mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.CStarAlgebra.GelfandDuality

Goal state before:
pp: A : Type u_1
inst✝ : NonUnitalCommCStarAlgebra A
a b : A
h : a * b = 0
⊢ ‖a + b‖ = max ‖a‖ ‖b‖

Tactic:
let f := gelfandStarTransform A⁺¹ ∘ inrNonUnitalAlgHom ℂ A

Goal state after:
A : Type u_1
inst✝ : NonUnitalCommCStarAlgebra A
a b : A
h : a * b = 0
f : A → C(↑(characterSpace ℂ (Unitization ℂ A)), ℂ) :=
  ⇑(gelfandStarTransform (Unitization ℂ A)) ∘ ⇑(inrNonUnitalAlgHom ℂ A)
⊢ ‖a + b‖ = max ‖a‖ ‖b‖
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: case neg.inr
A : Type u_1
inst✝⁸ : NonUnitalRing A
inst✝⁷ : Module ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹ : T2Space A
inst✝ : StarModule ℝ A
r : ℝ≥0
a : A
ha : ¬IsSelfAdjoint a
hr : r ≠ 0
⊢ (r • a)⁺ = r • a⁺

Tactic:
have := (not_iff_not.mpr <| (IsSelfAdjoint.all r).smul_iff hr.isUnit (x := a)) |>.mpr ha

Goal state after:
case neg.inr
A : Type u_1
inst✝⁸ : NonUnitalRing A
inst✝⁷ : Module ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹ : T2Space A
inst✝ : StarModule ℝ A
r : ℝ≥0
a : A
ha : ¬IsSelfAdjoint a
hr : r ≠ 0
this : ¬IsSelfAdjoint (r • a)
⊢ (r • a)⁺ = r • a⁺
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R1Space X
s t : Set X
hs : IsOpen[inst✝¹] s
hscp : IsCompact (closure[inst✝¹] sᶜᶜ)
ht : IsClosed[inst✝¹] t
hst : t ⊆ s
u v : Set X
huIsOpen : IsOpen[inst✝¹] u
hvIsOpen : IsOpen[inst✝¹] v
hscompl_subset_u : sᶜ ⊆ u
ht_subset_v : t ⊆ v
hDisjointuv : u ⊆ vᶜ
huvc : closure[inst✝¹] u ⊆ vᶜ
P : Set X → Set X → Prop := fun C x ↦ sᶜ ⊆ C
c u0 : Set X
cIsClosed : IsClosed[inst✝¹] c
Pc : P c u0
u0IsOpen : IsOpen[inst✝¹] u0
csubu0 : c ⊆ u0
u1 v1 : Set X
hu1 : IsOpen[inst✝¹] u1
hv1 : IsOpen[inst✝¹] v1
hcu1 : c ⊆ u1
hv1u : v1ᶜ ⊆ u0
hu1v1 : u1 ⊆ v1ᶜ
⊢ ∃ v, IsOpen[inst✝¹] v ∧ c ⊆ v ∧ closure[inst✝¹] v ⊆ u0 ∧ P c v ∧ P (closure[inst✝¹] v) u0

Tactic:
refine ⟨u1, hu1, hcu1, ?_, Pc, (Pc.trans hcu1).trans subset_closure⟩

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R1Space X
s t : Set X
hs : IsOpen[inst✝¹] s
hscp : IsCompact (closure[inst✝¹] sᶜᶜ)
ht : IsClosed[inst✝¹] t
hst : t ⊆ s
u v : Set X
huIsOpen : IsOpen[inst✝¹] u
hvIsOpen : IsOpen[inst✝¹] v
hscompl_subset_u : sᶜ ⊆ u
ht_subset_v : t ⊆ v
hDisjointuv : u ⊆ vᶜ
huvc : closure[inst✝¹] u ⊆ vᶜ
P : Set X → Set X → Prop := fun C x ↦ sᶜ ⊆ C
c u0 : Set X
cIsClosed : IsClosed[inst✝¹] c
Pc : P c u0
u0IsOpen : IsOpen[inst✝¹] u0
csubu0 : c ⊆ u0
u1 v1 : Set X
hu1 : IsOpen[inst✝¹] u1
hv1 : IsOpen[inst✝¹] v1
hcu1 : c ⊆ u1
hv1u : v1ᶜ ⊆ u0
hu1v1 : u1 ⊆ v1ᶜ
⊢ closure[inst✝¹] u1 ⊆ u0
Module: Mathlib.Topology.MetricSpace.UniformConvergence

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ : PseudoEMetricSpace γ
inst✝ : PseudoEMetricSpace β
f : α → γ → β
K : ℝ≥0
h : ∀ (c : α), LipschitzWith K (f c)
⊢ UniformContinuous (⇑ofFun ∘ Function.swap f)

Tactic:
rw [← lipschitzWith_ofFun_iff] at h

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ : PseudoEMetricSpace γ
inst✝ : PseudoEMetricSpace β
f : α → γ → β
K : ℝ≥0
h : LipschitzWith K fun x ↦ ofFun fun c ↦ f c x
⊢ UniformContinuous (⇑ofFun ∘ Function.swap f)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
x✝ : PartialOrder A := spectralOrder A
x y : A
h : x ≤ y
s : A
hs₁ : IsSelfAdjoint s
left✝ : QuasispectrumRestricts s ⇑ContinuousMap.realToNNReal
hs₂ : s * s = y - x
⊢ (fun s ↦ star s * s) s = s * s

Tactic:
by simp [hs₁.star_eq]

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

Goal state before:
pp: A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : Ring A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a : Aˣ
x : ℝ
ha' : 0 ≤ ↑a
h₁ : ContinuousOn (fun z ↦ z ^ x) (Inv.inv '' spectrum ℝ≥0 ↑a)
⊢ ↑a ^ (-x) = ↑a⁻¹ ^ x

Tactic:
by
    rw [← cfc_inv_id (R := ℝ≥0) a, rpow_def, rpow_def,
        ← cfc_comp' (fun z => z ^ x) (Inv.inv : ℝ≥0 → ℝ≥0) (a : A) h₁]
    refine cfc_congr fun _ _ => ?_
    simp [NNReal.rpow_neg, NNReal.inv_rpow]

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

Goal state before:
pp: A : Type u_1
inst✝⁹ : TopologicalSpace A
inst✝⁸ : Ring A
inst✝⁷ : StarRing A
inst✝⁶ : PartialOrder A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : IsTopologicalRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : T2Space A
inst✝ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
f g : ℝ≥0 → ℝ≥0
a : A
ha_spec : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
hf : ContinuousOn f (spectrum ℝ≥0 a)
hg : ContinuousOn g (spectrum ℝ≥0 a)
ha : 0 ≤ a
hf' : ContinuousOn (fun x ↦ ↑(f x.toNNReal)) (spectrum ℝ a)
hg' : ContinuousOn (fun x ↦ ↑(g x.toNNReal)) (spectrum ℝ a)
⊢ (∀ x ∈ spectrum ℝ a, ↑(f x.toNNReal) ≤ ↑(g x.toNNReal)) ↔ ∀ x ∈ spectrum ℝ≥0 a, f x ≤ g x

Tactic:
simp [NNReal.coe_le_coe, ← ha_spec.image]

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

Goal state before:
pp: A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : Ring A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a : A
ha : ¬IsStrictlyPositive a
⊢ ¬IsUnit (sqrt a)

Tactic:
CFC.isUnit_sqrt_iff_isStrictlyPositive

Goal state after:
A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : Ring A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a : A
ha : ¬IsStrictlyPositive a
⊢ ¬IsStrictlyPositive a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ → ℝ
s : Set A
hf : ConcaveOn ℝ (inr '' s) (cfc f)
⊢ ConcaveOn ℝ s (cfcₙ f)

Tactic:
by
  have : ConcaveOn ℝ s (- -cfcₙ f) := by
    rw [← cfcₙ_neg' f]
    refine (convexOn_cfcₙ_of_convexOn_cfc ?_).neg
    rw [cfc_neg']
    exact hf.neg
  simpa using this

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

Goal state before:
pp: A : Type u_1
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
ι : Type u_2
E : ι → Type u_3
inst✝⁵ : Fintype ι
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → Module ℂ (E i)
inst✝² : (i : ι) → SMul A (E i)
inst✝¹ : (i : ι) → CStarModule A (E i)
inst✝ : StarOrderedRing A
x : C⋆ᵐᵒᵈ(A, (i : ι) → E i)
x✝ : NormedAddCommGroup C⋆ᵐᵒᵈ(A, (i : ι) → E i) := normedAddCommGroup A
⊢ ‖(equiv A ((i : ι) → E i)) x‖ ≤ ‖x‖

Tactic:
rw [pi_norm_le_iff_of_nonneg (by positivity)]

Goal state after:
A : Type u_1
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
ι : Type u_2
E : ι → Type u_3
inst✝⁵ : Fintype ι
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → Module ℂ (E i)
inst✝² : (i : ι) → SMul A (E i)
inst✝¹ : (i : ι) → CStarModule A (E i)
inst✝ : StarOrderedRing A
x : C⋆ᵐᵒᵈ(A, (i : ι) → E i)
x✝ : NormedAddCommGroup C⋆ᵐᵒᵈ(A, (i : ι) → E i) := normedAddCommGroup A
⊢ ∀ (i : ι), ‖(equiv A ((i : ι) → E i)) x i‖ ≤ ‖x‖
Module: Mathlib.Analysis.Matrix.Normed

Goal state before:
pp: n : Type u_4
α : Type u_5
inst✝² : Fintype n
inst✝¹ : SeminormedAddCommGroup α
inst✝ : DecidableEq n
v : n → α
s : Finset (n × n) := Finset.map { toFun := fun i ↦ (i, i), inj' := ⋯ } Finset.univ
⊢ (∑ x, ‖diagonal v ({ toFun := fun i ↦ (i, i), inj' := ⋯ } x).1 ({ toFun := fun i ↦ (i, i), inj' := ⋯ } x).2‖₊ ^ 2) ^
      (1 / 2) =
    (∑ x, ‖v x‖₊ ^ 2) ^ (1 / 2)

Tactic:
dsimp

Goal state after:
n : Type u_4
α : Type u_5
inst✝² : Fintype n
inst✝¹ : SeminormedAddCommGroup α
inst✝ : DecidableEq n
v : n → α
s : Finset (n × n) := Finset.map { toFun := fun i ↦ (i, i), inj' := ⋯ } Finset.univ
⊢ (∑ x, ‖diagonal v x x‖₊ ^ 2) ^ (1 / 2) = (∑ x, ‖v x‖₊ ^ 2) ^ (1 / 2)
Module: Mathlib.Analysis.Matrix.Normed

Goal state before:
pp: n : Type u_4
α : Type u_5
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : SeminormedAddCommGroup α
inst✝ : One α
⊢ ↑(Fintype.card n) ^ (1 / OfNat.ofNat 2).toReal * ‖1‖₊ = NNReal.sqrt ↑(Fintype.card n) * ‖1‖₊

Tactic:
simp [NNReal.sqrt_eq_rpow]

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

Goal state before:
pp: l : Type u_2
m : Type u_3
n : Type u_4
α : Type u_5
inst✝³ : Fintype l
inst✝² : Fintype m
inst✝¹ : Fintype n
inst✝ : RCLike α
A : Matrix l m α
B : Matrix m n α
⊢ (∑ i, ∑ j, ‖∑ j_1, A i j_1 * B j_1 j‖₊ ^ 2) ^ (1 / 2) ≤ ((∑ i, ∑ j, ‖A i j‖₊ ^ 2) * ∑ y, ∑ x, ‖B x y‖₊ ^ 2) ^ (1 / 2)

Tactic:
Finset.sum_mul_sum

Goal state after:
l : Type u_2
m : Type u_3
n : Type u_4
α : Type u_5
inst✝³ : Fintype l
inst✝² : Fintype m
inst✝¹ : Fintype n
inst✝ : RCLike α
A : Matrix l m α
B : Matrix m n α
⊢ (∑ i, ∑ j, ‖∑ j_1, A i j_1 * B j_1 j‖₊ ^ 2) ^ (1 / 2) ≤ (∑ i, ∑ j, (∑ j, ‖A i j‖₊ ^ 2) * ∑ x, ‖B x j‖₊ ^ 2) ^ (1 / 2)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Commute

Goal state before:
pp: case pos
A : Type u_2
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra ℝ A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁴ : IsTopologicalRing A
inst✝³ : T2Space A
inst✝² : PartialOrder A
inst✝¹ : NonnegSpectrumClass ℝ A
inst✝ : StarOrderedRing A
a b : A
hb : Commute a b
f : ℝ≥0 → ℝ≥0
ha : 0 ≤ a
⊢ Commute (cfc f a) b

Tactic:
rw [cfc_nnreal_eq_real ..]
    exact hb.cfc_real _

Goal state after:
Module: Mathlib.MeasureTheory.SpecificCodomains.ContinuousMapZero

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝⁴ : MeasurableSpace X
μ : Measure X
inst✝³ : TopologicalSpace Y
E : Type u_3
inst✝² : NormedAddCommGroup E
s : Set Y
inst✝¹ : CompactSpace ↑s
inst✝ : Zero ↑s
f : X → Y → E
g : C(↑s, E)₀
f_ae_contOn : ∀ᵐ (x : X) ∂μ, ContinuousOn (f x) s
f_ae_zero : ∀ᵐ (x : X) ∂μ, f x ↑0 = 0
bound : X → ℝ
bound_int : HasFiniteIntegral bound μ
bound_ge : ∀ᵐ (x : X) ∂μ, ∀ y ∈ s, ‖f x y‖ ≤ bound x
⊢ HasFiniteIntegral (fun x ↦ mkD (s.domRestrict (f x)) g) μ

Tactic:
refine hasFiniteIntegral_mkD_of_bound _ _ ?_ f_ae_zero bound bound_int ?_

Goal state after:
case refine_1
X : Type u_1
Y : Type u_2
inst✝⁴ : MeasurableSpace X
μ : Measure X
inst✝³ : TopologicalSpace Y
E : Type u_3
inst✝² : NormedAddCommGroup E
s : Set Y
inst✝¹ : CompactSpace ↑s
inst✝ : Zero ↑s
f : X → Y → E
g : C(↑s, E)₀
f_ae_contOn : ∀ᵐ (x : X) ∂μ, ContinuousOn (f x) s
f_ae_zero : ∀ᵐ (x : X) ∂μ, f x ↑0 = 0
bound : X → ℝ
bound_int : HasFiniteIntegral bound μ
bound_ge : ∀ᵐ (x : X) ∂μ, ∀ y ∈ s, ‖f x y‖ ≤ bound x
⊢ ∀ᵐ (x : X) ∂μ, Continuous (s.domRestrict (f x))

case refine_2
X : Type u_1
Y : Type u_2
inst✝⁴ : MeasurableSpace X
μ : Measure X
inst✝³ : TopologicalSpace Y
E : Type u_3
inst✝² : NormedAddCommGroup E
s : Set Y
inst✝¹ : CompactSpace ↑s
inst✝ : Zero ↑s
f : X → Y → E
g : C(↑s, E)₀
f_ae_contOn : ∀ᵐ (x : X) ∂μ, ContinuousOn (f x) s
f_ae_zero : ∀ᵐ (x : X) ∂μ, f x ↑0 = 0
bound : X → ℝ
bound_int : HasFiniteIntegral bound μ
bound_ge : ∀ᵐ (x : X) ∂μ, ∀ y ∈ s, ‖f x y‖ ≤ bound x
⊢ ∀ᵐ (x : X) ∂μ, ∀ (y : ↑s), ‖s.domRestrict (f x) y‖ ≤ bound x
Module: Mathlib.Analysis.CStarAlgebra.CStarMatrix

Goal state before:
pp: case inl
m : Type u_1
n : Type u_2
A : Type u_5
inst✝⁴ : Fintype m
inst✝³ : NonUnitalCStarAlgebra A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Fintype n
M : CStarMatrix m n A
C : ℝ≥0
h : ∀ (v : C⋆ᵐᵒᵈ(A, m → A)) (w : C⋆ᵐᵒᵈ(A, n → A)), ‖⟪w, (toCLM M) v⟫_A‖ ≤ ↑C * ‖v‖ * ‖w‖
v : C⋆ᵐᵒᵈ(A, m → A)
h₀ : 0 = ‖(toCLM M) v‖
⊢ ‖(toCLM M) v‖ ≤ ↑C * ‖v‖

Tactic:
rw [← h₀]
    positivity

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

Goal state before:
pp: A : Type u_1
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra ℂ A
inst✝¹ : StarModule ℂ A
inst✝ : ContinuousFunctionalCalculus ℂ A IsStarNormal
a : A
hp : IsStarNormal a
this : cfc (fun z ↦ ↑z.re) a + cfc (fun z ↦ I * ↑z.im) a = ↑(ℜ a) + I • ↑(ℑ a)
⊢ cfc (fun x ↦ ↑x.im) a = ↑(ℑ a)

Tactic:
cfc_const_mul ..,

Goal state after:
A : Type u_1
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra ℂ A
inst✝¹ : StarModule ℂ A
inst✝ : ContinuousFunctionalCalculus ℂ A IsStarNormal
a : A
hp : IsStarNormal a
this : cfc (fun z ↦ ↑z.re) a + I • cfc (fun z ↦ ↑z.im) a = ↑(ℜ a) + I • ↑(ℑ a)
⊢ cfc (fun x ↦ ↑x.im) a = ↑(ℑ a)
Module: Mathlib.Analysis.InnerProductSpace.Dual

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_3
x y : E
b : Basis ι 𝕜 E
h : ∀ (i : ι), ⟪b i, x⟫ = ⟪b i, y⟫
i : ι
⊢ ⟪x, b i⟫ = ⟪y, b i⟫

Tactic:
rw [← inner_conj_symm]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_3
x y : E
b : Basis ι 𝕜 E
h : ∀ (i : ι), ⟪b i, x⟫ = ⟪b i, y⟫
i : ι
⊢ (starRingEnd 𝕜) ⟪b i, x⟫ = ⟪y, b i⟫
Module: Mathlib.Analysis.InnerProductSpace.Dual

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_3
x y : E
b : Basis ι 𝕜 E
h : ∀ (i : ι), ⟪b i, x⟫ = ⟪b i, y⟫
i : ι
| ⟪y, b i⟫

Tactic:
rw [← inner_conj_symm]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_3
x y : E
b : Basis ι 𝕜 E
h : ∀ (i : ι), ⟪b i, x⟫ = ⟪b i, y⟫
i : ι
| (starRingEnd 𝕜) ⟪b i, y⟫
Module: Mathlib.Analysis.InnerProductSpace.Dual

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_3
x y : E
b : Basis ι 𝕜 E
h : ∀ (i : ι), ⟪x, b i⟫ = ⟪y, b i⟫
i : ι
⊢ ⟪b i, x⟫ = ⟪b i, y⟫

Tactic:
rw [← inner_conj_symm]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_3
x y : E
b : Basis ι 𝕜 E
h : ∀ (i : ι), ⟪x, b i⟫ = ⟪y, b i⟫
i : ι
⊢ (starRingEnd 𝕜) ⟪x, b i⟫ = ⟪b i, y⟫
Module: Mathlib.Analysis.InnerProductSpace.Dual

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_3
x y : E
b : Basis ι 𝕜 E
h : ∀ (i : ι), ⟪x, b i⟫ = ⟪y, b i⟫
i : ι
| ⟪b i, y⟫

Tactic:
rw [← inner_conj_symm]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_3
x y : E
b : Basis ι 𝕜 E
h : ∀ (i : ι), ⟪x, b i⟫ = ⟪y, b i⟫
i : ι
| (starRingEnd 𝕜) ⟪y, b i⟫
Module: Mathlib.Analysis.InnerProductSpace.Completion

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
⊢ Continuous (uncurry (inner 𝕜))

Tactic:
let inner' : E →+ E →+ 𝕜 :=
    { toFun := fun x => (innerₛₗ 𝕜 x).toAddMonoidHom
      map_zero' := by ext x; exact inner_zero_left _
      map_add' := fun x y => by ext z; exact inner_add_left _ _ _ }
  have : Continuous fun p : E × E => inner' p.1 p.2 := continuous_inner
  rw [Completion.toInner, inner, uncurry_curry _]
  change
    Continuous
      (((isDenseInducing_toCompl E).prodMap (isDenseInducing_toCompl E)).extend fun p : E × E =>
        inner' p.1 p.2)
  exact (isDenseInducing_toCompl E).extend_Z_bilin (isDenseInducing_toCompl E) this

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

Goal state before:
pp: α : Type u_1
β : Type u_2
R₁ : Type u_3
R₂ : Type u_4
inst✝¹³ : UniformSpace α
inst✝¹² : AddCommGroup α
inst✝¹¹ : IsUniformAddGroup α
inst✝¹⁰ : Semiring R₁
inst✝⁹ : Module R₁ α
inst✝⁸ : UniformContinuousConstSMul R₁ α
inst✝⁷ : Semiring R₂
inst✝⁶ : UniformSpace β
inst✝⁵ : AddCommGroup β
inst✝⁴ : IsUniformAddGroup β
inst✝³ : Module R₂ β
inst✝² : UniformContinuousConstSMul R₂ β
σ : R₁ →+* R₂
inst✝¹ : T0Space β
inst✝ : CompleteSpace β
f : α →SL[σ] β
c : R₁
a : Completion α
⊢ Continuous fun a ↦ σ c • (↑((↑f).toAddMonoidHom.extension ⋯)).toFun a

Tactic:
dsimp

Goal state after:
α : Type u_1
β : Type u_2
R₁ : Type u_3
R₂ : Type u_4
inst✝¹³ : UniformSpace α
inst✝¹² : AddCommGroup α
inst✝¹¹ : IsUniformAddGroup α
inst✝¹⁰ : Semiring R₁
inst✝⁹ : Module R₁ α
inst✝⁸ : UniformContinuousConstSMul R₁ α
inst✝⁷ : Semiring R₂
inst✝⁶ : UniformSpace β
inst✝⁵ : AddCommGroup β
inst✝⁴ : IsUniformAddGroup β
inst✝³ : Module R₂ β
inst✝² : UniformContinuousConstSMul R₂ β
σ : R₁ →+* R₂
inst✝¹ : T0Space β
inst✝ : CompleteSpace β
f : α →SL[σ] β
c : R₁
a : Completion α
⊢ Continuous fun a ↦ σ c • ((↑f).toAddMonoidHom.extension ⋯) a
Module: Mathlib.Analysis.CStarAlgebra.Extreme

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
e : A
x✝ :
  (0 ≤ e ∧ ‖e‖ ≤ 1) ∧ ∀ (x₁ : A), 0 ≤ x₁ → ‖x₁‖ ≤ 1 → ∀ (x₂ : A), 0 ≤ x₂ → ‖x₂‖ ≤ 1 → e ∈ openSegment ℝ x₁ x₂ → x₁ = e
h1 : 0 ≤ e
h2 : ‖e‖ ≤ 1
h3 : ∀ (x₁ : A), 0 ≤ x₁ → ‖x₁‖ ≤ 1 → ∀ (x₂ : A), 0 ≤ x₂ → ‖x₂‖ ≤ 1 → e ∈ openSegment ℝ x₁ x₂ → x₁ = e
this : ↑e ≤ 1
⊢ 1 ≤ 2

Tactic:
one_le_two

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
e : A
x✝ :
  (0 ≤ e ∧ ‖e‖ ≤ 1) ∧ ∀ (x₁ : A), 0 ≤ x₁ → ‖x₁‖ ≤ 1 → ∀ (x₂ : A), 0 ≤ x₂ → ‖x₂‖ ≤ 1 → e ∈ openSegment ℝ x₁ x₂ → x₁ = e
h1 : 0 ≤ e
h2 : ‖e‖ ≤ 1
h3 : ∀ (x₁ : A), 0 ≤ x₁ → ‖x₁‖ ≤ 1 → ∀ (x₂ : A), 0 ≤ x₂ → ‖x₂‖ ≤ 1 → e ∈ openSegment ℝ x₁ x₂ → x₁ = e
this : ↑e ≤ 1
⊢ 2 ≤ 2
Module: Mathlib.Topology.OpenPartialHomeomorph.Constructions

Goal state before:
pp: X✝ : Type u_1
X'✝ : Type u_2
Y : Type u_3
Y' : Type u_4
Z✝ : Type u_5
Z' : Type u_6
inst✝⁹ : TopologicalSpace X✝
inst✝⁸ : TopologicalSpace X'✝
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : TopologicalSpace Y'
inst✝⁵ : TopologicalSpace Z✝
inst✝⁴ : TopologicalSpace Z'
e✝ : OpenPartialHomeomorph X✝ Y
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
⊢ ∀ ⦃x : X'⦄, x ∈ f '' e.source → (f ∘ e.invFun) (extend f (↑e) (fun x ↦ Classical.arbitrary Z) x) = x

Tactic:
intro x ⟨x₀, hx₀, hxx₀⟩

Goal state after:
X✝ : Type u_1
X'✝ : Type u_2
Y : Type u_3
Y' : Type u_4
Z✝ : Type u_5
Z' : Type u_6
inst✝⁹ : TopologicalSpace X✝
inst✝⁸ : TopologicalSpace X'✝
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : TopologicalSpace Y'
inst✝⁵ : TopologicalSpace Z✝
inst✝⁴ : TopologicalSpace Z'
e✝ : OpenPartialHomeomorph X✝ Y
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
x : X'
x₀ : X
hx₀ : x₀ ∈ e.source
hxx₀ : f x₀ = x
⊢ (f ∘ e.invFun) (extend f (↑e) (fun x ↦ Classical.arbitrary Z) x) = x
Module: Mathlib.Topology.OpenPartialHomeomorph.Constructions

Goal state before:
pp: X✝ : Type u_1
X'✝ : Type u_2
Y : Type u_3
Y' : Type u_4
Z✝ : Type u_5
Z' : Type u_6
inst✝⁹ : TopologicalSpace X✝
inst✝⁸ : TopologicalSpace X'✝
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : TopologicalSpace Y'
inst✝⁵ : TopologicalSpace Z✝
inst✝⁴ : TopologicalSpace Z'
e✝ : OpenPartialHomeomorph X✝ Y
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
h✝ : Nonempty X
F : X' → Z := extend f ↑e fun x ↦ Classical.arbitrary Z
F_eq : F = extend f ↑e fun x ↦ Classical.arbitrary Z
⊢ EqOn F (↑e ∘ ↑(IsOpenEmbedding.toOpenPartialHomeomorph f hf).symm) (f '' e.source)

Tactic:
intro x ⟨x₀, hx₀, hxx₀⟩

Goal state after:
X✝ : Type u_1
X'✝ : Type u_2
Y : Type u_3
Y' : Type u_4
Z✝ : Type u_5
Z' : Type u_6
inst✝⁹ : TopologicalSpace X✝
inst✝⁸ : TopologicalSpace X'✝
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : TopologicalSpace Y'
inst✝⁵ : TopologicalSpace Z✝
inst✝⁴ : TopologicalSpace Z'
e✝ : OpenPartialHomeomorph X✝ Y
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
h✝ : Nonempty X
F : X' → Z := extend f ↑e fun x ↦ Classical.arbitrary Z
F_eq : F = extend f ↑e fun x ↦ Classical.arbitrary Z
x : X'
x₀ : X
hx₀ : x₀ ∈ e.source
hxx₀ : f x₀ = x
⊢ F x = (↑e ∘ ↑(IsOpenEmbedding.toOpenPartialHomeomorph f hf).symm) x
Module: Mathlib.Topology.OpenPartialHomeomorph.Constructions

Goal state before:
pp: X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
x : X
⊢ ↑(e.lift_openEmbedding hf) (f x) = ↑e x

Tactic:
e.lift_openEmbedding_toFun

Goal state after:
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
x : X
⊢ extend f (↑e) (fun x ↦ Classical.arbitrary Z) (f x) = ↑e x
Module: Mathlib.Topology.FiberBundle.Basic

Goal state before:
pp: case a
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
Z : FiberBundleCore ι B F
i j : ι
x : B
v : F
hx : x ∈ Z.baseSet i ∧ x ∈ Z.baseSet j
⊢ x ∈ Z.baseSet i

pp: case a
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
Z : FiberBundleCore ι B F
i j : ι
x : B
v : F
hx : x ∈ Z.baseSet i ∧ x ∈ Z.baseSet j
⊢ x ∈ Z.baseSet i ∩ Z.baseSet j ∩ Z.baseSet i

Tactic:
exacts [hx.1, ⟨⟨hx.1, hx.2⟩, hx.1⟩]

Goal state after:
Module: Mathlib.Topology.FiberBundle.Trivialization

Goal state before:
pp: B : Type u_1
F : Type u_2
E : B → Type u_3
inst✝³ : TopologicalSpace B
inst✝² : TopologicalSpace F
inst✝¹ : TopologicalSpace (TotalSpace F E)
inst✝ : ∀ (x : B), Nonempty (E x)
e : Trivialization F TotalSpace.proj
this : ∀ z ∈ e.baseSet ×ˢ univ, { proj := z.1, snd := e.symm z.1 z.2 } = ↑e.symm z
⊢ ContinuousOn (↑e.symm) e.target

Tactic:
exact e.toOpenPartialHomeomorph.continuousOn_symm

Goal state after:
Module: Mathlib.Topology.Covering.Quotient

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace X
f : E → X
G : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G E
hf : IsQuotientMap f
inst✝² : ContinuousConstSMul G E
hfG : ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂
inst✝¹ : TopologicalSpace G
inst✝ : DiscreteTopology G
U : Set E
open_U : IsOpen[inst✝⁶] U
disjoint : ∀ (g : G), ((fun x ↦ g • x) '' U ∩ U).Nonempty → g = 1
pGE : ∀ (g : G) (e : E), f (g • e) = f e
preim_im : f ⁻¹' f '' U = ⋃ g, (fun x ↦ g • x) ⁻¹' U
⊢ Trivialization G f

Tactic:
have : Nonempty (X → E) := ⟨Function.surjInv hf.surjective⟩

Goal state after:
E : Type u_1
X : Type u_2
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace X
f : E → X
G : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G E
hf : IsQuotientMap f
inst✝² : ContinuousConstSMul G E
hfG : ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂
inst✝¹ : TopologicalSpace G
inst✝ : DiscreteTopology G
U : Set E
open_U : IsOpen[inst✝⁶] U
disjoint : ∀ (g : G), ((fun x ↦ g • x) '' U ∩ U).Nonempty → g = 1
pGE : ∀ (g : G) (e : E), f (g • e) = f e
preim_im : f ⁻¹' f '' U = ⋃ g, (fun x ↦ g • x) ⁻¹' U
this : Nonempty (X → E)
⊢ Trivialization G f