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}
⊢ gauge U ((bernsteinApproximation n f) x - f x) = gauge U (∑ k, (bernstein n ↑k) x • (f k/ₙ - f x))

Tactic:
by
      simp [bernsteinApproximation.apply, smul_sub, ← Finset.sum_smul]

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

Goal state before:
pp: case neg
X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : Zero X
A : Type u_2
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
r✝ : ℝ
f : C(X, ℝ)₀
r : ℝ≥0
hr : 0 < ↑r
⊢ r • φ (-f).toNNReal - r • φ f.toNNReal = -(r • φ f.toNNReal) + r • φ (-f).toNNReal

Tactic:
sub_eq_add_neg,

Goal state after:
case neg
X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : Zero X
A : Type u_2
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
r✝ : ℝ
f : C(X, ℝ)₀
r : ℝ≥0
hr : 0 < ↑r
⊢ r • φ (-f).toNNReal + -(r • φ f.toNNReal) = -(r • φ f.toNNReal) + r • φ (-f).toNNReal
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: case refine_2
F : Type u_2
S : Type u_3
K : Type u_4
A : Type u_5
inst✝¹³ : CommRing K
inst✝¹² : Ring A
inst✝¹¹ : Algebra K A
inst✝¹⁰ : TopologicalSpace K
inst✝⁹ : T1Space K
inst✝⁸ : TopologicalSpace A
inst✝⁷ : ContinuousSub A
inst✝⁶ : ContinuousSMul K A
inst✝⁵ : FunLike F A K
inst✝⁴ : AlgHomClass F K A K
inst✝³ : SetLike S A
inst✝² : OneMemClass S A
inst✝¹ : AddSubgroupClass S A
inst✝ : SMulMemClass S K A
φ : F
hφ : Continuous[inst✝⁸, inst✝¹⁰] ⇑φ
s : S
⊢ closure[inst✝⁸] ↑s ∩ ↑(RingHom.ker φ) ⊆ closure[inst✝⁸] (↑s ∩ ↑(RingHom.ker φ))

Tactic:
intro x ⟨hxs, (hxφ : φ x = 0)⟩

Goal state after:
case refine_2
F : Type u_2
S : Type u_3
K : Type u_4
A : Type u_5
inst✝¹³ : CommRing K
inst✝¹² : Ring A
inst✝¹¹ : Algebra K A
inst✝¹⁰ : TopologicalSpace K
inst✝⁹ : T1Space K
inst✝⁸ : TopologicalSpace A
inst✝⁷ : ContinuousSub A
inst✝⁶ : ContinuousSMul K A
inst✝⁵ : FunLike F A K
inst✝⁴ : AlgHomClass F K A K
inst✝³ : SetLike S A
inst✝² : OneMemClass S A
inst✝¹ : AddSubgroupClass S A
inst✝ : SMulMemClass S K A
φ : F
hφ : Continuous[inst✝⁸, inst✝¹⁰] ⇑φ
s : S
x : A
hxs : x ∈ closure[inst✝⁸] ↑s
hxφ : φ x = 0
⊢ x ∈ closure[inst✝⁸] (↑s ∩ ↑(RingHom.ker φ))
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: F : Type u_1
R : Type u_2
S : Type u_3
A : Type u_4
B : Type u_5
p : A → Prop
q : B → Prop
inst✝²⁴ : CommSemiring R
inst✝²³ : StarRing R
inst✝²² : MetricSpace R
inst✝²¹ : IsTopologicalSemiring R
inst✝²⁰ : ContinuousStar R
inst✝¹⁹ : Ring A
inst✝¹⁸ : StarRing A
inst✝¹⁷ : TopologicalSpace A
inst✝¹⁶ : Algebra R A
inst✝¹⁵ : Ring B
inst✝¹⁴ : StarRing B
inst✝¹³ : TopologicalSpace B
inst✝¹² : Algebra R B
inst✝¹¹ : CommSemiring S
inst✝¹⁰ : Algebra R S
inst✝⁹ : Algebra S A
inst✝⁸ : Algebra S B
inst✝⁷ : IsScalarTower R S A
inst✝⁶ : IsScalarTower R S B
inst✝⁵ : ContinuousFunctionalCalculus R A p
inst✝⁴ : ContinuousFunctionalCalculus R B q
inst✝³ : ContinuousMap.UniqueHom R B
inst✝² : FunLike F A B
inst✝¹ : AlgHomClass F S A B
inst✝ : StarHomClass F A B
φ : F
f : R → R
a : A
hf : ContinuousOn f (spectrum R a)
hφ : Continuous[inst✝¹⁷, inst✝¹³] ⇑φ
ha : p a
hφa : q (φ a)
ψ : A →⋆ₐ[R] B := StarAlgHom.restrictScalars R ↑φ
this✝ : Continuous[inst✝¹⁷, inst✝¹³] ⇑ψ
h_spec : spectrum R (ψ a) ⊆ spectrum R a
hψa : q (ψ a)
ι : C(↑(spectrum R (ψ a)), ↑(spectrum R a)) := { toFun := Set.inclusion h_spec, continuous_toFun := ⋯ }
this : ψ.comp (cfcHom ha) = (cfcHom hψa).comp (ContinuousMap.compStarAlgHom' R R ι)
hf' : ContinuousOn f (spectrum R (ψ a))
⊢ φ ((cfcHom ha) { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ }) = cfc f (φ a)

Tactic:
cfc_apply ..

Goal state after:
F : Type u_1
R : Type u_2
S : Type u_3
A : Type u_4
B : Type u_5
p : A → Prop
q : B → Prop
inst✝²⁴ : CommSemiring R
inst✝²³ : StarRing R
inst✝²² : MetricSpace R
inst✝²¹ : IsTopologicalSemiring R
inst✝²⁰ : ContinuousStar R
inst✝¹⁹ : Ring A
inst✝¹⁸ : StarRing A
inst✝¹⁷ : TopologicalSpace A
inst✝¹⁶ : Algebra R A
inst✝¹⁵ : Ring B
inst✝¹⁴ : StarRing B
inst✝¹³ : TopologicalSpace B
inst✝¹² : Algebra R B
inst✝¹¹ : CommSemiring S
inst✝¹⁰ : Algebra R S
inst✝⁹ : Algebra S A
inst✝⁸ : Algebra S B
inst✝⁷ : IsScalarTower R S A
inst✝⁶ : IsScalarTower R S B
inst✝⁵ : ContinuousFunctionalCalculus R A p
inst✝⁴ : ContinuousFunctionalCalculus R B q
inst✝³ : ContinuousMap.UniqueHom R B
inst✝² : FunLike F A B
inst✝¹ : AlgHomClass F S A B
inst✝ : StarHomClass F A B
φ : F
f : R → R
a : A
hf : ContinuousOn f (spectrum R a)
hφ : Continuous[inst✝¹⁷, inst✝¹³] ⇑φ
ha : p a
hφa : q (φ a)
ψ : A →⋆ₐ[R] B := StarAlgHom.restrictScalars R ↑φ
this✝ : Continuous[inst✝¹⁷, inst✝¹³] ⇑ψ
h_spec : spectrum R (ψ a) ⊆ spectrum R a
hψa : q (ψ a)
ι : C(↑(spectrum R (ψ a)), ↑(spectrum R a)) := { toFun := Set.inclusion h_spec, continuous_toFun := ⋯ }
this : ψ.comp (cfcHom ha) = (cfcHom hψa).comp (ContinuousMap.compStarAlgHom' R R ι)
hf' : ContinuousOn f (spectrum R (ψ a))
⊢ φ ((cfcHom ha) { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ }) =
    (cfcHom hψa) { toFun := (spectrum R (φ a)).domRestrict f, continuous_toFun := ⋯ }
Module: Mathlib.Topology.Semicontinuity.Hemicontinuity

Goal state before:
pp: α : Type u_3
β : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → Set β
s : Set α
x : α
γ : Type u_5
inst✝ : TopologicalSpace γ
i : γ → β
hf : UpperHemicontinuousWithinAt f s x
hi : IsInducing i
h_cl : IsClosed[inst✝¹] (range i)
⊢ UpperHemicontinuousWithinAt (fun x ↦ i ⁻¹' f x) s x

Tactic:
refine .of_forall_isOpen fun u hu hifu ↦ ?_
  obtain ⟨v, hv, rfl⟩ := hi.isOpen_iff.mp hu
  simp_rw [← preimage_inter_range (s := f _), preimage_subset_preimage_iff inter_subset_right]
    at hifu ⊢
  exact hf.inter h_cl |>.forall_isOpen v hv hifu

Goal state after:
Module: Mathlib.Topology.Semicontinuity.Hemicontinuity

Goal state before:
pp: α : Type u_3
β : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : TopologicalSpace β
f : α → Set β
ι : Type u_5
l : Filter ι
inst✝² : l.NeBot
inst✝¹ : l.IsCountablyGenerated
x₀ : α
hf : LowerHemicontinuousAt f x₀
x : ι → α
hx : Tendsto x l (𝓝 x₀)
y₀ : β
hy₀ : y₀ ∈ f x₀
inst✝ : (𝓝 y₀).IsCountablyGenerated
⊢ ∃ u y, Tendsto u atTop l ∧ (∀ᶠ (k : ℕ) in atTop, y k ∈ f (x (u k))) ∧ Tendsto y atTop (𝓝 y₀)

Tactic:
obtain ⟨u, hu⟩ := Filter.exists_seq_tendsto l
  obtain ⟨y, hy_mem, hy_lim⟩ := hf.exists_seq_tendsto (hx.comp hu) hy₀
  exact ⟨u, y, hu, hy_mem, hy_lim⟩

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
⊢ ∀ ⦃x : Set (Set α)⦄,
    x ∈ {u | u.Finite ∧ ∀ U ∈ u, IsOpen[inst✝] U} →
      ∀ x_1 ∈ {s | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty},
        ∃
          v ∈
            (fun p ↦ {s | s ⊆ p.1 ∧ ∀ U ∈ p.2, (s ∩ U).Nonempty}) ''
              {p | IsOpen[inst✝] p.1 ∧ p.2.Finite ∧ p.2 ⊆ B ∧ ∀ U ∈ p.2, U ⊆ p.1},
          x_1 ∈ v ∧ v ⊆ {s | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty}

Tactic:
intro u ⟨hu₁, hu₂⟩ s ⟨hs₁, hs₂⟩

Goal state after:
case refine_2
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
u : Set (Set α)
hu₁ : u.Finite
hu₂ : ∀ U ∈ u, IsOpen[inst✝] U
s : Set α
hs₁ : s ⊆ ⋃₀ u
hs₂ : ∀ U ∈ u, (s ∩ U).Nonempty
⊢ ∃
    v ∈
      (fun p ↦ {s | s ⊆ p.1 ∧ ∀ U ∈ p.2, (s ∩ U).Nonempty}) ''
        {p | IsOpen[inst✝] p.1 ∧ p.2.Finite ∧ p.2 ⊆ B ∧ ∀ U ∈ p.2, U ⊆ p.1},
    s ∈ v ∧ v ⊆ {s | s ⊆ ⋃₀ u ∧ ∀ U ∈ u, (s ∩ U).Nonempty}
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝³ : NormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
k : 𝕜
h : ‖a‖ * ‖1‖ < ‖k‖
a✝ : Nontrivial A
hk : k ≠ 0
ku : Aˣ := (Units.map ↑↑ₐ) (Units.mk0 k hk)
⊢ IsUnit (k • 1 - a)

Tactic:
rw [← inv_inv ‖(1 : A)‖,
    mul_inv_lt_iff₀' (inv_pos.2 <| norm_pos_iff.2 (one_ne_zero : (1 : A) ≠ 0))] at h

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝³ : NormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
k : 𝕜
h : ‖a‖ < ‖1‖⁻¹ * ‖k‖
a✝ : Nontrivial A
hk : k ≠ 0
ku : Aˣ := (Units.map ↑↑ₐ) (Units.mk0 k hk)
⊢ IsUnit (k • 1 - a)
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
inst✝ : UniformSpace α
⊢ UniformContinuous closure[inst✝.toTopologicalSpace]

Tactic:
simp_rw [UniformContinuous, (𝓤 α).basis_sets.uniformity_hausdorff.tendsto_iff
    (𝓤 α).basis_sets.uniformity_hausdorff, Function.comp_id, mem_hausdorffEntourage]
  intro U hU
  obtain ⟨V : SetRel α α, hV, hVU⟩ := comp_mem_uniformity_sets hU
  refine ⟨V, hV, fun ⟨s, t⟩ ⟨hst, hts⟩ => ?_⟩
  simp only at *
  constructor
  · grw [closure_subset_preimage hV s, hst, ← subset_closure, ← hVU, SetRel.preimage_comp]
  · grw [closure_subset_image hV t, hts, ← subset_closure, ← hVU, SetRel.image_comp]

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

Goal state before:
pp: α : Type u_1
inst✝ : UniformSpace α
t : Set α
ht : TotallyBounded t
⊢ ∀ i ∈ 𝓤 α, ∃ t_1, t_1.Finite ∧ ∀ x ⊆ t, x ∈ ⋃ y ∈ t_1, {x | (x, y) ∈ hausdorffEntourage i}

Tactic:
Set.mem_iUnion

Goal state after:
α : Type u_1
inst✝ : UniformSpace α
t : Set α
ht : TotallyBounded t
⊢ ∀ i ∈ 𝓤 α, ∃ t_1, t_1.Finite ∧ ∀ x ⊆ t, ∃ i_1, ∃ (_ : i_1 ∈ t_1), x ∈ {x | (x, i_1) ∈ hausdorffEntourage i}
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
inst✝ : UniformSpace α
t : Set α
ht : TotallyBounded t
⊢ TotallyBounded (𝒫 t)

Tactic:
by
  simp_rw [(𝓤 α).basis_sets.uniformity_hausdorff.totallyBounded_iff, Function.comp_id,
    Set.powerset, Set.ofPred_subset, Set.mem_iUnion]
  intro (U : SetRel α α) hU
  obtain ⟨u, hu, ht⟩ := ht U hU
  refine ⟨u.powerset, hu.powerset, fun s hs => ⟨u ∩ U.image s, by grind, fun x hx => ?_,
    fun x ⟨_, hx⟩ => hx⟩⟩
  obtain ⟨y, hy, hxy⟩ := Set.mem_iUnion₂.mp (ht (hs hx))
  exact ⟨y, ⟨hy, ⟨x, hx, hxy⟩⟩, hxy⟩

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
z : 𝕜
hz : z ∈ spectrum 𝕜 a
this : NormedAlgebra ℚ A
hexpmul : exp a = exp (a - ↑ₐ z) * ↑ₐ (exp z)
b : A := ∑' (n : ℕ), (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ n
hb : Summable fun n ↦ (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ n
h₀ : ∑' (n : ℕ), (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ (n + 1) = (a - ↑ₐ z) * b
⊢ ∑' (n : ℕ), (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ (n + 1) = b * (a - ↑ₐ z)

Tactic:
simpa only [pow_succ, Algebra.smul_mul_assoc] using hb.tsum_mul_right (a - ↑ₐ z)

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedRing A
inst✝² : StarRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : IsometricContinuousFunctionalCalculus 𝕜 A p
a : A
f : C(↑(σ 𝕜 a), 𝕜)
ha : p a
⊢ ‖(cfcHom ⋯) f‖ = ‖f‖

Tactic:
refine isometry_cfcHom a |>.norm_map_of_map_zero (map_zero _) f

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

Goal state before:
pp: case inr
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedRing A
inst✝² : StarRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : IsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
hc : 0 ≤ c
h : ∀ x ∈ σ 𝕜 a, ‖f x‖ ≤ c
h✝ : Nontrivial A
⊢ ‖cfc f a‖ ≤ c

Tactic:
refine cfc_cases (‖·‖ ≤ c) a f (by simpa) fun hf ha ↦ ?_
    simp only [← cfc_apply f a, isLUB_le_iff (IsGreatest.norm_cfc f a hf ha |>.isLUB)]
    rintro - ⟨x, hx, rfl⟩
    exact h x hx

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

Goal state before:
pp: A : Type u_3
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
t : ℝ≥0
ht : spectralRadius ℝ a ≤ ↑t
this : spectrum ℝ a ⊆ Set.Icc (-↑t) ↑t
h : ∀ x ∈ spectrum ℝ a, 0 ≤ x
x : ℝ
hx : x ∈ {↑t} - spectrum ℝ a
⊢ ∃ y ∈ spectrum ℝ a, ↑t - y = x

Tactic:
simpa using hx

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

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a✝ : A
x✝ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A
this : IsAddTorsionFree A
ha : ∀ (x : ℝ), NormedSpace.exp (x • a✝) * NormedSpace.exp (-x • star a✝) ∈ unitary A
key :
  ∀ (x : ℝ),
    NormedSpace.exp (-x • a✝) * NormedSpace.exp (x • star a✝) =
      NormedSpace.exp (x • star a✝) * NormedSpace.exp (-x • a✝)
h_deriv :
  ∀ (a b c : A) (y : ℝ),
    deriv (fun x ↦ NormedSpace.exp (x • a) * c * NormedSpace.exp (x • b)) y =
      NormedSpace.exp (y • a) * (a * c + c * b) * NormedSpace.exp (y • b)
a b : A
| fun y ↦ deriv (fun x ↦ NormedSpace.exp (x • a) * NormedSpace.exp (x • b)) y

Tactic:
y,

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
a✝ : A
x✝ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A
this : IsAddTorsionFree A
ha : ∀ (x : ℝ), NormedSpace.exp (x • a✝) * NormedSpace.exp (-x • star a✝) ∈ unitary A
key :
  ∀ (x : ℝ),
    NormedSpace.exp (-x • a✝) * NormedSpace.exp (x • star a✝) =
      NormedSpace.exp (x • star a✝) * NormedSpace.exp (-x • a✝)
h_deriv :
  ∀ (a b c : A) (y : ℝ),
    deriv (fun x ↦ NormedSpace.exp (x • a) * c * NormedSpace.exp (x • b)) y =
      NormedSpace.exp (y • a) * (a * c + c * b) * NormedSpace.exp (y • b)
a b : A
y : ℝ
| deriv (fun x ↦ NormedSpace.exp (x • a) * NormedSpace.exp (x • b)) y
Module: Mathlib.Analysis.Normed.Operator.Completeness

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NontriviallyNormedField 𝕜₂
inst✝³ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
E' : Type u_6
inst✝² : SeminormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
inst✝ : RingHomIsometric σ₁₂
f : ℕ → E' →SL[σ₁₂] F
g : E' →SL[σ₁₂] F
hg : Tendsto (fun n x ↦ (f n) x) atTop (𝓝 ⇑g)
hf : CauchySeq f
b : ℕ → ℝ
hb₀ : ∀ (n : ℕ), 0 ≤ b n
hb_lim : Tendsto b atTop (𝓝 0)
hfb : ∀ (n m N : ℕ), N ≤ n → N ≤ m → ‖f n - f m‖ ≤ b N
n : ℕ
x : E'
this : Tendsto (fun m ↦ ‖(f n) x - (f m) x‖) atTop (𝓝 ‖(f n) x - g x‖)
m : ℕ
hm : n ≤ m
⊢ ‖(f n) x - (f m) x‖ ≤ b n * ‖x‖

Tactic:
exact (f n - f m).le_of_opNorm_le (hfb _ _ _ le_rfl hm) _

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor V P
𝕜 : Type u_5
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 V
inst✝ : Invertible 2
p₁ p₂ : P
⊢ dist p₁ (midpoint 𝕜 p₁ p₂) = dist p₂ (midpoint 𝕜 p₁ p₂)

Tactic:
dist_left_midpoint p₁ p₂,

Goal state after:
V : Type u_1
P : Type u_2
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor V P
𝕜 : Type u_5
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 V
inst✝ : Invertible 2
p₁ p₂ : P
⊢ ‖2‖⁻¹ * dist p₁ p₂ = dist p₂ (midpoint 𝕜 p₁ p₂)
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝² : SeminormedAddCommGroup V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
p q : P
⊢ dist ((Equiv.pointReflection p) q) p = dist p q

Tactic:
simp [dist_eq_norm_vsub V, Equiv.pointReflection_vsub_left (G := V)]

Goal state after:
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
h0 : 0 < 2⁻¹
h1234 : 2⁻¹ < 3 / 4
h1 : 3 / 4 < 1
x : X
x✝ : True
n : ℕ
ihn : ∀ (c : CU P), ∀ᶠ (x_1 : X) in 𝓝 x, dist (c.lim x_1) (c.lim x) ≤ (3 / 4) ^ n
c : CU P
hxl : x ∈ c.left.U
a✝ : X
hyl : a✝ ∈ c.left.U
hyd : dist (c.left.lim a✝) (c.left.lim x) ≤ (3 / 4) ^ n
⊢ dist (midpoint ℝ (c.left.lim a✝) 0) (midpoint ℝ (c.left.lim x) (c.right.lim x)) ≤ 3 / 4 * (3 / 4) ^ n

Tactic:
c.right.lim_of_mem_C _ (c.left_U_subset_right_C hxl)

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
h0 : 0 < 2⁻¹
h1234 : 2⁻¹ < 3 / 4
h1 : 3 / 4 < 1
x : X
x✝ : True
n : ℕ
ihn : ∀ (c : CU P), ∀ᶠ (x_1 : X) in 𝓝 x, dist (c.lim x_1) (c.lim x) ≤ (3 / 4) ^ n
c : CU P
hxl : x ∈ c.left.U
a✝ : X
hyl : a✝ ∈ c.left.U
hyd : dist (c.left.lim a✝) (c.left.lim x) ≤ (3 / 4) ^ n
⊢ dist (midpoint ℝ (c.left.lim a✝) 0) (midpoint ℝ (c.left.lim x) 0) ≤ 3 / 4 * (3 / 4) ^ n
Module: Mathlib.LinearAlgebra.AffineSpace.Ordered

Goal state before:
pp: k : Type u_1
E : Type u_2
inst✝⁸ : Field k
inst✝⁷ : LinearOrder k
inst✝⁶ : IsStrictOrderedRing 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
a b : E
r r' : k
h : r < r'
⊢ (lineMap a b) r ≤ (lineMap a b) r' ↔ a ≤ b

Tactic:
simp only [lineMap_apply_module]
  rw [← le_sub_iff_add_le, add_sub_assoc, ← sub_le_iff_le_add', ← sub_smul, ← sub_smul,
    sub_sub_sub_cancel_left, smul_le_smul_iff_of_pos_left (sub_pos.2 h)]

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

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
b : A
a : A := star b * b
a_def : a = star b * b
ha : IsSelfAdjoint a
c : A := b * a⁻
h_eq_negPart_a : -(star c * c) = a⁻ ^ 3
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x ↦ ContinuousMap.realToNNReal x
⊢ SpectrumRestricts (2 • (↑(ℜ c) * ↑(ℜ c) + ↑(ℑ c) * ↑(ℑ c)) - star c * c) ⇑ContinuousMap.realToNNReal

Tactic:
sub_eq_add_neg,

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
b : A
a : A := star b * b
a_def : a = star b * b
ha : IsSelfAdjoint a
c : A := b * a⁻
h_eq_negPart_a : -(star c * c) = a⁻ ^ 3
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x ↦ ContinuousMap.realToNNReal x
⊢ SpectrumRestricts (2 • (↑(ℜ c) * ↑(ℜ c) + ↑(ℑ c) * ↑(ℑ c)) + -(star c * c)) ⇑ContinuousMap.realToNNReal
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: case pos
A : Type u_1
inst✝¹³ : NonUnitalCStarAlgebra A
R : Type u_2
inst✝¹² : Semifield R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : Algebra R ℂ
inst✝³ : IsScalarTower R ℂ A
p : A → Prop
p' : Unitization ℂ A → Prop
inst✝² : NonUnitalContinuousFunctionalCalculus R A p
inst✝¹ : ContinuousFunctionalCalculus R (Unitization ℂ A) p'
inst✝ : ContinuousMapZero.UniqueHom R (Unitization ℂ A)
hp : ∀ {a : A}, p' ↑a ↔ p a
a : A
f : R → R
hf₀ : f 0 = 0
h : ContinuousOn f (σₙ R a) ∧ p a
⊢ ↑(cfcₙ f a) = cfc f ↑a

Tactic:
obtain ⟨hf, ha⟩ := h
    rw [← cfcₙ_eq_cfc (quasispectrum_inr_eq R ℂ a ▸ hf)]
    exact (inrNonUnitalStarAlgHom ℂ A).map_cfcₙ f a

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: ι : Type u_2
C : ι → Type u_3
inst✝¹⁴ : (i : ι) → PartialOrder (C i)
inst✝¹³ : (i : ι) → NonUnitalRing (C i)
inst✝¹² : (i : ι) → TopologicalSpace (C i)
inst✝¹¹ : (i : ι) → StarRing (C i)
inst✝¹⁰ : ∀ (i : ι), StarOrderedRing (C i)
inst✝⁹ : StarOrderedRing ((i : ι) → C i)
inst✝⁸ : (i : ι) → Module ℝ (C i)
inst✝⁷ : ∀ (i : ι), SMulCommClass ℝ (C i) (C i)
inst✝⁶ : ∀ (i : ι), IsScalarTower ℝ (C i) (C i)
inst✝⁵ : ∀ (i : ι), NonUnitalContinuousFunctionalCalculus ℝ (C i) IsSelfAdjoint
inst✝⁴ : NonUnitalContinuousFunctionalCalculus ℝ ((i : ι) → C i) IsSelfAdjoint
inst✝³ : ∀ (i : ι), IsSemitopologicalRing (C i)
inst✝² : ∀ (i : ι), T2Space (C i)
inst✝¹ : NonnegSpectrumClass ℝ ((i : ι) → C i)
inst✝ : ∀ (i : ι), NonnegSpectrumClass ℝ (C i)
c : (i : ι) → C i
hc : ∀ (i : ι), 0 ≤ c i
⊢ c ^ (1 / 2) = fun i ↦ c i ^ (1 / 2)

Tactic:
exact nnrpow_map_pi

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

Goal state before:
pp: case pos
X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R A p
l : Filter X
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f l (spectrum R a)
hF : ∀ᶠ (x : X) in l, ContinuousOn (F x) (spectrum R a)
hl : l.NeBot
hf : ContinuousOn f (spectrum R a)
ha : p a
s : Set X := {x | ContinuousOn (F x) (spectrum R a)}
⊢ Tendsto (fun x ↦ (cfcHom ha) { toFun := (spectrum R a).domRestrict (F ↑x), continuous_toFun := ⋯ })
    (comap Subtype.val l) (𝓝 (cfc f a))

Tactic:
cfc_apply ..

Goal state after:
case pos
X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R A p
l : Filter X
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f l (spectrum R a)
hF : ∀ᶠ (x : X) in l, ContinuousOn (F x) (spectrum R a)
hl : l.NeBot
hf : ContinuousOn f (spectrum R a)
ha : p a
s : Set X := {x | ContinuousOn (F x) (spectrum R a)}
⊢ Tendsto (fun x ↦ (cfcHom ha) { toFun := (spectrum R a).domRestrict (F ↑x), continuous_toFun := ⋯ })
    (comap Subtype.val l) (𝓝 ((cfcHom ha) { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ }))
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: A : Type u_1
inst✝⁷ : PartialOrder A
inst✝⁶ : Ring A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : StarOrderedRing A
inst✝² : Algebra ℝ A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
a : A
x y : ℝ
ha : IsUnit a
⊢ a ^ (x + y) = a ^ x * a ^ y

Tactic:
have ha' : 0 ∉ spectrum ℝ≥0 a := spectrum.zero_notMem _ ha

Goal state after:
A : Type u_1
inst✝⁷ : PartialOrder A
inst✝⁶ : Ring A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : StarOrderedRing A
inst✝² : Algebra ℝ A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
a : A
x y : ℝ
ha : IsUnit a
ha' : 0 ∉ spectrum ℝ≥0 a
⊢ a ^ (x + y) = a ^ x * a ^ y
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: A : Type u_1
inst✝⁷ : PartialOrder A
inst✝⁶ : Ring A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : StarOrderedRing A
inst✝² : Algebra ℝ A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
a : A
ha : IsStrictlyPositive a
⊢ Ring.inverse a = a ^ (-1)

Tactic:
obtain ⟨ax, hax⟩ := ha.isUnit

Goal state after:
A : Type u_1
inst✝⁷ : PartialOrder A
inst✝⁶ : Ring A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : StarOrderedRing A
inst✝² : Algebra ℝ A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
a : A
ha : IsStrictlyPositive a
ax : Aˣ
hax : ↑ax = a
⊢ Ring.inverse a = a ^ (-1)