹² : IsStrictOrderedRing 𝕜
inst✝¹¹ : TopologicalSpace 𝕜
inst✝¹⁰ : OrderClosedTopology 𝕜
inst✝⁹ : CompactIccSpace 𝕜
inst✝⁸ : ContinuousAdd 𝕜
inst✝⁷ : AddCommGroup V
inst✝⁶ : TopologicalSpace V
inst✝⁵ : IsTopologicalAddGroup V
inst✝⁴ : Module 𝕜 V
inst✝³ : ContinuousSMul 𝕜 V
inst✝² : AffineSpace V P
inst✝¹ : TopologicalSpace P
inst✝ : IsTopologicalAddTorsor P
n : ℕ
s : Simplex 𝕜 P n
this : IsCompact (⇑↑(AffineEquiv.vaddConst 𝕜 (s.points 0)).symm '' s.closedInterior)
⊢ IsCompact (⇑(Homeomorph.vaddConst (s.points 0)).symm '' s.closedInterior)
Module: Mathlib.Topology.Connected.LocallyPathConnected

Goal state before:
pp: case refine_1.hsF
X✝ : Type u_1
Y : Type u_2
inst✝⁴ : TopologicalSpace X✝
inst✝³ : TopologicalSpace Y
x✝ y z : X✝
ι : Type u_3
F : Set X✝
inst✝² : LocallyPathConnectedSpace X✝
X : ι → Type u_4
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : ∀ (i : ι), LocallyPathConnectedSpace (X i)
x : (i : ι) × X i
u : Set ((i : ι) × X i)
hu : IsOpen[instTopologicalSpaceSigma] u
hxu : x ∈ u
⊢ mk x.fst '' pathComponentIn (mk x.fst ⁻¹' u) x.snd ⊆ u

Tactic:
exact (image_mono pathComponentIn_subset).trans (u.image_preimage_subset _)

Goal state after:
Module: Mathlib.Topology.Algebra.Module.LocallyConvex

Goal state before:
pp: case inl
R : Type u_1
inst✝⁴ : TopologicalSpace R
inst✝³ : Semiring R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : OrderTopology R
x : R
hl : IsBot x
⊢ (𝓝 x).HasBasis (fun s ↦ s ∈ 𝓝 x ∧ Convex R s) id

Tactic:
refine hl.rec ?_ _

Goal state after:
case inl
R : Type u_1
inst✝⁴ : TopologicalSpace R
inst✝³ : Semiring R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : OrderTopology R
x : R
hl : IsBot x
⊢ ∀ [inst : OrderBot R], (𝓝 ⊥).HasBasis (fun s ↦ s ∈ 𝓝 ⊥ ∧ Convex R s) id
Module: Mathlib.Topology.Algebra.Module.LocallyConvex

Goal state before:
pp: E : Type u_1
𝕜 : Type u_2
inst✝⁵ : Semiring 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : AddCommMonoid E
inst✝² : Module 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : LocallyConvexSpace 𝕜 E
s : Set E
hs : Convex 𝕜 s
x₀ : E
hx₀s : x₀ ∈ s
p : E → Prop
h : ∀ᶠ (x : E) in 𝓝[s] x₀, p x
⊢ ∀ᶠ (x : E) in 𝓝[s] x₀, ∀ y ∈ [x₀ -[𝕜] x], p y

Tactic:
rw [eventually_nhdsWithin_iff, (LocallyConvexSpace.convex_basis (𝕜 := 𝕜) x₀).eventually_iff]
    at h ⊢
  obtain ⟨u, ⟨hu_nhds, hu_convex⟩, h⟩ := h
  refine ⟨u, ⟨hu_nhds, hu_convex⟩, fun x hxu hxs y hy ↦ h ?_ (hs.segment_subset hx₀s hxs hy)⟩
  suffices segment 𝕜 x₀ x ⊆ u from this hy
  exact hu_convex.segment_subset (mem_of_mem_nhds hu_nhds) hxu

Goal state after:
Module: Mathlib.Topology.Algebra.SeparationQuotient.Section

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝⁵ : DivisionRing K
inst✝⁴ : AddCommGroup E
inst✝³ : Module K E
inst✝² : UniformSpace E
inst✝¹ : IsUniformAddGroup E
inst✝ : ContinuousConstSMul K E
⊢ IsUniformInducing (mk ∘ ⇑(outCLM K E))

Tactic:
mk_comp_outCLM

Goal state after:
K : Type u_1
E : Type u_2
inst✝⁵ : DivisionRing K
inst✝⁴ : AddCommGroup E
inst✝³ : Module K E
inst✝² : UniformSpace E
inst✝¹ : IsUniformAddGroup E
inst✝ : ContinuousConstSMul K E
⊢ IsUniformInducing id
Module: Mathlib.Topology.Algebra.Module.UniformConvergence

Goal state before:
pp: case hsmul
𝕜 : Type u_1
α : Type u_2
E : Type u_3
H : Type u_4
hom : Type u_5
inst✝¹⁰ : NormedField 𝕜
inst✝⁹ : AddCommGroup H
inst✝⁸ : Module 𝕜 H
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : TopologicalSpace H
inst✝⁴ : UniformSpace E
inst✝³ : IsUniformAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : FunLike hom H (α → E)
inst✝ : LinearMapClass hom 𝕜 H (α → E)
φ : hom
hφ : IsInducing (⇑ofFun ∘ ⇑φ)
h : ∀ (u : H), Bornology.IsVonNBounded 𝕜 (Set.range (φ u))
this✝ : IsTopologicalAddGroup H
hb : (𝓝 0).HasBasis (fun x ↦ x ∈ 𝓝 0) fun V ↦ {u | ∀ (x : α), φ u x ∈ V}
U : Set E
hU : U ∈ 𝓝 0
this : Tendsto (fun x ↦ x.1 • x.2) (𝓝 0) (𝓝 0)
V : Set 𝕜
W : Set E
hVW : Set.MapsTo (fun x ↦ x.1 • x.2) (id (V, W).1 ×ˢ id (V, W).2) U
hV : (V, W).1 ∈ 𝓝 0
hW : (V, W).2 ∈ 𝓝 0
⊢ ∃ V ∈ 𝓝 0, ∃ j ∈ 𝓝 0, V • {u | ∀ (x : α), φ u x ∈ j} ⊆ {u | ∀ (x : α), φ u x ∈ U}

Tactic:
refine ⟨V, hV, W, hW, Set.smul_subset_iff.2 fun a ha u hu x ↦ ?_⟩

Goal state after:
case hsmul
𝕜 : Type u_1
α : Type u_2
E : Type u_3
H : Type u_4
hom : Type u_5
inst✝¹⁰ : NormedField 𝕜
inst✝⁹ : AddCommGroup H
inst✝⁸ : Module 𝕜 H
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : TopologicalSpace H
inst✝⁴ : UniformSpace E
inst✝³ : IsUniformAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : FunLike hom H (α → E)
inst✝ : LinearMapClass hom 𝕜 H (α → E)
φ : hom
hφ : IsInducing (⇑ofFun ∘ ⇑φ)
h : ∀ (u : H), Bornology.IsVonNBounded 𝕜 (Set.range (φ u))
this✝ : IsTopologicalAddGroup H
hb : (𝓝 0).HasBasis (fun x ↦ x ∈ 𝓝 0) fun V ↦ {u | ∀ (x : α), φ u x ∈ V}
U : Set E
hU : U ∈ 𝓝 0
this : Tendsto (fun x ↦ x.1 • x.2) (𝓝 0) (𝓝 0)
V : Set 𝕜
W : Set E
hVW : Set.MapsTo (fun x ↦ x.1 • x.2) (id (V, W).1 ×ˢ id (V, W).2) U
hV : (V, W).1 ∈ 𝓝 0
hW : (V, W).2 ∈ 𝓝 0
a : 𝕜
ha : a ∈ V
u : H
hu : u ∈ {u | ∀ (x : α), φ u x ∈ W}
x : α
⊢ φ (a • u) x ∈ U
Module: Mathlib.Topology.Algebra.Module.Spaces.UniformConvergenceCLM

Goal state before:
pp: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝⁸ : NormedField 𝕜₁
inst✝⁷ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_3
F : Type u_4
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜₁ E
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜₂ F
inst✝¹ : UniformSpace F
inst✝ : IsUniformAddGroup F
𝔖 : Set (Set E)
⊢ instTopologicalSpace σ F 𝔖 =
    TopologicalSpace.induced (⇑(UniformOnFun.ofFun 𝔖) ∘ DFunLike.coe) (UniformOnFun.topologicalSpace E F 𝔖)

Tactic:
instTopologicalSpace

Goal state after:
𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝⁸ : NormedField 𝕜₁
inst✝⁷ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_3
F : Type u_4
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜₁ E
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜₂ F
inst✝¹ : UniformSpace F
inst✝ : IsUniformAddGroup F
𝔖 : Set (Set E)
⊢ TopologicalSpace.induced DFunLike.coe (UniformOnFun.topologicalSpace E F 𝔖) =
    TopologicalSpace.induced (⇑(UniformOnFun.ofFun 𝔖) ∘ DFunLike.coe) (UniformOnFun.topologicalSpace E F 𝔖)
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: case mp
𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
p : SeminormFamily 𝕜 E ι
inst✝ : TopologicalSpace E
s : Set E
hp : WithSeminorms p
⊢ (∀ (I : Finset ι), ∃ r > 0, ∀ x ∈ s, (I.sup p) x < r) → ∀ (i : ι), ∃ r > 0, ∀ x ∈ s, (p i) x < r

Tactic:
intro hI i

Goal state after:
case mp
𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
p : SeminormFamily 𝕜 E ι
inst✝ : TopologicalSpace E
s : Set E
hp : WithSeminorms p
hI : ∀ (I : Finset ι), ∃ r > 0, ∀ x ∈ s, (I.sup p) x < r
i : ι
⊢ ∃ r > 0, ∀ x ∈ s, (p i) x < r
Module: Mathlib.Analysis.Normed.Operator.Basic

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
𝓕 : Type u_8
inst✝⁸ : SeminormedAddCommGroup E
inst✝⁷ : SeminormedAddCommGroup F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝² : FunLike 𝓕 E F
inst✝¹ : SemilinearMapClass 𝓕 σ₁₂ E F
inst✝ : RingHomSurjective σ₁₂
f : 𝓕
x : F
r : ℝ
hr : 0 < r
h : ∀ x_1 ∈ ball x r, x_1 ∈ Set.range ⇑f
⊢ ∀ x ∈ ball 0 r, x ∈ (↑f).range

Tactic:
intro _

Goal state after:
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
𝓕 : Type u_8
inst✝⁸ : SeminormedAddCommGroup E
inst✝⁷ : SeminormedAddCommGroup F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝² : FunLike 𝓕 E F
inst✝¹ : SemilinearMapClass 𝓕 σ₁₂ E F
inst✝ : RingHomSurjective σ₁₂
f : 𝓕
x : F
r : ℝ
hr : 0 < r
h : ∀ x_1 ∈ ball x r, x_1 ∈ Set.range ⇑f
x✝ : F
⊢ x✝ ∈ ball 0 r → x✝ ∈ (↑f).range
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: case mpr
𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
p : SeminormFamily 𝕜 E ι
inst✝ : TopologicalSpace E
s : Set E
hp : WithSeminorms p
i : ι
⊢ (∃ x, ∀ ⦃x_1 : E⦄, x_1 ∈ s → (p i) x_1 ≤ x) → ∃ r > 0, ∀ x ∈ s, (p i) x < r

Tactic:
rintro ⟨r, _⟩
    use 1 + max r 0
    grind

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Basic

Goal state before:
pp: case refine_1
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
inst✝⁶ : SeminormedAddCommGroup E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜₂
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝ : RingHomIsometric σ₁₂
A : ∀ (f : (E →SL[σ₁₂] F) × (E →SL[σ₁₂] F)), ‖-f.1 + f.2‖ = ‖f.1 - f.2‖
⊢ ∃ r, ContinuousLinearMap.seminorm.closedBall 0 r ∈ 𝓝 0

Tactic:
rcases NormedField.exists_lt_norm 𝕜 1 with ⟨c, hc⟩
    refine ⟨‖c‖, ContinuousLinearMap.hasBasis_nhds_zero.mem_iff.2
      ⟨(closedBall 0 1, closedBall 0 1), ?_⟩⟩
    suffices ∀ f : E →SL[σ₁₂] F, (∀ x, ‖x‖ ≤ 1 → ‖f x‖ ≤ 1) → ‖f‖ ≤ ‖c‖ by
      simpa [NormedSpace.isVonNBounded_closedBall, closedBall_mem_nhds, subset_def] using! this
    intro f hf
    refine opNorm_le_of_shell (f := f) one_pos (norm_nonneg c) hc fun x hcx hx ↦ ?_
    exact (hf x hx.le).trans ((div_le_iff₀' <| one_pos.trans hc).1 hcx)

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

Goal state before:
pp: 𝕜 : Type u_2
F : Type u_7
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
q : Seminorm 𝕜 F
hq : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ⇑q
hq' : Tendsto (⇑q) (𝓝 0) (𝓝 0)
ε : ℝ
ε_pos : 0 < ε
hε : {y | ‖y‖ < ε} ⊆ ⇑q ⁻¹' Iio 1
c : 𝕜
hc : 1 < ‖c‖
this : 0 < ‖c‖ / ε
x : F
⊢ q x ≤ ‖c‖ / ε * ‖x‖

Tactic:
by_cases hx : ‖x‖ = 0

Goal state after:
case pos
𝕜 : Type u_2
F : Type u_7
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
q : Seminorm 𝕜 F
hq : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ⇑q
hq' : Tendsto (⇑q) (𝓝 0) (𝓝 0)
ε : ℝ
ε_pos : 0 < ε
hε : {y | ‖y‖ < ε} ⊆ ⇑q ⁻¹' Iio 1
c : 𝕜
hc : 1 < ‖c‖
this : 0 < ‖c‖ / ε
x : F
hx : ‖x‖ = 0
⊢ q x ≤ ‖c‖ / ε * ‖x‖

case neg
𝕜 : Type u_2
F : Type u_7
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
q : Seminorm 𝕜 F
hq : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ⇑q
hq' : Tendsto (⇑q) (𝓝 0) (𝓝 0)
ε : ℝ
ε_pos : 0 < ε
hε : {y | ‖y‖ < ε} ⊆ ⇑q ⁻¹' Iio 1
c : 𝕜
hc : 1 < ‖c‖
this : 0 < ‖c‖ / ε
x : F
hx : ¬‖x‖ = 0
⊢ q x ≤ ‖c‖ / ε * ‖x‖
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_6
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
t₁ t₂ : TopologicalSpace E
ht₁ : PolynormableSpace 𝕜 E
ht₂ : PolynormableSpace 𝕜 E
⊢ ∀ t ∈ {t₁, t₂}, PolynormableSpace 𝕜 E

Tactic:
simp [ht₁, ht₂]

Goal state after:
Module: Mathlib.Topology.UniformSpace.CompactConvergence

Goal state before:
pp: α : Type u₁
β : Type u₂
inst✝² : TopologicalSpace α
inst✝¹ : UniformSpace β
f : C(α, β)
ι : Type u₃
p : Filter ι
F : ι → C(α, β)
inst✝ : WeaklyLocallyCompactSpace α
h : Tendsto F p (𝓝 f)
V : Set (β × β)
hV : V ∈ 𝓤 β
x : α
⊢ ∃ t ∈ 𝓝 x, ∀ᶠ (n : ι) in p, ∀ y ∈ t, (f y, (fun i a ↦ (F i) a) n y) ∈ V

Tactic:
rw [tendsto_iff_forall_isCompact_tendstoUniformlyOn] at h

Goal state after:
α : Type u₁
β : Type u₂
inst✝² : TopologicalSpace α
inst✝¹ : UniformSpace β
f : C(α, β)
ι : Type u₃
p : Filter ι
F : ι → C(α, β)
inst✝ : WeaklyLocallyCompactSpace α
h : ∀ (K : Set α), IsCompact K → TendstoUniformlyOn (fun i a ↦ (F i) a) (⇑f) p K
V : Set (β × β)
hV : V ∈ 𝓤 β
x : α
⊢ ∃ t ∈ 𝓝 x, ∀ᶠ (n : ι) in p, ∀ y ∈ t, (f y, (fun i a ↦ (F i) a) n y) ∈ V
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
⊢ (IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f fun i ↦ ↑(u i)) ↔ IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u

Tactic:
by
  simp only [IsBoundedUnder, IsBounded, eventually_map, ← coe_le_coe, NNReal.exists, coe_mk]
  constructor
  · rintro ⟨b, hb⟩
    exact ⟨b.toNNReal, by simp, by filter_upwards [hb]; simp +contextual⟩
  · rintro ⟨b, -, hb⟩
    exact ⟨b, hb⟩

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Algebra

Goal state before:
pp: α : Type u_1
inst✝⁵ : TopologicalSpace α
R : Type u_2
inst✝⁴ : CommSemiring R
A : Type u_3
inst✝³ : TopologicalSpace A
inst✝² : Semiring A
inst✝¹ : Algebra R A
inst✝ : IsTopologicalSemiring A
s s' : Subalgebra R C(α, A)
r : s ≤ s'
h : (fun s ↦ s.SeparatesPoints) s
x y : α
n : x ≠ y
⊢ ∃ f ∈ (fun f ↦ ⇑f) '' ↑s', f x ≠ f y

Tactic:
obtain ⟨f, m, w⟩ := h n
  rcases m with ⟨f, ⟨m, rfl⟩⟩
  exact ⟨_, ⟨f, ⟨r m, rfl⟩⟩, w⟩

Goal state after:
Module: Mathlib.MeasureTheory.Function.AEEqFun

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : TopologicalSpace β
f : α →ₘ[μ] β
⊢ f.toGerm = ↑↑f

Tactic:
rw [← mk_toGerm f f.aestronglyMeasurable, mk_coeFn]

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: case inr.inl
ι : Type u_1
α : ι → Type u_3
inst✝ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
s : (i : ι) → Set (α i)
h✝ : Nonempty ι
h : univ.pi s = ∅
⊢ piPremeasure m (univ.pi s) = ∏ i, (m i) (s i)

Tactic:
rcases univ_pi_eq_empty_iff.mp h with ⟨i, hi⟩

Goal state after:
case inr.inl
ι : Type u_1
α : ι → Type u_3
inst✝ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
s : (i : ι) → Set (α i)
h✝ : Nonempty ι
h : univ.pi s = ∅
i : ι
hi : s i = ∅
⊢ piPremeasure m (univ.pi s) = ∏ i, (m i) (s i)
Module: Mathlib.MeasureTheory.Function.SpecialFunctions.Basic

Goal state before:
pp: α : Type u_1
x✝¹ : MeasurableSpace α
f : α → ℝ
hf : Measurable fun x ↦ rexp (f x)
x✝ : α
⊢ f x✝ = log (rexp (f x✝))

Tactic:
rw [log_exp]

Goal state after:
Module: Mathlib.MeasureTheory.Function.SpecialFunctions.Basic

Goal state before:
pp: α : Type u_1
x✝¹ : MeasurableSpace α
f : α → ℝ
μ : Measure α
hf : AEMeasurable (fun x ↦ rexp (f x)) μ
x✝ : α
⊢ f x✝ = log (rexp (f x✝))

Tactic:
rw [log_exp]

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι✝ : Type u_1
ι' : Type u_2
α : ι✝ → Type u_3
inst✝⁷ : Fintype ι✝
m : (i : ι✝) → OuterMeasure (α i)
inst✝⁶ : (i : ι✝) → MeasurableSpace (α i)
μ✝ : (i : ι✝) → Measure (α i)
inst✝⁵ : ∀ (i : ι✝), SigmaFinite (μ✝ i)
ι : Type u_4
inst✝⁴ : Fintype ι
X : ι → Type u_5
inst✝³ : (i : ι) → PseudoMetricSpace (X i)
inst✝² : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
inst✝¹ : ∀ (i : ι), SigmaFinite (μ i)
inst✝ : ∀ (i : ι), IsUnifLocDoublingMeasure (μ i)
r : ℝ
hr :
  ∀ (i : ι) (x : X i),
    (μ i) (Metric.closedBall x (2 * r)) ≤
      ↑(IsUnifLocDoublingMeasure.doublingConstant (μ i)) * (μ i) (Metric.closedBall x r)
⊢ r ∈ Ioi 0 →
    ∀ (x : (i : ι) → X i),
      (Measure.pi μ) (Metric.closedBall x (2 * r)) ≤
        ↑(∏ i, IsUnifLocDoublingMeasure.doublingConstant (μ i)) * (Measure.pi μ) (Metric.closedBall x r)

Tactic:
(hr₀ : 0 < r)

Goal state after:
ι✝ : Type u_1
ι' : Type u_2
α : ι✝ → Type u_3
inst✝⁷ : Fintype ι✝
m : (i : ι✝) → OuterMeasure (α i)
inst✝⁶ : (i : ι✝) → MeasurableSpace (α i)
μ✝ : (i : ι✝) → Measure (α i)
inst✝⁵ : ∀ (i : ι✝), SigmaFinite (μ✝ i)
ι : Type u_4
inst✝⁴ : Fintype ι
X : ι → Type u_5
inst✝³ : (i : ι) → PseudoMetricSpace (X i)
inst✝² : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
inst✝¹ : ∀ (i : ι), SigmaFinite (μ i)
inst✝ : ∀ (i : ι), IsUnifLocDoublingMeasure (μ i)
r : ℝ
hr :
  ∀ (i : ι) (x : X i),
    (μ i) (Metric.closedBall x (2 * r)) ≤
      ↑(IsUnifLocDoublingMeasure.doublingConstant (μ i)) * (μ i) (Metric.closedBall x r)
hr₀ : 0 < r
⊢ ∀ (x : (i : ι) → X i),
    (Measure.pi μ) (Metric.closedBall x (2 * r)) ≤
      ↑(∏ i, IsUnifLocDoublingMeasure.doublingConstant (μ i)) * (Measure.pi μ) (Metric.closedBall x r)
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: α : Type u_1
F : Type u_5
m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ : Measure α
inst✝ : NormedAddCommGroup F
f : α → F
hq_pos : 0 < q
⊢ eLpNorm (fun x ↦ ‖f x‖ₑ ^ q) p μ = eLpNorm (fun x ↦ ‖f x‖ ^ q) p μ

Tactic:
convert! eLpNorm_ofReal (fun x ↦ ‖f x‖ ^ q) (by filter_upwards with x using by positivity)

Goal state after:
case e'_2.e'_5
α : Type u_1
F : Type u_5
m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ : Measure α
inst✝ : NormedAddCommGroup F
f : α → F
hq_pos : 0 < q
x✝ : α
⊢ ‖f x✝‖ₑ ^ q = (ENNReal.ofReal ∘ fun x ↦ ‖f x‖ ^ q) x✝
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup E
f : α → E
hf : AEStronglyMeasurable f μ
C : ℝ
hfC : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C
⊢ eLpNorm f ∞ μ < ∞

Tactic:
by
    rw [eLpNorm_exponent_top]
    exact eLpNormEssSup_lt_top_of_ae_bound hfC

Goal state after: