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

Goal state before:
pp: case inr
k : Type u_1
V : Type u_2
P : Type u_4
inst✝⁷ : Ring k
inst✝⁶ : AddCommGroup V
inst✝⁵ : Module k V
inst✝⁴ : AffineSpace V P
inst✝³ : LinearOrder k
inst✝² : IsOrderedAddMonoid k
inst✝¹ : ZeroLEOneClass k
n : ℕ
inst✝ : NeZero n
s : Simplex k P n
w : Fin (n + 1) → k
hw1 : ∑ i, w i = 1
hp : ∀ (i : Fin (n + 1)), w i ∈ Set.Icc 0 1
hp' : (affineCombination k univ s.points) w ∈ ↑(affineSpan k (Set.range s.points))
i : Fin (n + 1)
hi : 1 ≤ w i
hi1 : w i = 1
⊢ ∃ j, w j = 0

Tactic:
← hi1,

Goal state after:
case inr
k : Type u_1
V : Type u_2
P : Type u_4
inst✝⁷ : Ring k
inst✝⁶ : AddCommGroup V
inst✝⁵ : Module k V
inst✝⁴ : AffineSpace V P
inst✝³ : LinearOrder k
inst✝² : IsOrderedAddMonoid k
inst✝¹ : ZeroLEOneClass k
n : ℕ
inst✝ : NeZero n
s : Simplex k P n
w : Fin (n + 1) → k
hp : ∀ (i : Fin (n + 1)), w i ∈ Set.Icc 0 1
hp' : (affineCombination k univ s.points) w ∈ ↑(affineSpan k (Set.range s.points))
i : Fin (n + 1)
hw1 : ∑ i, w i = w i
hi : 1 ≤ w i
hi1 : w i = 1
⊢ ∃ j, w j = 0
Module: Mathlib.Topology.NhdsKer

Goal state before:
pp: X : Type u_2
inst✝¹ : TopologicalSpace X
Y : Type u_3
inst✝ : TopologicalSpace Y
s : Set X
t : Set Y
⊢ ⋃ p ∈ s ×ˢ t, nhdsKer {p} = ⋃ p ∈ s ×ˢ t, nhdsKer {p.1} ×ˢ nhdsKer {p.2}

Tactic:
congr! with ⟨x, y⟩ _

Goal state after:
case f
X : Type u_2
inst✝¹ : TopologicalSpace X
Y : Type u_3
inst✝ : TopologicalSpace Y
s : Set X
t : Set Y
x : X
y : Y
x✝ : (x, y) ∈ s ×ˢ t
⊢ nhdsKer {(x, y)} = nhdsKer {(x, y).1} ×ˢ nhdsKer {(x, y).2}
Module: Mathlib.LinearAlgebra.AffineSpace.Independent

Goal state before:
pp: case inr.refine_2.hd
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 = vectorSpan k s
hb₃ : AffineIndependent k fun p_1 ↦ ↑p_1
hb₀ : ∀ v ∈ b, v ≠ 0
⊢ (affineSpan k ({p} ∪ ⇑(Equiv.vaddConst p) '' b)).direction = (affineSpan k s).direction

Tactic:
have : Submodule.span k b = Submodule.span k (insert 0 b) := by simp

Goal state after:
case inr.refine_2.hd
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 = vectorSpan k s
hb₃ : AffineIndependent k fun p_1 ↦ ↑p_1
hb₀ : ∀ v ∈ b, v ≠ 0
this : Submodule.span k b = Submodule.span k (insert 0 b)
⊢ (affineSpan k ({p} ∪ ⇑(Equiv.vaddConst p) '' b)).direction = (affineSpan k s).direction
Module: Mathlib.Topology.Algebra.Module.LocallyConvex

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
F : Type u_4
inst✝⁸ : Semiring 𝕜
inst✝⁷ : PartialOrder 𝕜
inst✝⁶ : AddCommMonoid E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : AddCommMonoid F
inst✝³ : Module 𝕜 F
inst✝² : TopologicalSpace F
inst✝¹ : LocallyConvexSpace 𝕜 F
inst✝ : TopologicalSpace E
f : E →ₗ[𝕜] F
hf : IsInducing ⇑f
⊢ LocallyConvexSpace 𝕜 E

Tactic:
hf.eq_induced

Goal state after:
𝕜 : Type u_2
E : Type u_3
F : Type u_4
inst✝⁸ : Semiring 𝕜
inst✝⁷ : PartialOrder 𝕜
inst✝⁶ : AddCommMonoid E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : AddCommMonoid F
inst✝³ : Module 𝕜 F
inst✝² : TopologicalSpace F
inst✝¹ : LocallyConvexSpace 𝕜 F
inst✝ : TopologicalSpace E
f : E →ₗ[𝕜] F
hf : IsInducing ⇑f
⊢ LocallyConvexSpace 𝕜 E
Module: Mathlib.Analysis.Convex.Combination

Goal state before:
pp: R : Type u_1
E : Type u_3
ι : Type u_5
inst✝⁵ : Field R
inst✝⁴ : AddCommGroup E
inst✝³ : Module R E
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
s : Set E
x : E
inst✝ : Fintype ι
w : ι → R
z : ι → E
hw₀ : ∀ (i : ι), 0 ≤ w i
hw₁ : ∑ i, w i = 1
hz : ∀ (i : ι), z i ∈ s
hx : ∑ i, w i • z i = x
⊢ x ∈ (convexHull R) s

Tactic:
by
  rw [← hx, ← centerMass_eq_of_sum_1 _ _ hw₁]
  exact centerMass_mem_convexHull _ (by simpa using hw₀) (by simp [hw₁]) (by simpa using hz)

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

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : Set E
α : Type u_2
f : Filter α
x : E
y z : α → E
r : α → E → Prop
hy : Tendsto y f (𝓝 x)
hz : Tendsto z f (𝓝 x)
hr : ∀ᶠ (p : α × E) in f ×ˢ 𝓝[s] x, r p.1 p.2
seg : ∀ᶠ (χ : α) in f, [y χ -[ℝ] z χ] ⊆ s
⊢ ∀ᶠ (χ : α) in f, ∀ v ∈ [y χ -[ℝ] z χ], r χ v

Tactic:
refine seg.mp <| .mono ?_ (fun _ => forall₂_imp)

Goal state after:
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : Set E
α : Type u_2
f : Filter α
x : E
y z : α → E
r : α → E → Prop
hy : Tendsto y f (𝓝 x)
hz : Tendsto z f (𝓝 x)
hr : ∀ᶠ (p : α × E) in f ×ˢ 𝓝[s] x, r p.1 p.2
seg : ∀ᶠ (χ : α) in f, [y χ -[ℝ] z χ] ⊆ s
⊢ ∀ᶠ (x : α) in f, ∀ a ∈ [y x -[ℝ] z x], a ∈ s → r x a
Module: Mathlib.Topology.Algebra.Module.UniformConvergence

Goal state before:
pp: case hsmul_left
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}
c : 𝕜
⊢ ∀ {i : Set E}, i ∈ 𝓝 0 → ∃ j ∈ 𝓝 0, Set.MapsTo (fun x ↦ c • x) {u | ∀ (x : α), φ u x ∈ j} {u | ∀ (x : α), φ u x ∈ i}

Tactic:
U

Goal state after:
case hsmul_left
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}
c : 𝕜
U : Set E
⊢ U ∈ 𝓝 0 → ∃ j ∈ 𝓝 0, Set.MapsTo (fun x ↦ c • x) {u | ∀ (x : α), φ u x ∈ j} {u | ∀ (x : α), φ u x ∈ U}
Module: Mathlib.Topology.Algebra.Module.UniformConvergence

Goal state before:
pp: case hsmul_right
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 : H
⊢ ∀ {i : Set E}, i ∈ 𝓝 0 → ∀ᶠ (x : 𝕜) in 𝓝 0, x • u ∈ {u | ∀ (x : α), φ u x ∈ i}

Tactic:
U

Goal state after:
case hsmul_right
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 : H
U : Set E
⊢ U ∈ 𝓝 0 → ∀ᶠ (x : 𝕜) in 𝓝 0, x • u ∈ {u | ∀ (x : α), φ u x ∈ U}
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: case h
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
U V : Set E
hU✝ : U ∈ p.basisSets
hV✝ : V ∈ p.basisSets
s : Finset ι
r₁ : ℝ
hr₁ : 0 < r₁
hU : U = (s.sup p).ball 0 r₁
t : Finset ι
r₂ : ℝ
hr₂ : 0 < r₂
hV : V = (t.sup p).ball 0 r₂
⊢ ⋂ i ∈ s ∪ t, (p i).ball 0 (min r₁ r₂) ⊆ (⋂ i ∈ s, (p i).ball 0 r₁) ∩ ⋂ i ∈ t, (p i).ball 0 r₂

Tactic:
exact
      Set.subset_inter
        (Set.iInter₂_mono' fun i hi =>
          ⟨i, Finset.subset_union_left hi, ball_mono <| min_le_left _ _⟩)
        (Set.iInter₂_mono' fun i hi =>
          ⟨i, Finset.subset_union_right hi, ball_mono <| min_le_right _ _⟩)

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

Goal state before:
pp: case neg
𝕜 : Type u_2
F : Type u_7
ι : Type u_9
inst✝² : NormedDivisionRing 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
p : SeminormFamily 𝕜 F ι
x : 𝕜
U : Set F
hU✝ : U ∈ p.basisSets
s : Finset ι
r : ℝ
hr : 0 < r
hU : U = (s.sup p).ball 0 r
h : ¬x ≠ 0
⊢ ∃ V ∈ p.addGroupFilterBasis.sets, V ⊆ (fun y ↦ x • y) ⁻¹' (s.sup p).ball 0 r

Tactic:
refine ⟨(s.sup p).ball 0 r, p.basisSets_mem s hr, ?_⟩

Goal state after:
case neg
𝕜 : Type u_2
F : Type u_7
ι : Type u_9
inst✝² : NormedDivisionRing 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
p : SeminormFamily 𝕜 F ι
x : 𝕜
U : Set F
hU✝ : U ∈ p.basisSets
s : Finset ι
r : ℝ
hr : 0 < r
hU : U = (s.sup p).ball 0 r
h : ¬x ≠ 0
⊢ (s.sup p).ball 0 r ⊆ (fun y ↦ x • y) ⁻¹' (s.sup p).ball 0 r
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)
s : Set E
hs : s ∈ 𝔖
⊢ ∀ i ∈ 𝓝 0, {f | MapsTo (⇑f) s i} ∈ absorbing R S

Tactic:
U

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)
s : Set E
hs : s ∈ 𝔖
U : Set F
⊢ U ∈ 𝓝 0 → {f | MapsTo (⇑f) s U} ∈ absorbing R S
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_6
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousConstSMul 𝕜 E
p : Seminorm 𝕜 E
h : p.ball 0 1 ∈ 𝓝 0
h' : IsVonNBounded 𝕜 (p.ball 0 1)
s : Set E
hs : s ∈ (SeminormFamily.moduleFilterBasis fun x ↦ p).filter
t : Set E
ht : t ∈ (SeminormFamily.moduleFilterBasis fun x ↦ p).toFilterBasis
ts : t ⊆ s
w : Finset (Fin 1)
r : ℝ
r_pos : 0 < r
hw : t = (w.sup fun x ↦ p).ball 0 r
w_ne : w ≠ ∅
⊢ w = Finset.univ

Tactic:
rcases Finset.nonempty_of_ne_empty w_ne with ⟨i, hi⟩

Goal state after:
𝕜 : Type u_2
E : Type u_6
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousConstSMul 𝕜 E
p : Seminorm 𝕜 E
h : p.ball 0 1 ∈ 𝓝 0
h' : IsVonNBounded 𝕜 (p.ball 0 1)
s : Set E
hs : s ∈ (SeminormFamily.moduleFilterBasis fun x ↦ p).filter
t : Set E
ht : t ∈ (SeminormFamily.moduleFilterBasis fun x ↦ p).toFilterBasis
ts : t ⊆ s
w : Finset (Fin 1)
r : ℝ
r_pos : 0 < r
hw : t = (w.sup fun x ↦ p).ball 0 r
w_ne : w ≠ ∅
i : Fin 1
hi : i ∈ w
⊢ w = Finset.univ
Module: Mathlib.Topology.Algebra.Module.Spaces.UniformConvergenceCLM

Goal state before:
pp: 𝕜₂ : Type u_2
inst✝¹⁶ : NormedField 𝕜₂
E : Type u_3
F : Type u_4
G : Type u_5
inst✝¹⁵ : AddCommGroup E
inst✝¹⁴ : TopologicalSpace E
inst✝¹³ : AddCommGroup F
inst✝¹² : Module 𝕜₂ F
𝕜₁ : Type u_6
inst✝¹¹ : NontriviallyNormedField 𝕜₁
σ : 𝕜₁ →+* 𝕜₂
inst✝¹⁰ : Module 𝕜₁ E
inst✝⁹ : AddCommGroup G
𝕜₃ : Type u_7
inst✝⁸ : NormedField 𝕜₃
inst✝⁷ : Module 𝕜₃ G
τ : 𝕜₃ →+* 𝕜₂
inst✝⁶ : RingHomSurjective τ
inst✝⁵ : TopologicalSpace F
inst✝⁴ : IsTopologicalAddGroup F
inst✝³ : ContinuousConstSMul 𝕜₂ F
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜₃ G
𝔖 : Set (Set E)
h𝔖 : ∀ s ∈ 𝔖, IsVonNBounded 𝕜₁ s
B : G →ₛₗ[τ] E →SLᵤ[σ, 𝔖] F
hB : Continuous[instTopologicalSpaceProd, inst✝⁵] fun p ↦ (B p.1) p.2
S : Set E
hS : S ∈ 𝔖
⊢ ∀ U ∈ 𝓝 0, ∀ᶠ (g : G) in 𝓝 0, ∀ e ∈ S, (B g) e ∈ U

Tactic:
U

Goal state after:
𝕜₂ : Type u_2
inst✝¹⁶ : NormedField 𝕜₂
E : Type u_3
F : Type u_4
G : Type u_5
inst✝¹⁵ : AddCommGroup E
inst✝¹⁴ : TopologicalSpace E
inst✝¹³ : AddCommGroup F
inst✝¹² : Module 𝕜₂ F
𝕜₁ : Type u_6
inst✝¹¹ : NontriviallyNormedField 𝕜₁
σ : 𝕜₁ →+* 𝕜₂
inst✝¹⁰ : Module 𝕜₁ E
inst✝⁹ : AddCommGroup G
𝕜₃ : Type u_7
inst✝⁸ : NormedField 𝕜₃
inst✝⁷ : Module 𝕜₃ G
τ : 𝕜₃ →+* 𝕜₂
inst✝⁶ : RingHomSurjective τ
inst✝⁵ : TopologicalSpace F
inst✝⁴ : IsTopologicalAddGroup F
inst✝³ : ContinuousConstSMul 𝕜₂ F
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜₃ G
𝔖 : Set (Set E)
h𝔖 : ∀ s ∈ 𝔖, IsVonNBounded 𝕜₁ s
B : G →ₛₗ[τ] E →SLᵤ[σ, 𝔖] F
hB : Continuous[instTopologicalSpaceProd, inst✝⁵] fun p ↦ (B p.1) p.2
S : Set E
hS : S ∈ 𝔖
U : Set F
⊢ U ∈ 𝓝 0 → ∀ᶠ (g : G) in 𝓝 0, ∀ e ∈ S, (B g) e ∈ U
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
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‖
⊢ 𝓤 (E →SL[σ₁₂] F) = ⨅ r, ⨅ (_ : r > 0), 𝓟 {f | ‖-f.1 + f.2‖ < r}

Tactic:
simp only [A]

Goal state after:
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‖
⊢ 𝓤 (E →SL[σ₁₂] F) = ⨅ r, ⨅ (_ : r > 0), 𝓟 {f | ‖f.1 - f.2‖ < r}
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‖
c : 𝕜
hc : 1 < ‖c‖
⊢ ∀ (f : E →SL[σ₁₂] F), (∀ (x : E), ‖x‖ ≤ 1 → ‖f x‖ ≤ 1) → ‖f‖ ≤ ‖c‖

Tactic:
intro f hf

Goal state after:
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‖
c : 𝕜
hc : 1 < ‖c‖
f : E →SL[σ₁₂] F
hf : ∀ (x : E), ‖x‖ ≤ 1 → ‖f x‖ ≤ 1
⊢ ‖f‖ ≤ ‖c‖
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: 𝕜 : Type u_2
𝕜₂ : Type u_3
E : Type u_6
F : Type u_7
ι : Type u_9
inst✝⁸ : NormedField 𝕜
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : NormedField 𝕜₂
inst✝⁴ : AddCommGroup F
inst✝³ : Module 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝² : RingHomIsometric σ₁₂
inst✝¹ : TopologicalSpace F
q : SeminormFamily 𝕜₂ F ι
hq : WithSeminorms q
inst✝ : TopologicalSpace E
f : E →ₛₗ[σ₁₂] F
hf : IsInducing ⇑f
⊢ WithSeminorms (q.comp f)

Tactic:
hf.eq_induced

Goal state after:
𝕜 : Type u_2
𝕜₂ : Type u_3
E : Type u_6
F : Type u_7
ι : Type u_9
inst✝⁸ : NormedField 𝕜
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : NormedField 𝕜₂
inst✝⁴ : AddCommGroup F
inst✝³ : Module 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝² : RingHomIsometric σ₁₂
inst✝¹ : TopologicalSpace F
q : SeminormFamily 𝕜₂ F ι
hq : WithSeminorms q
inst✝ : TopologicalSpace E
f : E →ₛₗ[σ₁₂] F
hf : IsInducing ⇑f
⊢ WithSeminorms (q.comp f)
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✝ g✝ h : α → β
op : β → γ → δ
f f' : α → β
Hf✝ : f ≈ f'
g g' : α → γ
Hg✝ : g ≈ g'
x : α
Hf : f x = f' x
Hg : g x = g' x
⊢ (fun x ↦ op (f x) (g x)) x = (fun x ↦ op (f' x) (g' x)) x

Tactic:
by simp only [Hf, Hg]

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
F : Type u_6
G : Type u_8
inst✝⁹ : SeminormedAddCommGroup E
inst✝⁸ : SeminormedAddCommGroup F
inst✝⁷ : SeminormedAddCommGroup G
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
inst✝⁴ : NontriviallyNormedField 𝕜₃
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace 𝕜₂ F
inst✝¹ : NormedSpace 𝕜₃ G
σ₁₂ : 𝕜 →+* 𝕜₂
σ₁₃ : 𝕜 →+* 𝕜₃
inst✝ : RingHomIsometric σ₁₃
f : E →SL[σ₁₂] F
g : E →SL[σ₁₃] G
h : ∀ (x : E), ‖f x‖ = ‖g x‖
c : ℝ
hc : c ≥ 0
h₂ : ∀ (x : E), ‖f x‖ ≤ c * ‖x‖
z : E
⊢ ‖f z‖ ≤ c * ‖z‖

Tactic:
exact h₂ z

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

Goal state before:
pp: case mpr
α : Type u₁
β : Type u₂
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
ι : Type u₃
p : Filter ι
F : ι → C(α, β)
f : C(α, β)
h : ∀ (K : Set α), IsCompact K → TendstoUniformlyOn (fun i a ↦ (F i) a) (⇑f) p K
K : Set α
hK : IsCompact K
⊢ ∀ (U : Set β), IsOpen[inst✝.toTopologicalSpace] U → MapsTo (⇑f) K U → ∀ᶠ (a : ι) in p, MapsTo (⇑(F a)) K U

Tactic:
U

Goal state after:
case mpr
α : Type u₁
β : Type u₂
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
ι : Type u₃
p : Filter ι
F : ι → C(α, β)
f : C(α, β)
h : ∀ (K : Set α), IsCompact K → TendstoUniformlyOn (fun i a ↦ (F i) a) (⇑f) p K
K : Set α
hK : IsCompact K
U : Set β
⊢ IsOpen[inst✝.toTopologicalSpace] U → MapsTo (⇑f) K U → ∀ᶠ (a : ι) in p, MapsTo (⇑(F a)) K U
Module: Mathlib.Topology.UniformSpace.CompactConvergence

Goal state before:
pp: α : Type u₁
β : Type u₂
inst✝² : TopologicalSpace α
inst✝¹ : UniformSpace β
ι : Type u₃
δ : ι → Type u_1
inst✝ : (i : ι) → TopologicalSpace (δ i)
φ : (i : ι) → C(δ i, α)
h_proper : ∀ (i : ι), IsProperMap ⇑(φ i)
h_lf : LocallyFinite fun i ↦ range ⇑(φ i)
h_cover : ⋃ i, range ⇑(φ i) = univ
𝔖 : Set (Set α) := {K | IsCompact K}
𝔗 : (i : ι) → Set (Set (δ i)) := fun i ↦ {K | IsCompact K}
h_image : ∀ (i : ι), MapsTo (fun x ↦ ⇑(φ i) '' x) (𝔗 i) 𝔖
⊢ inferInstance = ⨅ i, UniformSpace.comap (fun x ↦ x.comp (φ i)) inferInstance

Tactic:
have h_preimage : ∀ i, MapsTo (φ i ⁻¹' ·) 𝔖 (𝔗 i) := fun i K ↦ (h_proper i).isCompact_preimage

Goal state after:
α : Type u₁
β : Type u₂
inst✝² : TopologicalSpace α
inst✝¹ : UniformSpace β
ι : Type u₃
δ : ι → Type u_1
inst✝ : (i : ι) → TopologicalSpace (δ i)
φ : (i : ι) → C(δ i, α)
h_proper : ∀ (i : ι), IsProperMap ⇑(φ i)
h_lf : LocallyFinite fun i ↦ range ⇑(φ i)
h_cover : ⋃ i, range ⇑(φ i) = univ
𝔖 : Set (Set α) := {K | IsCompact K}
𝔗 : (i : ι) → Set (Set (δ i)) := fun i ↦ {K | IsCompact K}
h_image : ∀ (i : ι), MapsTo (fun x ↦ ⇑(φ i) '' x) (𝔗 i) 𝔖
h_preimage : ∀ (i : ι), MapsTo (fun x ↦ ⇑(φ i) ⁻¹' x) 𝔖 (𝔗 i)
⊢ inferInstance = ⨅ i, UniformSpace.comap (fun x ↦ x.comp (φ i)) inferInstance
Module: Mathlib.Topology.ContinuousMap.Algebra

Goal state before:
pp: α : Type u_1
inst✝⁹ : TopologicalSpace α
R : Type u_2
inst✝⁸ : CommSemiring R
A : Type u_3
inst✝⁷ : TopologicalSpace A
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
inst✝⁴ : IsTopologicalSemiring A
A₂ : Type u_4
inst✝³ : TopologicalSpace A₂
inst✝² : Semiring A₂
inst✝¹ : Algebra R A₂
inst✝ : IsTopologicalSemiring A₂
c : R
f : C(α, A)
x : α
⊢ (c • f) x = (C c * f) x

Tactic:
exact Algebra.smul_def' _ _

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

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), IsPiSystem (C i)
⊢ IsPiSystem (univ.pi '' univ.pi C)

Tactic:
rintro _ ⟨s₁, hs₁, rfl⟩ _ ⟨s₂, hs₂, rfl⟩ hst
  rw [← pi_inter_distrib] at hst ⊢; rw [univ_pi_nonempty_iff] at hst
  exact mem_image_of_mem _ fun i _ => hC i _ (hs₁ i (mem_univ i)) _ (hs₂ i (mem_univ i)) (hst i)

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

Goal state before:
pp: case refine_3
ι : Type u_1
α : ι → Type u_3
inst✝¹ : Fintype ι
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
i : ι
s : Set (α i)
hs : MeasurableSet s
t : Set ((a : ι) → α a)
⊢ ∀ j ∈ Finset.univ,
    j ≠ i → (μ j).toOuterMeasure (eval j '' (t \ (fun b ↦ b i) ⁻¹' s)) ≤ (μ j).toOuterMeasure (eval j '' t)

Tactic:
· rintro j - _; gcongr; apply sdiff_subset

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Indicator

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
s : Set α
f : α → ε
⊢ eLpNormEssSup (s.indicator f) μ ≤ eLpNormEssSup f μ

Tactic:
by
  refine essSup_mono_ae (.of_forall fun x => ?_)
  simp_rw [enorm_indicator_eq_indicator_enorm]
  exact Set.indicator_le_self s _ x

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✝⁴ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝³ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝² : ∀ (i : ι), SigmaFinite (μ i)
inst✝¹ : (i : ι) → TopologicalSpace (α i)
inst✝ : ∀ (i : ι), (μ i).IsOpenPosMeasure
U : Set ((i : ι) → α i)
U_open : IsOpen[Pi.topologicalSpace] U
a : (i : ι) → α i
ha : a ∈ U
s : (a : ι) → Set (α a)
hs : ∀ (a_1 : ι), IsOpen[inst✝¹ a_1] (s a_1) ∧ a a_1 ∈ s a_1
hsU : univ.pi s ⊆ U
i : ι
a✝ : i ∈ Finset.univ
⊢ 0 < (μ i) (s i)

Tactic:
apply (hs i).1.measure_pos (μ i) ⟨a i, (hs i).2⟩

Goal state after:
Module: Mathlib.Analysis.Convex.SpecificFunctions.Basic

Goal state before:
pp: case inl
s : ℝ
hs✝ : -1 ≤ s
hs' : s ≠ 0
p : ℝ
hp1 : 0 < p
hp2 : p < 1
hs : -1 < s
hs1 : 0 < 1 + s
h : s < 0
⊢ -1 < p * s

pp: case inr
s : ℝ
hs✝ : -1 ≤ s
hs' : s ≠ 0
p : ℝ
hp1 : 0 < p
hp2 : p < 1
hs : -1 < s
hs1 : 0 < 1 + s
h : 0 < s
⊢ -1 < p * s

Tactic:
· exact hs.trans (lt_mul_of_lt_one_left h hp2)

Goal state after:
case inr
s : ℝ
hs✝ : -1 ≤ s
hs' : s ≠ 0
p : ℝ
hp1 : 0 < p
hp2 : p < 1
hs : -1 < s
hs1 : 0 < 1 + s
h : 0 < s
⊢ -1 < p * s
Module: Mathlib.Analysis.Convex.SpecificFunctions.Basic

Goal state before:
pp: s : ℝ
hs✝ : -1 ≤ s
p : ℝ
hp1 : 0 < p
hp2 : p < 1
hs : -1 < s
hs1 : 0 < 1 + s
hs2 : 0 < 1 + p * s
hs3 : 1 + s ≠ 1
hs' : 1 + p * s = 1
⊢ s = 0

Tactic:
rwa [add_eq_left, mul_eq_zero, eq_false_intro hp1.ne', false_or] at hs'

Goal state after:
Module: Mathlib.Data.Real.ConjExponents

Goal state before:
pp: p q r : ℝ≥0∞
h : p.toNNReal.HolderTriple q.toNNReal r.toNNReal
⊢ p.toReal.HolderTriple q.toReal r.toReal

Tactic:
simpa only [coe_toNNReal_eq_toReal] using h.coe

Goal state after:
Module: Mathlib.Data.Real.ConjExponents

Goal state before:
pp: p q : ℝ≥0∞
h : p.toNNReal.HolderConjugate q.toNNReal
⊢ p.toReal.HolderConjugate q.toReal

Tactic:
simpa only [coe_toNNReal_eq_toReal] using h.coe

Goal state after:
Module: Mathlib.Analysis.Convex.SpecificFunctions.Basic

Goal state before:
pp: p : ℝ
hp : 1 < p
x y z : ℝ
hx : 0 ≤ x
hz : 0 ≤ z
hxy : x < y
hyz : y < z
hy : 0 < y
hy' : 0 < y ^ p
q : 0 < y - x
⊢ -1 + 1 ≤ x / y

Tactic:
neg_add_cancel,

Goal state after:
p : ℝ
hp : 1 < p
x y z : ℝ
hx : 0 ≤ x
hz : 0 ≤ z
hxy : x < y
hyz : y < z
hy : 0 < y
hy' : 0 < y ^ p
q : 0 < y - x
⊢ 0 ≤ x / y
Module: Mathlib.Data.Real.ConjExponents

Goal state before:
pp: p q : ℝ≥0∞
h : p.HolderConjugate q
⊢ p / q = p - 1

Tactic:
obtain rfl | hq := eq_or_ne q ∞
  · obtain rfl := unique ∞ p 1
    simp
  refine ENNReal.eq_sub_of_add_eq one_ne_top ?_
  rw [← ENNReal.div_self (ne_zero q p) hq, ← ENNReal.add_div, ← h.mul_eq_add, mul_div_assoc,
    ENNReal.div_self (ne_zero q p) hq, mul_one]

Goal state after: