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

Goal state before:
pp: case inr.hn
k : Type u_1
V₁ : Type u_2
P₁ : Type u_3
V₂ : Type u_4
P₂ : Type u_5
inst✝⁶ : Ring k
inst✝⁵ : AddCommGroup V₁
inst✝⁴ : Module k V₁
inst✝³ : AffineSpace V₁ P₁
inst✝² : AddCommGroup V₂
inst✝¹ : Module k V₂
inst✝ : AffineSpace V₂ P₂
f : P₁ →ᵃ[k] P₂
s : Set P₁
p : P₁
hp : p ∈ s
⊢ (↑(map f (affineSpan k s)) ∩ ↑(affineSpan k (⇑f '' s))).Nonempty

Tactic:
exact ⟨f p, mem_image_of_mem f (subset_affineSpan k _ hp),
          subset_affineSpan k _ (mem_image_of_mem f hp)⟩

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic

Goal state before:
pp: k : Type u_1
V₁ : Type u_2
P₁ : Type u_3
V₂ : Type u_4
P₂ : Type u_5
inst✝⁶ : Ring k
inst✝⁵ : AddCommGroup V₁
inst✝⁴ : Module k V₁
inst✝³ : AffineSpace V₁ P₁
inst✝² : AddCommGroup V₂
inst✝¹ : Module k V₂
inst✝ : AffineSpace V₂ P₂
f : P₁ →ᵃ[k] P₂
hf : Function.Injective ⇑f
s₁ s₂ : AffineSubspace k P₁
⊢ map f (s₁ ⊓ s₂) = map f s₁ ⊓ map f s₂

Tactic:
ext p
  simp [mem_inf_iff]
  grind

Goal state after:
Module: Mathlib.Analysis.Seminorm

Goal state before:
pp: R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
ι : Type u_11
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q✝ : Seminorm 𝕜 E
x✝ : E
p q : Seminorm 𝕜 E
x : E
⊢ p 0 + q (x - 0) ≤ q x

Tactic:
simp only [map_zero, zero_add, sub_zero]; rfl

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

Goal state before:
pp: 𝕜 : Type u_1
α : Type u_4
β : Type u_5
inst✝⁷ : Semiring 𝕜
inst✝⁶ : PartialOrder 𝕜
inst✝⁵ : AddCommMonoid α
inst✝⁴ : PartialOrder α
inst✝³ : SMul 𝕜 α
inst✝² : AddCommMonoid β
inst✝¹ : PartialOrder β
inst✝ : SMul 𝕜 β
f : α ≃o β
hf : ConcaveOn 𝕜 univ ⇑f
x : β
x✝¹ : x ∈ univ
y : β
x✝ : y ∈ univ
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
x' : α
hx'' : x = f x'
y' : α
hy'' : y = f y'
⊢ f.symm (a • x + b • y) ≤ a • f.symm x + b • f.symm y

Tactic:
simp only [hx'', hy'', OrderIso.symm_apply_apply]

Goal state after:
𝕜 : Type u_1
α : Type u_4
β : Type u_5
inst✝⁷ : Semiring 𝕜
inst✝⁶ : PartialOrder 𝕜
inst✝⁵ : AddCommMonoid α
inst✝⁴ : PartialOrder α
inst✝³ : SMul 𝕜 α
inst✝² : AddCommMonoid β
inst✝¹ : PartialOrder β
inst✝ : SMul 𝕜 β
f : α ≃o β
hf : ConcaveOn 𝕜 univ ⇑f
x : β
x✝¹ : x ∈ univ
y : β
x✝ : y ∈ univ
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
x' : α
hx'' : x = f x'
y' : α
hy'' : y = f y'
⊢ f.symm (a • f x' + b • f y') ≤ a • x' + b • y'
Module: Mathlib.Analysis.Seminorm

Goal state before:
pp: case inl
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
ι : Type u_11
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q✝ : Seminorm 𝕜 E
x✝ x y : E
q : E → ℝ
hq : q ∈ upperBounds (DFunLike.coe '' ∅)
⊢ (⨆ p, ⇑↑p) (x + y) ≤ (⨆ p, ⇑↑p) x + (⨆ p, ⇑↑p) y

pp: case inr
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
ι : Type u_11
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q✝ : Seminorm 𝕜 E
x✝ : E
s : Set (Seminorm 𝕜 E)
x y : E
q : E → ℝ
hq : q ∈ upperBounds (DFunLike.coe '' s)
h : s.Nonempty
⊢ (⨆ p, ⇑↑p) (x + y) ≤ (⨆ p, ⇑↑p) x + (⨆ p, ⇑↑p) y

Tactic:
· simp [Real.iSup_of_isEmpty]

Goal state after:
case inr
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
ι : Type u_11
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q✝ : Seminorm 𝕜 E
x✝ : E
s : Set (Seminorm 𝕜 E)
x y : E
q : E → ℝ
hq : q ∈ upperBounds (DFunLike.coe '' s)
h : s.Nonempty
⊢ (⨆ p, ⇑↑p) (x + y) ≤ (⨆ p, ⇑↑p) x + (⨆ p, ⇑↑p) y
Module: Mathlib.Analysis.Convex.Function

Goal state before:
pp: 𝕜 : Type u_1
α : Type u_4
β : Type u_5
inst✝⁷ : Semiring 𝕜
inst✝⁶ : PartialOrder 𝕜
inst✝⁵ : AddCommMonoid α
inst✝⁴ : PartialOrder α
inst✝³ : SMul 𝕜 α
inst✝² : AddCommMonoid β
inst✝¹ : PartialOrder β
inst✝ : SMul 𝕜 β
f : α ≃o β
hf : ConvexOn 𝕜 univ ⇑f
x : β
x✝¹ : x ∈ univ
y : β
x✝ : y ∈ univ
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
x' : α
hx'' : x = f x'
y' : α
hy'' : y = f y'
⊢ a • f.symm x + b • f.symm y ≤ f.symm (a • x + b • y)

Tactic:
simp only [hx'', hy'', OrderIso.symm_apply_apply]

Goal state after:
𝕜 : Type u_1
α : Type u_4
β : Type u_5
inst✝⁷ : Semiring 𝕜
inst✝⁶ : PartialOrder 𝕜
inst✝⁵ : AddCommMonoid α
inst✝⁴ : PartialOrder α
inst✝³ : SMul 𝕜 α
inst✝² : AddCommMonoid β
inst✝¹ : PartialOrder β
inst✝ : SMul 𝕜 β
f : α ≃o β
hf : ConvexOn 𝕜 univ ⇑f
x : β
x✝¹ : x ∈ univ
y : β
x✝ : y ∈ univ
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
x' : α
hx'' : x = f x'
y' : α
hy'' : y = f y'
⊢ a • x' + b • y' ≤ f.symm (a • f x' + b • f y')
Module: Mathlib.Analysis.Seminorm

Goal state before:
pp: 𝕜 : Type u_3
E : Type u_7
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
ι : Sort u_12
p : ι → Seminorm 𝕜 E
⊢ BddAbove (range p) ↔ ∀ (x : E), BddAbove (range fun i ↦ (p i) x)

Tactic:
Seminorm.bddAbove_iff,

Goal state after:
𝕜 : Type u_3
E : Type u_7
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
ι : Sort u_12
p : ι → Seminorm 𝕜 E
⊢ BddAbove (DFunLike.coe '' range p) ↔ ∀ (x : E), BddAbove (range fun i ↦ (p i) x)
Module: Mathlib.Analysis.Seminorm

Goal state before:
pp: case refine_1
𝕜 : Type u_3
E : Type u_7
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set (Seminorm 𝕜 E)
hs₁ : BddAbove s
hs₂ : s.Nonempty
p : Seminorm 𝕜 E
hp : p ∈ s
x : E
this : Nonempty ↑s
⊢ p x ≤ ⨆ i, ↑i x

Tactic:
rcases hs₁ with ⟨q, hq⟩

Goal state after:
case refine_1
𝕜 : Type u_3
E : Type u_7
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set (Seminorm 𝕜 E)
hs₂ : s.Nonempty
p : Seminorm 𝕜 E
hp : p ∈ s
x : E
this : Nonempty ↑s
q : Seminorm 𝕜 E
hq : q ∈ upperBounds s
⊢ p x ≤ ⨆ i, ↑i x
Module: Mathlib.Analysis.Seminorm

Goal state before:
pp: 𝕜 : Type u_3
E : Type u_7
inst✝⁶ : NormedField 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : SMul ℝ 𝕜
inst✝³ : NormSMulClass ℝ 𝕜
inst✝² : Module 𝕜 E
inst✝¹ : SMul ℝ E
inst✝ : IsScalarTower ℝ 𝕜 E
p : Seminorm 𝕜 E
x : E
x✝² : x ∈ univ
y : E
x✝¹ : y ∈ univ
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
x✝ : a + b = 1
⊢ p (a • x + b • y) ≤ a • p x + b • p y

Tactic:
calc
    p (a • x + b • y) ≤ p (a • x) + p (b • y) := map_add_le_add p _ _
    _ = ‖a • (1 : 𝕜)‖ * p x + ‖b • (1 : 𝕜)‖ * p y := by
      rw [← map_smul_eq_mul p, ← map_smul_eq_mul p, smul_one_smul, smul_one_smul]
    _ = a * p x + b * p y := by
      rw [norm_smul, norm_smul, norm_one, mul_one, mul_one, Real.norm_of_nonneg ha,
        Real.norm_of_nonneg hb]

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

Goal state before:
pp: case refine_2
k : Type u_1
V : Type u_2
P : Type u_5
inst✝⁴ : Ring k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
n : ℕ
inst✝ : NeZero n
s : Simplex k P n
w : Fin (n + 1) → k
hw : ∑ i, w i = 1
i : Fin (n + 1)
h : w i = 0
⊢ (affineCombination k univ s.points) w ∈ affineSpan k (Set.range (s.faceOpposite i).points)

Tactic:
rw [range_faceOpposite_points]

Goal state after:
case refine_2
k : Type u_1
V : Type u_2
P : Type u_5
inst✝⁴ : Ring k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
n : ℕ
inst✝ : NeZero n
s : Simplex k P n
w : Fin (n + 1) → k
hw : ∑ i, w i = 1
i : Fin (n + 1)
h : w i = 0
⊢ (affineCombination k univ s.points) w ∈ affineSpan k (s.points '' {i}ᶜ)
Module: Mathlib.Topology.NhdsKer

Goal state before:
pp: X : Type u_2
inst✝ : TopologicalSpace X
⊢ nhdsKer ∅ = ∅

Tactic:
exact nhdsKer_empty

Goal state after:
Module: Mathlib.Topology.AlexandrovDiscrete

Goal state before:
pp: α : Type u_3
inst✝ : TopologicalSpace α
hα : ∀ (a : α), 𝓝 a = 𝓟 (nhdsKer {a})
S : Set (Set α)
hS : ∀ s ∈ S, ∀ (a : α), (nhdsKer {a} ∩ s).Nonempty → a ∈ s
a : α
s : Set α
hs : s ∈ S
has : (nhdsKer {a} ∩ s).Nonempty
⊢ a ∈ ⋃₀ S

Tactic:
specialize hS s hs a has

Goal state after:
α : Type u_3
inst✝ : TopologicalSpace α
hα : ∀ (a : α), 𝓝 a = 𝓟 (nhdsKer {a})
S : Set (Set α)
a : α
s : Set α
hS : a ∈ s
hs : s ∈ S
has : (nhdsKer {a} ∩ s).Nonempty
⊢ a ∈ ⋃₀ S
Module: Mathlib.Topology.Connected.LocallyPathConnected

Goal state before:
pp: case refine_1
ι : Type u_3
Z : ι → Type u_4
inst✝ : (i : ι) → TopologicalSpace (Z i)
h : LocallyPathConnectedSpace ((i : ι) → Z i)
⊢ IsEmpty ((i : ι) → Z i) ∨ (∀ (i : ι), LocallyPathConnectedSpace (Z i)) ∧ {i | ¬PathConnectedSpace (Z i)}.Finite

Tactic:
rcases isEmpty_or_nonempty (∀ i, Z i) with he | hne

Goal state after:
case refine_1.inl
ι : Type u_3
Z : ι → Type u_4
inst✝ : (i : ι) → TopologicalSpace (Z i)
h : LocallyPathConnectedSpace ((i : ι) → Z i)
he : IsEmpty ((i : ι) → Z i)
⊢ IsEmpty ((i : ι) → Z i) ∨ (∀ (i : ι), LocallyPathConnectedSpace (Z i)) ∧ {i | ¬PathConnectedSpace (Z i)}.Finite

case refine_1.inr
ι : Type u_3
Z : ι → Type u_4
inst✝ : (i : ι) → TopologicalSpace (Z i)
h : LocallyPathConnectedSpace ((i : ι) → Z i)
hne : Nonempty ((i : ι) → Z i)
⊢ IsEmpty ((i : ι) → Z i) ∨ (∀ (i : ι), LocallyPathConnectedSpace (Z i)) ∧ {i | ¬PathConnectedSpace (Z i)}.Finite
Module: Mathlib.Topology.Algebra.Module.LocallyConvex

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁸ : Field 𝕜
inst✝⁷ : PartialOrder 𝕜
inst✝⁶ : ZeroLEOneClass 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousConstSMul 𝕜 E
inst✝ : LocallyConvexSpace 𝕜 E
s t : Set E
disj : Disjoint s t
hs₁ : Convex 𝕜 s
hs₂ : IsCompact s
ht₁ : Convex 𝕜 t
ht₂ : IsClosed[inst✝³] t
this✝¹ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
this✝ : IsUniformAddGroup E
this : (uniformity E).HasBasis (fun s ↦ 0 ∈ s ∧ IsOpen[inst✝³] s ∧ Convex 𝕜 s) fun i ↦ (fun x ↦ x.2 - x.1) ⁻¹' id i
V : Set E
hV :
  Disjoint (⋃ x ∈ s, UniformSpace.ball x ((fun x ↦ x.2 - x.1) ⁻¹' id V))
    (⋃ x ∈ t, UniformSpace.ball x ((fun x ↦ x.2 - x.1) ⁻¹' id V))
hV0 : 0 ∈ V
hVopen : IsOpen[inst✝³] V
hVconvex : Convex 𝕜 V
⊢ Disjoint (⋃ a ∈ s, (fun x ↦ -a + x) ⁻¹' V) (⋃ a ∈ t, (fun x ↦ -a + x) ⁻¹' V)

Tactic:
simp_rw [UniformSpace.ball, ← preimage_comp, sub_eq_neg_add] at hV

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁸ : Field 𝕜
inst✝⁷ : PartialOrder 𝕜
inst✝⁶ : ZeroLEOneClass 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousConstSMul 𝕜 E
inst✝ : LocallyConvexSpace 𝕜 E
s t : Set E
disj : Disjoint s t
hs₁ : Convex 𝕜 s
hs₂ : IsCompact s
ht₁ : Convex 𝕜 t
ht₂ : IsClosed[inst✝³] t
this✝¹ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
this✝ : IsUniformAddGroup E
this : (uniformity E).HasBasis (fun s ↦ 0 ∈ s ∧ IsOpen[inst✝³] s ∧ Convex 𝕜 s) fun i ↦ (fun x ↦ x.2 - x.1) ⁻¹' id i
V : Set E
hV0 : 0 ∈ V
hVopen : IsOpen[inst✝³] V
hVconvex : Convex 𝕜 V
hV : Disjoint (⋃ x ∈ s, (fun x ↦ -x.1 + x.2) ∘ Prod.mk x ⁻¹' id V) (⋃ x ∈ t, (fun x ↦ -x.1 + x.2) ∘ Prod.mk x ⁻¹' id V)
⊢ Disjoint (⋃ a ∈ s, (fun x ↦ -a + x) ⁻¹' V) (⋃ a ∈ t, (fun x ↦ -a + x) ⁻¹' V)
Module: Mathlib.LinearAlgebra.AffineSpace.Independent

Goal state before:
pp: case refine_2
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : Ring k
inst✝⁴ : LinearOrder k
inst✝³ : IsStrictOrderedRing 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 : ι → k
s : Finset ι
hw : ∑ i ∈ s, w i = 1
i₁ i₂ i₃ : ι
h₁ : i₁ ∈ s
h₂ : i₂ ∈ s
h₃ : i₃ ∈ s
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
c : k
hc0 : 0 < c
hc1 : c < 1
hs :
  (affineCombination k s p) w ∈
    line[k, (affineCombination k s p) (Pi.single i₁ 1),
      (affineCombination k s p) (affineCombinationLineMapWeights i₂ i₃ c)]
⊢ SignType.sign (affineCombinationLineMapWeights i₂ i₃ c i₂) =
    SignType.sign (affineCombinationLineMapWeights i₂ i₃ c i₃)

Tactic:
rw [Finset.affineCombinationLineMapWeights_apply_left h₂₃,
      Finset.affineCombinationLineMapWeights_apply_right h₂₃]

Goal state after:
case refine_2
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : Ring k
inst✝⁴ : LinearOrder k
inst✝³ : IsStrictOrderedRing 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 : ι → k
s : Finset ι
hw : ∑ i ∈ s, w i = 1
i₁ i₂ i₃ : ι
h₁ : i₁ ∈ s
h₂ : i₂ ∈ s
h₃ : i₃ ∈ s
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
c : k
hc0 : 0 < c
hc1 : c < 1
hs :
  (affineCombination k s p) w ∈
    line[k, (affineCombination k s p) (Pi.single i₁ 1),
      (affineCombination k s p) (affineCombinationLineMapWeights i₂ i₃ c)]
⊢ SignType.sign (1 - c) = SignType.sign c
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: case neg
R : Type u_1
E : Type u_6
ι : Type u_9
inst✝² : SeminormedRing R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
p : SeminormFamily R E ι
v : E
U : Set E
hU✝ : U ∈ p.basisSets
s : Finset ι
r : ℝ
hr : 0 < r
hU : U = (s.sup p).ball 0 r
h : (s.sup p) v ≤ 0
⊢ {x | ‖x‖ * 0 < r} ∈ 𝓝 0

Tactic:
mul_zero,

Goal state after:
case neg
R : Type u_1
E : Type u_6
ι : Type u_9
inst✝² : SeminormedRing R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
p : SeminormFamily R E ι
v : E
U : Set E
hU✝ : U ∈ p.basisSets
s : Finset ι
r : ℝ
hr : 0 < r
hU : U = (s.sup p).ball 0 r
h : (s.sup p) v ≤ 0
⊢ {x | 0 < r} ∈ 𝓝 0
Module: Mathlib.Analysis.Convex.Jensen

Goal state before:
pp: 𝕜 : Type u_1
β : Type u_4
inst✝⁷ : Field 𝕜
inst✝⁶ : LinearOrder 𝕜
inst✝⁵ : IsStrictOrderedRing 𝕜
inst✝⁴ : AddCommGroup β
inst✝³ : LinearOrder β
inst✝² : IsOrderedAddMonoid β
inst✝¹ : Module 𝕜 β
inst✝ : IsStrictOrderedModule 𝕜 β
s : Set 𝕜
f : 𝕜 → β
x y z : 𝕜
hf : ConvexOn 𝕜 s f
hx : x ∈ s
hy : y ∈ s
hz : z ∈ Set.Icc x y
⊢ f z ≤ max (f x) (f y)

Tactic:
by
  rw [← segment_eq_Icc (hz.1.trans hz.2)] at hz; exact hf.le_max_of_mem_segment hx hy hz

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

Goal state before:
pp: 𝕜 : Type u_11
E : Type u_12
ι : Type u_13
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : TopologicalSpace E
p : SeminormFamily 𝕜 E ι
hp : ∀ (i : ι), Continuous[inst✝, _] ⇑(p i)
s : Finset ι
r : ℝ
hr : 0 < r
⊢ Continuous[inst✝, _] ⇑(s.sup p)

Tactic:
classical
  induction s using Finset.induction_on with
  | empty => simpa using continuous_zero
  | insert a s _ hs =>
    simp only [Finset.sup_insert, coe_sup]
    exact Continuous.max (hp a) hs

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

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : TopologicalSpace E
p : SeminormFamily 𝕜 E ι
hp : WithSeminorms p
x : E
this✝ : IsTopologicalAddGroup E
sr : Finset ι × ℝ
this : (sr.1.sup p).ball (x +ᵥ 0) sr.2 = x +ᵥ (sr.1.sup p).ball 0 sr.2
⊢ (sr.1.sup p).ball x sr.2 = (fun x_1 ↦ x + x_1) '' (sr.1.sup p).ball 0 sr.2

Tactic:
rwa [vadd_eq_add, add_zero] at this

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

Goal state before:
pp: 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), ∃ r > 0, ∀ x ∈ s, (p i) x < r) ↔ ∀ (i : ι), BddAbove (⇑(p i) '' s)

Tactic:
bddAbove_def,

Goal state after:
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), ∃ r > 0, ∀ x ∈ s, (p i) x < r) ↔ ∀ (i : ι), ∃ x, ∀ y ∈ ⇑(p i) '' s, y ≤ x
Module: Mathlib.Topology.Algebra.Module.Spaces.UniformConvergenceCLM

Goal state before:
pp: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝¹² : NormedField 𝕜₁
inst✝¹¹ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_3
F : Type u_4
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module 𝕜₁ E
inst✝⁸ : TopologicalSpace E
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜₂ F
R : Type u_6
inst✝⁵ : NormedDivisionRing R
inst✝⁴ : TopologicalSpace F
inst✝³ : IsTopologicalAddGroup F
inst✝² : Module R F
inst✝¹ : ContinuousConstSMul R F
inst✝ : SMulCommClass 𝕜₂ R F
𝔖 : Set (Set E)
S : Set (E →SLᵤ[σ, 𝔖] F)
h : ∀ s ∈ 𝔖, IsVonNBounded R (image2 (fun f x ↦ f x) S s)
⊢ IsVonNBounded R S

Tactic:
simp_rw [isVonNBounded_iff_absorbing_le, nhds_zero_eq, le_iInf_iff, le_principal_iff]

Goal state after:
𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝¹² : NormedField 𝕜₁
inst✝¹¹ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_3
F : Type u_4
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module 𝕜₁ E
inst✝⁸ : TopologicalSpace E
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜₂ F
R : Type u_6
inst✝⁵ : NormedDivisionRing R
inst✝⁴ : TopologicalSpace F
inst✝³ : IsTopologicalAddGroup F
inst✝² : Module R F
inst✝¹ : ContinuousConstSMul R F
inst✝ : SMulCommClass 𝕜₂ R F
𝔖 : Set (Set E)
S : Set (E →SLᵤ[σ, 𝔖] F)
h : ∀ s ∈ 𝔖, IsVonNBounded R (image2 (fun f x ↦ f x) S s)
⊢ ∀ i ∈ 𝔖, ∀ i_2 ∈ 𝓝 0, {f | MapsTo (⇑f) i i_2} ∈ absorbing R S
Module: Mathlib.Topology.Algebra.Module.Spaces.ContinuousLinearMap

Goal state before:
pp: R : Type u_1
𝕜₂ : Type u_3
𝕜₃ : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
inst✝¹³ : Semiring R
inst✝¹² : NormedField 𝕜₂
inst✝¹¹ : NormedField 𝕜₃
inst✝¹⁰ : AddCommMonoid E
inst✝⁹ : Module R E
inst✝⁸ : TopologicalSpace E
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜₂ F
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommGroup G
inst✝³ : Module 𝕜₃ G
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜₃ G
σ₁₃ : R →+* 𝕜₃
σ₂₃ : 𝕜₂ →+* 𝕜₃
f : E →SL[σ₁₃] F →SL[σ₂₃] G
c : R
x : E
y : F
⊢ (σ₁₃ c • f x) y = σ₁₃ c • (f x) y

Tactic:
smul_apply

Goal state after:
R : Type u_1
𝕜₂ : Type u_3
𝕜₃ : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
inst✝¹³ : Semiring R
inst✝¹² : NormedField 𝕜₂
inst✝¹¹ : NormedField 𝕜₃
inst✝¹⁰ : AddCommMonoid E
inst✝⁹ : Module R E
inst✝⁸ : TopologicalSpace E
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜₂ F
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommGroup G
inst✝³ : Module 𝕜₃ G
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜₃ G
σ₁₃ : R →+* 𝕜₃
σ₂₃ : 𝕜₂ →+* 𝕜₃
f : E →SL[σ₁₃] F →SL[σ₂₃] G
c : R
x : E
y : F
⊢ σ₁₃ c • (f x) y = σ₁₃ c • (f x) y
Module: Mathlib.Topology.Algebra.Module.Spaces.ContinuousLinearMap

Goal state before:
pp: 𝕜₂ : Type u_3
𝕜₃ : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
inst✝¹² : NormedField 𝕜₂
inst✝¹¹ : NormedField 𝕜₃
inst✝¹⁰ : AddCommMonoid E
inst✝⁹ : Module 𝕜₃ E
inst✝⁸ : TopologicalSpace E
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜₂ F
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommGroup G
inst✝³ : Module 𝕜₃ G
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜₃ G
σ₂₃ : 𝕜₂ →+* 𝕜₃
f : E →L[𝕜₃] F →SL[σ₂₃] G
c : 𝕜₃
x : E
y : F
⊢ (c • f x) y = c • (f x) y

Tactic:
smul_apply

Goal state after:
𝕜₂ : Type u_3
𝕜₃ : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
inst✝¹² : NormedField 𝕜₂
inst✝¹¹ : NormedField 𝕜₃
inst✝¹⁰ : AddCommMonoid E
inst✝⁹ : Module 𝕜₃ E
inst✝⁸ : TopologicalSpace E
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜₂ F
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommGroup G
inst✝³ : Module 𝕜₃ G
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜₃ G
σ₂₃ : 𝕜₂ →+* 𝕜₃
f : E →L[𝕜₃] F →SL[σ₂₃] G
c : 𝕜₃
x : E
y : F
⊢ c • (f x) y = c • (f x) y
Module: Mathlib.Order.Filter.Germ.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
l : Filter α
f✝¹ g h : α → β
M : Type u_5
N : Type u_6
R : Type u_7
inst✝¹ : Monoid M
inst✝ : MulAction M β
c₁ c₂ : M
f✝ : l.Germ β
f : α → β
⊢ (c₁ * c₂) • f =ᶠ[l] c₁ • c₂ • f

Tactic:
simp [mul_smul]

Goal state after:
Module: Mathlib.Order.Filter.Germ.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
l : Filter α
f✝¹ g h : α → β
M : Type u_5
N : Type u_6
R : Type u_7
inst✝¹ : Monoid M
inst✝ : MulAction M β
c₁✝ c₂✝ : l.Germ M
f✝ : l.Germ β
c₁ c₂ : α → M
f : α → β
⊢ (c₁ * c₂) • f =ᶠ[l] c₁ • c₂ • f

Tactic:
simp [mul_smul]

Goal state after:
Module: Mathlib.Topology.Compactness.CompactlyCoherentSpace

Goal state before:
pp: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompactlyCoherentSpace X
s : Set X
⊢ IsOpen[inst✝¹] s ↔
    ∀ (K : Type u) [inst : TopologicalSpace K] [CompactSpace K] (f : K → X), Continuous[_, inst✝¹] f → IsOpen (f ⁻¹' s)

Tactic:
refine ⟨fun hs _ _ _ _ hf ↦ hs.preimage hf, fun hs ↦ isOpen_iff |>.mpr ?_⟩
  intro K hK
  have : CompactSpace K := isCompact_iff_compactSpace.mp hK
  exact hs K Subtype.val continuous_subtype_val

Goal state after:
Module: Mathlib.Topology.Compactness.CompactlyCoherentSpace

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
A : Set (𝐤X)
⊢ IsOpen[instTopologicalSpace] A ↔
    ∀ (K : Set X), IsCompact K → IsOpen[instTopologicalSpaceSubtype] (K ↓∩ ⇑(CompactCoherentification.mk X) ⁻¹' A)

Tactic:
simp_rw [isOpen_coinduced, isOpen_iSup_iff]

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
A : Set (𝐤X)
⊢ (∀ (i : Set X),
      IsCompact i →
        IsOpen[TopologicalSpace.coinduced Subtype.val
              { IsOpen := fun s ↦ ∃ t, IsOpen[inst✝] t ∧ Subtype.val ⁻¹' t = s, isOpen_univ := ⋯, isOpen_inter := ⋯,
                isOpen_sUnion := ⋯ }]
          (⇑(CompactCoherentification.mk X) ⁻¹' A)) ↔
    ∀ (K : Set X), IsCompact K → IsOpen[instTopologicalSpaceSubtype] (K ↓∩ ⇑(CompactCoherentification.mk X) ⁻¹' A)
Module: Mathlib.Topology.Compactness.CompactlyCoherentSpace

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
K : Set X
⊢ IsCompact (⇑(CompactCoherentification.mk X) '' K) ↔ IsCompact K

Tactic:
by
  rw [isCompact_iff, Equiv.preimage_image]

Goal state after:
Module: Mathlib.Probability.ConditionalProbability

Goal state before:
pp: Ω : Type u_1
α : Type u_3
m : MeasurableSpace Ω
inst✝³ : Fintype α
inst✝² : MeasurableSpace α
inst✝¹ : DiscreteMeasurableSpace α
X : Ω → α
hX : Measurable X
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
E : Set Ω
hE : MeasurableSet E
⊢ ∑ x, μ (X ⁻¹' {x}) * μ[E | X ⁻¹' {x}] = ∑ x, μ (X ⁻¹' {x} ∩ E)

Tactic:
mul_comm (μ _),

Goal state after:
Ω : Type u_1
α : Type u_3
m : MeasurableSpace Ω
inst✝³ : Fintype α
inst✝² : MeasurableSpace α
inst✝¹ : DiscreteMeasurableSpace α
X : Ω → α
hX : Measurable X
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
E : Set Ω
hE : MeasurableSet E
⊢ ∑ x, μ[E | X ⁻¹' {x}] * μ (X ⁻¹' {x}) = ∑ x, μ (X ⁻¹' {x} ∩ E)
Module: Mathlib.MeasureTheory.Function.EssSup

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : ConditionallyCompleteLattice β
γ : Type u_3
mγ : MeasurableSpace γ
f : α → γ
g : γ → β
hf : AEMeasurable f μ
hgf : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) (ae μ) (g ∘ f)
hg : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (ae (Measure.map f μ)) g
⊢ essSup (g ∘ f) μ ≤ essSup g (Measure.map f μ)

Tactic:
refine limsSup_le_limsSup_of_le ?_ hgf hg

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : ConditionallyCompleteLattice β
γ : Type u_3
mγ : MeasurableSpace γ
f : α → γ
g : γ → β
hf : AEMeasurable f μ
hgf : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) (ae μ) (g ∘ f)
hg : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (ae (Measure.map f μ)) g
⊢ map (g ∘ f) (ae μ) ≤ map g (ae (Measure.map f μ))
Module: Mathlib.Probability.UniformOn

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
⊢ (uniformOn s) (t ∩ u) = (uniformOn (s ∩ u)) t * (uniformOn s) u

Tactic:
← Set.inter_comm

Goal state after:
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
⊢ (uniformOn s) (u ∩ t) = (uniformOn (s ∩ u)) t * (uniformOn s) u
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Defs

Goal state before:
pp: α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
p : ℝ≥0∞
inst✝ : ENorm ε
μ : Measure α
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
f : α → ε
⊢ eLpNorm' f p.toReal μ = (∫⁻ (x : α), ‖f x‖ₑ ^ p.toReal ∂μ) ^ (1 / p.toReal)

Tactic:
eLpNorm'_eq_lintegral_enorm

Goal state after:
α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
p : ℝ≥0∞
inst✝ : ENorm ε
μ : Measure α
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
f : α → ε
⊢ (∫⁻ (a : α), ‖f a‖ₑ ^ p.toReal ∂μ) ^ (1 / p.toReal) = (∫⁻ (x : α), ‖f x‖ₑ ^ p.toReal ∂μ) ^ (1 / p.toReal)