mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.SpecialFunctions.Bernstein

Goal state before:
pp: E : Type u_1
inst✝⁵ : AddCommGroup E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : Module ℝ E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : LocallyConvexSpace ℝ E
f : C(↑I, E)
this✝ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
this : IsUniformAddGroup E
U : Set E
hU₀ : U ∈ 𝓝 0
hUc : Convex ℝ U
C : ℝ
hC : ∀ (x y : ↑I), gauge U (f x - f y) ≤ C
hC₀ : 0 ≤ C
δ : ℝ
hδ₀ : δ > 0
hδ : ∀ (x y : ↑I), dist x y < δ → gauge U (f x - f y) < 1 / 2
nhds_zero : Tendsto (fun n ↦ C / δ ^ 2 / ↑n) atTop (𝓝 0)
n : ℕ
nh : C / δ ^ 2 / ↑n < 1 / 2
hn₀ : n ≠ 0
x : ↑I
S : Finset (Fin (n + 1)) := {k | dist k/ₙ x < δ}
⊢ 1 / 2 * 1 = 1 / 2

Tactic:
mul_one

Goal state after:
E : Type u_1
inst✝⁵ : AddCommGroup E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : Module ℝ E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : LocallyConvexSpace ℝ E
f : C(↑I, E)
this✝ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
this : IsUniformAddGroup E
U : Set E
hU₀ : U ∈ 𝓝 0
hUc : Convex ℝ U
C : ℝ
hC : ∀ (x y : ↑I), gauge U (f x - f y) ≤ C
hC₀ : 0 ≤ C
δ : ℝ
hδ₀ : δ > 0
hδ : ∀ (x y : ↑I), dist x y < δ → gauge U (f x - f y) < 1 / 2
nhds_zero : Tendsto (fun n ↦ C / δ ^ 2 / ↑n) atTop (𝓝 0)
n : ℕ
nh : C / δ ^ 2 / ↑n < 1 / 2
hn₀ : n ≠ 0
x : ↑I
S : Finset (Fin (n + 1)) := {k | dist k/ₙ x < δ}
⊢ 1 / 2 = 1 / 2
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁰ : Semifield R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Algebra R A
inst✝¹ : ContinuousFunctionalCalculus R A p
inst✝ : ContinuousMapZero.UniqueHom R A
f : R → R
a : A
hf : ContinuousOn f (σₙ R a)
hf0 : f 0 = 0
⊢ cfcₙ f a = cfc f a

Tactic:
by_cases ha : p a
  · have hf' := hf.mono <| spectrum_subset_quasispectrum R a
    rw [cfc_apply f a ha hf', cfcₙ_apply f a hf, cfcₙHom_eq_cfcₙHom_of_cfcHom, cfcₙHom_of_cfcHom]
    dsimp only [NonUnitalStarAlgHom.comp_apply,
      NonUnitalStarAlgHom.coe_coe, compStarAlgHom'_apply]
    congr
  · simp [cfc_apply_of_not_predicate a ha, cfcₙ_apply_of_not_predicate (R := R) a ha]

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: case inr
X : Type u_1
inst✝⁵ : TopologicalSpace X
A : Type u_2
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A
r : ℝ
hr : r ≤ 0
⊢ φ ((algebraMap ℝ C(X, ℝ)) r).toNNReal - φ (-(algebraMap ℝ C(X, ℝ)) r).toNNReal = (algebraMap ℝ A) r

Tactic:
rw [← neg_neg r, ← map_neg, neg_neg (-r)]

Goal state after:
case inr
X : Type u_1
inst✝⁵ : TopologicalSpace X
A : Type u_2
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A
r : ℝ
hr : r ≤ 0
⊢ φ ((algebraMap ℝ C(X, ℝ)) (- -r)).toNNReal - φ ((algebraMap ℝ C(X, ℝ)) (-r)).toNNReal = (algebraMap ℝ A) (- -r)
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: case mem
𝕜 : Type u_1
inst✝ : RCLike 𝕜
s : Set 𝕜
p : C(↑s, 𝕜) → Prop
const : ∀ (r : 𝕜), p (ContinuousMap.const (↑s) r)
id : p (restrict s (ContinuousMap.id 𝕜))
star_id : p (star (restrict s (ContinuousMap.id 𝕜)))
add : ∀ (f g : C(↑s, 𝕜)), p f → p g → p (f + g)
mul : ∀ (f g : C(↑s, 𝕜)), p f → p g → p (f * g)
closure : (∀ f ∈ (polynomialFunctions s).starClosure, p f) → ∀ (f : C(↑s, 𝕜)), p f
f✝¹ f✝ f : C(↑s, 𝕜)
hf : f = (toContinuousMapOnAlgHom s) X ∨ f = star ((toContinuousMapOnAlgHom s) X)
⊢ p f

Tactic:
obtain (rfl | rfl) := hf

Goal state after:
case mem.inl
𝕜 : Type u_1
inst✝ : RCLike 𝕜
s : Set 𝕜
p : C(↑s, 𝕜) → Prop
const : ∀ (r : 𝕜), p (ContinuousMap.const (↑s) r)
id : p (restrict s (ContinuousMap.id 𝕜))
star_id : p (star (restrict s (ContinuousMap.id 𝕜)))
add : ∀ (f g : C(↑s, 𝕜)), p f → p g → p (f + g)
mul : ∀ (f g : C(↑s, 𝕜)), p f → p g → p (f * g)
closure : (∀ f ∈ (polynomialFunctions s).starClosure, p f) → ∀ (f : C(↑s, 𝕜)), p f
f✝ f : C(↑s, 𝕜)
⊢ p ((toContinuousMapOnAlgHom s) X)

case mem.inr
𝕜 : Type u_1
inst✝ : RCLike 𝕜
s : Set 𝕜
p : C(↑s, 𝕜) → Prop
const : ∀ (r : 𝕜), p (ContinuousMap.const (↑s) r)
id : p (restrict s (ContinuousMap.id 𝕜))
star_id : p (star (restrict s (ContinuousMap.id 𝕜)))
add : ∀ (f g : C(↑s, 𝕜)), p f → p g → p (f + g)
mul : ∀ (f g : C(↑s, 𝕜)), p f → p g → p (f * g)
closure : (∀ f ∈ (polynomialFunctions s).starClosure, p f) → ∀ (f : C(↑s, 𝕜)), p f
f✝ f : C(↑s, 𝕜)
⊢ p (star ((toContinuousMapOnAlgHom s) X))
Module: Mathlib.Topology.Algebra.Module.Spaces.CharacterSpace

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁷ : CommRing 𝕜
inst✝⁶ : NoZeroDivisors 𝕜
inst✝⁵ : TopologicalSpace 𝕜
inst✝⁴ : ContinuousAdd 𝕜
inst✝³ : ContinuousConstSMul 𝕜 𝕜
inst✝² : TopologicalSpace A
inst✝¹ : Semiring A
inst✝ : Algebra 𝕜 A
map_one' : ∀ (φ : ↑(characterSpace 𝕜 A)), φ 1 = 1
φ : ↑(characterSpace 𝕜 A)
r : 𝕜
⊢ r * 1 = r

Tactic:
mul_one

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁷ : CommRing 𝕜
inst✝⁶ : NoZeroDivisors 𝕜
inst✝⁵ : TopologicalSpace 𝕜
inst✝⁴ : ContinuousAdd 𝕜
inst✝³ : ContinuousConstSMul 𝕜 𝕜
inst✝² : TopologicalSpace A
inst✝¹ : Semiring A
inst✝ : Algebra 𝕜 A
map_one' : ∀ (φ : ↑(characterSpace 𝕜 A)), φ 1 = 1
φ : ↑(characterSpace 𝕜 A)
r : 𝕜
⊢ r = r
Module: Mathlib.Topology.Algebra.Module.Spaces.CharacterSpace

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁷ : CommRing 𝕜
inst✝⁶ : NoZeroDivisors 𝕜
inst✝⁵ : TopologicalSpace 𝕜
inst✝⁴ : ContinuousAdd 𝕜
inst✝³ : ContinuousConstSMul 𝕜 𝕜
inst✝² : TopologicalSpace A
inst✝¹ : Ring A
inst✝ : Algebra 𝕜 A
φ ψ : ↑(characterSpace 𝕜 A)
h : RingHom.ker φ = RingHom.ker ψ
x : A
⊢ x - (algebraMap 𝕜 A) (ψ x) ∈ RingHom.ker φ

Tactic:
simpa only [h, RingHom.mem_ker, map_sub, AlgHomClass.commutes] using! sub_self (ψ x)

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

Goal state before:
pp: case mp
𝕜 : Type u_1
inst✝ : RCLike 𝕜
s : Set 𝕜
h0 : 0 ∈ s
r : 𝕜
f : C(↑s, 𝕜)
hf : f ∈ adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}
hf₂ : r * 1 = 0
this : f ⟨0, h0⟩ = 0
⊢ r • 1 + f ∈ adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}

Tactic:
mul_one

Goal state after:
case mp
𝕜 : Type u_1
inst✝ : RCLike 𝕜
s : Set 𝕜
h0 : 0 ∈ s
r : 𝕜
f : C(↑s, 𝕜)
hf : f ∈ adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}
hf₂ : r = 0
this : f ⟨0, h0⟩ = 0
⊢ r • 1 + f ∈ adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}
Module: Mathlib.Topology.Algebra.Module.Spaces.CharacterSpace

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁶ : CommRing 𝕜
inst✝⁵ : NoZeroDivisors 𝕜
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : IsTopologicalRing 𝕜
inst✝² : TopologicalSpace A
inst✝¹ : Semiring A
inst✝ : Algebra 𝕜 A
⊢ { toFun := fun φ ↦ φ 0, continuous_toFun := ⋯ } = 0

Tactic:
by ext; simp only [map_zero, coe_mk, coe_zero, Pi.zero_apply]

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

Goal state before:
pp: case mpr
𝕜 : Type u_1
inst✝ : RCLike 𝕜
s : Set 𝕜
h0 : 0 ∈ s
f : C(↑s, 𝕜)
hf : f ∈ adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}
⊢ f ∈ StarAlgebra.adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}

Tactic:
exact adjoin_le_starAlgebra_adjoin _ _ hf

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

Goal state before:
pp: case e'_3
𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
s : Set 𝕜
h0 : 0 ∈ s
inst✝ : CompactSpace ↑s
⊢ closure ↑(StarAlgebra.adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}) = Set.univ

Tactic:
rw [← Polynomial.toContinuousMapOn_X_eq_restrict_id, ← Polynomial.toContinuousMapOnAlgHom_apply,
    ← polynomialFunctions.starClosure_eq_adjoin_X s]

Goal state after:
case e'_3
𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
s : Set 𝕜
h0 : 0 ∈ s
inst✝ : CompactSpace ↑s
⊢ closure ↑(polynomialFunctions s).starClosure = Set.univ
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: case mem
𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
s : Set 𝕜
inst✝ : Fact (0 ∈ s)
p : C(↑s, 𝕜)₀ → Prop
zero : p 0
id : p (ContinuousMapZero.id s)
star_id : p (star (ContinuousMapZero.id s))
add : ∀ (f g : C(↑s, 𝕜)₀), p f → p g → p (f + g)
mul : ∀ (f g : C(↑s, 𝕜)₀), p f → p g → p (f * g)
smul : ∀ (r : 𝕜) (f : C(↑s, 𝕜)₀), p f → p (r • f)
closure : (∀ f ∈ adjoin 𝕜 {ContinuousMapZero.id s}, p f) → ∀ (f : C(↑s, 𝕜)₀), p f
f✝¹ f✝ f : C(↑s, 𝕜)₀
hf : f = ContinuousMapZero.id s ∨ star (ContinuousMapZero.id s) = f
⊢ p f

Tactic:
obtain (rfl | rfl) := hf

Goal state after:
case mem.inl
𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
s : Set 𝕜
inst✝ : Fact (0 ∈ s)
p : C(↑s, 𝕜)₀ → Prop
zero : p 0
id : p (ContinuousMapZero.id s)
star_id : p (star (ContinuousMapZero.id s))
add : ∀ (f g : C(↑s, 𝕜)₀), p f → p g → p (f + g)
mul : ∀ (f g : C(↑s, 𝕜)₀), p f → p g → p (f * g)
smul : ∀ (r : 𝕜) (f : C(↑s, 𝕜)₀), p f → p (r • f)
closure : (∀ f ∈ adjoin 𝕜 {ContinuousMapZero.id s}, p f) → ∀ (f : C(↑s, 𝕜)₀), p f
f✝ f : C(↑s, 𝕜)₀
⊢ p (ContinuousMapZero.id s)

case mem.inr
𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
s : Set 𝕜
inst✝ : Fact (0 ∈ s)
p : C(↑s, 𝕜)₀ → Prop
zero : p 0
id : p (ContinuousMapZero.id s)
star_id : p (star (ContinuousMapZero.id s))
add : ∀ (f g : C(↑s, 𝕜)₀), p f → p g → p (f + g)
mul : ∀ (f g : C(↑s, 𝕜)₀), p f → p g → p (f * g)
smul : ∀ (r : 𝕜) (f : C(↑s, 𝕜)₀), p f → p (r • f)
closure : (∀ f ∈ adjoin 𝕜 {ContinuousMapZero.id s}, p f) → ∀ (f : C(↑s, 𝕜)₀), p f
f✝ f : C(↑s, 𝕜)₀
⊢ p (star (ContinuousMapZero.id s))
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
U : SetRel α α
inst✝ : U.IsSymm
⊢ hausdorffEntourage U.inv = hausdorffEntourage U

Tactic:
inv_eq_self

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
U : SetRel α α
inst✝ : U.IsSymm
⊢ hausdorffEntourage U = hausdorffEntourage U
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
U : SetRel α α
inst✝ : U.IsTrans
⊢ hausdorffEntourage (U ○ U) ⊆ hausdorffEntourage U

Tactic:
comp_subset_self

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
U : SetRel α α
inst✝ : U.IsTrans
⊢ hausdorffEntourage U ⊆ hausdorffEntourage U
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝² : NormedField 𝕜
inst✝¹ : Ring A
inst✝ : Algebra 𝕜 A
a : A
n : ℕ
hn : n ≠ 0
⊢ spectralRadius 𝕜 a ^ n ≤ spectralRadius 𝕜 (a ^ n)

Tactic:
by
  simp only [spectralRadius, ENNReal.iSup₂_pow_of_ne_zero _ hn]
  refine iSup₂_le fun x hx ↦ ?_
  apply le_iSup₂_of_le (x ^ n) (spectrum.pow_mem_pow a n hx)
  simp

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝³ : NormedField 𝕜
inst✝² : Ring A
inst✝¹ : Algebra 𝕜 A
inst✝ : Nontrivial A
a : A
n : ℕ
⊢ spectralRadius 𝕜 a ^ n ≤ spectralRadius 𝕜 (a ^ n)

Tactic:
cases n
  · simp
  · exact spectralRadius_pow_le a _ (by simp)

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : NormedRing A
inst✝² : NormedAlgebra 𝕜 A
inst✝¹ : CompleteSpace A
inst✝ : NormOneClass A
a : A
k : 𝕜
h : ‖a‖ < ‖k‖
⊢ ‖a‖ * 1 < ‖k‖

Tactic:
mul_one

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : NormedRing A
inst✝² : NormedAlgebra 𝕜 A
inst✝¹ : CompleteSpace A
inst✝ : NormOneClass A
a : A
k : 𝕜
h : ‖a‖ < ‖k‖
⊢ ‖a‖ < ‖k‖
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
⊢ ∀ᶠ (z : 𝕜) in cobounded 𝕜, IsUnit (resolvent a z)

Tactic:
atTop_basis_Ioi.cobounded_of_norm.eventually_iff

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
⊢ ∃ i, True ∧ ∀ ⦃x : 𝕜⦄, x ∈ norm ⁻¹' Set.Ioi i → IsUnit (resolvent a x)
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: case neg
𝕜 : Type u_1
A : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
a : A
z : 𝕜
h : ↑‖z‖₊ < (spectralRadius 𝕜 a)⁻¹
hz : ¬z = 0
u : 𝕜ˣ := Units.mk0 z hz
⊢ spectralRadius 𝕜 a < ↑‖↑u⁻¹‖₊

Tactic:
Units.val_inv_eq_inv_val,

Goal state after:
case neg
𝕜 : Type u_1
A : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
a : A
z : 𝕜
h : ↑‖z‖₊ < (spectralRadius 𝕜 a)⁻¹
hz : ¬z = 0
u : 𝕜ˣ := Units.mk0 z hz
⊢ spectralRadius 𝕜 a < ↑‖(↑u)⁻¹‖₊
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case e'_3
α : Type u_1
inst✝ : TopologicalSpace α
⊢ {⊥} = SetLike.coe ⁻¹' SetLike.coe '' {⊥}

Tactic:
SetLike.coe_injective.preimage_image

Goal state after:
case e'_3
α : Type u_1
inst✝ : TopologicalSpace α
⊢ {⊥} = {⊥}
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case refine_1
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
u : Set (Set α)
⊢ u ∈ {u | u.Finite ∧ u ⊆ B} → IsOpen {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}

Tactic:
⟨hu, huB⟩

Goal state after:
case refine_1
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
u : Set (Set α)
hu : u.Finite
huB : u ⊆ B
⊢ IsOpen {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case refine_2
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst✝] V
hu : u.Finite
huB : u ⊆ B
huV : ∀ U ∈ u, U ⊆ V
K : Compacts α
hKu : ∀ U ∈ u, (↑K ∩ U).Nonempty
w : Set (Set α)
hKV : ↑K ⊆ ⋃₀ w
hwB : w ⊆ B
hwV : ⋃₀ w ⊆ V
⊢ ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u ⊆ B},
    K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}

Tactic:
wlog hwK : ∀ W ∈ w, ((K : Set α) ∩ W).Nonempty generalizing w

Goal state after:
case refine_2.inr
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst✝] V
hu : u.Finite
huB : u ⊆ B
huV : ∀ U ∈ u, U ⊆ V
K : Compacts α
hKu : ∀ U ∈ u, (↑K ∩ U).Nonempty
w : Set (Set α)
hKV : ↑K ⊆ ⋃₀ w
hwB : w ⊆ B
hwV : ⋃₀ w ⊆ V
this :
  ∀ (w : Set (Set α)),
    ↑K ⊆ ⋃₀ w →
      w ⊆ B →
        ⋃₀ w ⊆ V →
          (∀ W ∈ w, (↑K ∩ W).Nonempty) →
            ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u ⊆ B},
              K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}
hwK : ¬∀ W ∈ w, (↑K ∩ W).Nonempty
⊢ ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u ⊆ B},
    K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}

α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst✝] V
hu : u.Finite
huB : u ⊆ B
huV : ∀ U ∈ u, U ⊆ V
K : Compacts α
hKu : ∀ U ∈ u, (↑K ∩ U).Nonempty
w : Set (Set α)
hKV : ↑K ⊆ ⋃₀ w
hwB : w ⊆ B
hwV : ⋃₀ w ⊆ V
hwK : ∀ W ∈ w, (↑K ∩ W).Nonempty
⊢ ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u ⊆ B},
    K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : ProperSpace 𝕜
inst✝⁴ : NonUnitalNormedRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : CompleteSpace A
a : A
⊢ quasispectrum 𝕜 a = (spectrum 𝕜 ∘ fun x ↦ toLp 1 ↑x) a

Tactic:
Unitization.quasispectrum_eq_spectrum_inr,

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : ProperSpace 𝕜
inst✝⁴ : NonUnitalNormedRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : CompleteSpace A
a : A
⊢ spectrum 𝕜 ↑a = (spectrum 𝕜 ∘ fun x ↦ toLp 1 ↑x) a
Module: Mathlib.Analysis.CStarAlgebra.Spectrum

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
x✝ : ℂ
⊢ x✝ ∈ {z | z.im < 0} → x✝ ∈ {z | z.im ≤ 0}

Tactic:
by simpa using le_of_lt

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

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
x✝ : ℂ
⊢ x✝ ∈ {z | 0 < z.im} → x✝ ∈ {z | 0 ≤ z.im}

Tactic:
by simpa using le_of_lt

Goal state after:
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : T2Space α
K₁ K₂ : Compacts α
h : K₁ ≠ K₂
⊢ (fun x y ↦ ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v) K₁ K₂

Tactic:
by
    wlog h' : ¬(K₁ ≤ K₂) generalizing K₁ K₂
    · grind [Disjoint.symm, le_antisymm]
    rw [SetLike.not_le_iff_exists] at h'
    obtain ⟨x, hx₁, hx₂⟩ := h'
    obtain ⟨U, V, hU, hV, hU', hV', hUV⟩ := K₂.isCompact.separation_of_notMem hx₂
    exact ⟨_, _, isOpen_inter_nonempty_of_isOpen hV, isOpen_subsets_of_isOpen hU, ⟨x, hx₁, hV'⟩,
      hU', by grind [Set.Nonempty]⟩

Goal state after:
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
S : Set (Compacts α)
hS : IsCompact S
ι : Type u_1
U : ι → Set α
hU : ∀ (i : ι), IsOpen[inst✝] (U i)
h : ∀ i ∈ S, ↑i ⊆ ⋃ i, U i
s : Finset (Finset ι)
hs : S ⊆ ⋃ i ∈ s, {K | ↑K ⊆ ⋃ i_1 ∈ i, U i_1}
⊢ ∃ t, ⋃ K ∈ S, ↑K ⊆ ⋃ i ∈ t, U i

Tactic:
refine ⟨s.biUnion id, (biUnion_subset_biUnion_left hs).trans ?_⟩

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
S : Set (Compacts α)
hS : IsCompact S
ι : Type u_1
U : ι → Set α
hU : ∀ (i : ι), IsOpen[inst✝] (U i)
h : ∀ i ∈ S, ↑i ⊆ ⋃ i, U i
s : Finset (Finset ι)
hs : S ⊆ ⋃ i ∈ s, {K | ↑K ⊆ ⋃ i_1 ∈ i, U i_1}
⊢ ⋃ x ∈ ⋃ i ∈ s, {K | ↑K ⊆ ⋃ i_1 ∈ i, U i_1}, ↑x ⊆ ⋃ i ∈ s.biUnion id, U i
Module: Mathlib.Analysis.LocallyConvex.Polar

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedCommRing 𝕜
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
a : E
y : F
hy : y ∈ {y | ‖(B a) y‖ ≤ 1}
x✝ : E
hb : x✝ ∈ {a}
⊢ ‖(B a) y‖ ≤ 1

Tactic:
exact hy

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
h : B.SeparatingRight
y : F
hy : y ∈ B.polar Set.univ
x : E
ε : ℝ
hε : 0 < ε
c : 𝕜
hc : 0 < ‖c‖
hcε : ‖c‖ < ε
⊢ ‖(B x) y‖ = ‖c‖ * ‖(B (c⁻¹ • x)) y‖

Tactic:
rw [B.map_smul, LinearMap.smul_apply, smul_eq_mul, norm_mul, norm_inv,
        mul_inv_cancel_left₀ hc.ne']

Goal state after:
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case refine_2
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
u : Set (Set α)
⊢ u ∈ {u | u.Finite ∧ u ⊆ B} →
    ∀ x ∈ toCompacts ⁻¹' {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty},
      ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u.Nonempty ∧ u ⊆ B},
        x ∈ v ∧ v ⊆ toCompacts ⁻¹' {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}

Tactic:
⟨hu, huB⟩

Goal state after:
case refine_2
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
u : Set (Set α)
hu : u.Finite
huB : u ⊆ B
⊢ ∀ x ∈ toCompacts ⁻¹' {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty},
    ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u.Nonempty ∧ u ⊆ B},
      x ∈ v ∧ v ⊆ toCompacts ⁻¹' {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}
Module: Mathlib.Analysis.Normed.Module.HahnBanach

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : IsRCLikeNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : Subspace 𝕜 E
f : StrongDual 𝕜 ↥p
⊢ ∃ g, (∀ (x : ↥p), g ↑x = f x) ∧ ‖g‖ = ‖f‖

Tactic:
by
  obtain ⟨g, hg, hl⟩ := by
    refine Module.Dual.exists_continuous_extension_of_le_seminorm p f
      (show Continuous (‖f‖₊ • (normSeminorm 𝕜 E)) from ?_) fun x => f.le_opNorm x
    exact continuous_norm.const_smul ‖f‖₊
  refine ⟨g, hg, le_antisymm (g.opNorm_le_bound (norm_nonneg f) hl) ?_⟩
  exact f.opNorm_le_bound (norm_nonneg _) fun x => by simpa [hg x] using g.le_opNorm x

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

Goal state before:
pp: ι : Type u_4
𝕜 : Type u_5
E : Type u_6
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E →ₗ[𝕜] 𝕜
φ : E →ₗ[𝕜] 𝕜
t𝕜 : TopologicalSpace 𝕜 := inferInstance
t : TopologicalSpace E := ⨅ i, induced (⇑(f i)) t𝕜
⊢ φ ∈ Submodule.span 𝕜 (Set.range f) ↔ ∃ s c, φ.toSeminorm ≤ c • s.sup fun i ↦ (f i).toSeminorm

Tactic:
have : IsTopologicalAddGroup E := topologicalAddGroup_iInf fun _ ↦ topologicalAddGroup_induced _

Goal state after:
ι : Type u_4
𝕜 : Type u_5
E : Type u_6
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E →ₗ[𝕜] 𝕜
φ : E →ₗ[𝕜] 𝕜
t𝕜 : TopologicalSpace 𝕜 := inferInstance
t : TopologicalSpace E := ⨅ i, induced (⇑(f i)) t𝕜
this : IsTopologicalAddGroup E
⊢ φ ∈ Submodule.span 𝕜 (Set.range f) ↔ ∃ s c, φ.toSeminorm ≤ c • s.sup fun i ↦ (f i).toSeminorm
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: W : Type u_3
Q : Type u_4
inst✝⁴ : NormedAddCommGroup W
inst✝³ : MetricSpace Q
inst✝² : NormedAddTorsor W Q
𝕜 : Type u_5
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 W
p₁ p₂ : Q
h : p₁ ≠ p₂
c₁ c₂ : 𝕜
⊢ dist c₁ c₂ ≤ dist c₁ c₂ * 1

Tactic:
mul_one

Goal state after:
W : Type u_3
Q : Type u_4
inst✝⁴ : NormedAddCommGroup W
inst✝³ : MetricSpace Q
inst✝² : NormedAddTorsor W Q
𝕜 : Type u_5
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 W
p₁ p₂ : Q
h : p₁ ≠ p₂
c₁ c₂ : 𝕜
⊢ dist c₁ c₂ ≤ dist c₁ c₂
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
c : CU P
⊢ Continuous[inst✝, _] c.lim

Tactic:
obtain ⟨h0, h1234, h1⟩ : 0 < (2⁻¹ : ℝ) ∧ (2⁻¹ : ℝ) < 3 / 4 ∧ (3 / 4 : ℝ) < 1 := by norm_num

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
c : CU P
h0 : 0 < 2⁻¹
h1234 : 2⁻¹ < 3 / 4
h1 : 3 / 4 < 1
⊢ Continuous[inst✝, _] c.lim
Module: Mathlib.Analysis.Normed.Module.WeakDual

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : SeparableSpace E
K : Set (WeakDual 𝕜 E)
K_cpt : IsCompact K
⊢ MetrizableSpace ↑K

Tactic:
by
  have : CompactSpace K := isCompact_iff_compactSpace.mp K_cpt
  obtain ⟨gs, gs_cont, gs_sep⟩ := exists_countable_separating 𝕜 E
  exact Metric.PiNatEmbed.TopologicalSpace.MetrizableSpace.of_countable_separating
    (fun n k ↦ gs n k) (fun n ↦ (gs_cont n).comp continuous_subtype_val)
    fun x y hxy ↦ gs_sep <| Subtype.val_injective.ne hxy

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

Goal state before:
pp: k : Type u_1
E : Type u_2
inst✝⁷ : Field k
inst✝⁶ : LinearOrder k
inst✝⁵ : AddCommGroup E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : Module k E
inst✝¹ : IsStrictOrderedModule k E
inst✝ : PosSMulReflectLE k E
f : k → E
a b r' : k
h : 0 < r' * (b - a)
⊢ f (r' * (a - b) + b) ≤ r' • (f a - f b) + f b ↔
    (b - a)⁻¹ • (f b - f a) ≤ (-(r' * (a - b)))⁻¹ • (f b - f (r' * (a - b) + b))

Tactic:
neg_mul_eq_mul_neg,

Goal state after:
k : Type u_1
E : Type u_2
inst✝⁷ : Field k
inst✝⁶ : LinearOrder k
inst✝⁵ : AddCommGroup E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : Module k E
inst✝¹ : IsStrictOrderedModule k E
inst✝ : PosSMulReflectLE k E
f : k → E
a b r' : k
h : 0 < r' * (b - a)
⊢ f (r' * (a - b) + b) ≤ r' • (f a - f b) + f b ↔
    (b - a)⁻¹ • (f b - f a) ≤ (r' * -(a - b))⁻¹ • (f b - f (r' * (a - b) + b))
Module: Mathlib.LinearAlgebra.AffineSpace.Ordered

Goal state before:
pp: case h
k : Type u_1
E : Type u_2
inst✝⁷ : Field k
inst✝⁶ : LinearOrder k
inst✝⁵ : AddCommGroup E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : Module k E
inst✝¹ : IsStrictOrderedModule k E
inst✝ : PosSMulReflectLE k E
f : k → E
a b r' : k
h : 0 < r' * (b - a)
⊢ b - a ≠ 0

Tactic:
· exact right_ne_zero_of_mul h.ne'

Goal state after: