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

Goal state before:
pp: case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : ↑a ≤ ↑b
ha : 0 ≤ ↑a
hb : 0 ≤ ↑b
c : Unitization ℂ A
hc : 0 ≤ c
x✝ : ℝ≥0
⊢ x✝ ∈ σ ℝ≥0 ↑?a → x✝ ≠ 0

Tactic:
_

Goal state after:
case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : ↑a ≤ ↑b
ha : 0 ≤ ↑a
hb : 0 ≤ ↑b
c : Unitization ℂ A
hc : 0 ≤ c
x✝ : ℝ≥0
a✝ : x✝ ∈ σ ℝ≥0 ↑?a
⊢ x✝ ≠ 0
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : ↑a ≤ ↑b
ha : 0 ≤ ↑a
hb : 0 ≤ ↑b
c : Unitization ℂ A
hc : 0 ≤ c
x✝ : ℝ≥0
⊢ x✝ ≠ 0

Tactic:
_

Goal state after:
case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : ↑a ≤ ↑b
ha : 0 ≤ ↑a
hb : 0 ≤ ↑b
c : Unitization ℂ A
hc : 0 ≤ c
x✝ : ℝ≥0
a✝ : x✝ = 0
⊢ False
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : ↑a ≤ ↑b
ha : 0 ≤ ↑a
hb : 0 ≤ ↑b
c : Unitization ℂ A
hc : 0 ≤ c
x✝ : ℝ≥0
⊢ x✝ ∈ σ ℝ≥0 c → 1 + x✝ ≠ 0

Tactic:
_

Goal state after:
case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : ↑a ≤ ↑b
ha : 0 ≤ ↑a
hb : 0 ≤ ↑b
c : Unitization ℂ A
hc : 0 ≤ c
x✝ : ℝ≥0
a✝ : x✝ ∈ σ ℝ≥0 c
⊢ 1 + x✝ ≠ 0
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : Nontrivial R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : IsScalarTower R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
l : Filter X
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f l (quasispectrum R a)
hF : ∀ᶠ (x : X) in l, ContinuousOn (F x) (quasispectrum R a)
hF0 : ∀ᶠ (x : X) in l, F x 0 = 0
hl : l.NeBot
hf : ContinuousOn f (quasispectrum R a)
hf0 : f 0 = 0
ha : p a
s : Set X := {x | ContinuousOn (F x) (quasispectrum R a) ∧ F x 0 = 0}
hs : s ∈ l
| Tendsto ((fun x ↦ cfcₙ (F x) a) ∘ Subtype.val) (comap Subtype.val l) (𝓝 (cfcₙ f a))

Tactic:
1,

Goal state after:
X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : Nontrivial R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : IsScalarTower R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
l : Filter X
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f l (quasispectrum R a)
hF : ∀ᶠ (x : X) in l, ContinuousOn (F x) (quasispectrum R a)
hF0 : ∀ᶠ (x : X) in l, F x 0 = 0
hl : l.NeBot
hf : ContinuousOn f (quasispectrum R a)
hf0 : f 0 = 0
ha : p a
s : Set X := {x | ContinuousOn (F x) (quasispectrum R a) ∧ F x 0 = 0}
hs : s ∈ l
| (fun x ↦ cfcₙ (F x) a) ∘ Subtype.val
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ≥0 → ℝ≥0 := fun x ↦ 1 - (1 + x)⁻¹
g : ℝ≥0 → ℝ≥0 := fun x ↦ x * (1 - x)⁻¹
a b : A
ha₁ : 0 ≤ a
ha₂ : ‖a‖ < 1
x✝ : ℝ≥0
⊢ x✝ ∈ σ ℝ≥0 ↑?a → x✝ ≠ 0

Tactic:
_

Goal state after:
case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ≥0 → ℝ≥0 := fun x ↦ 1 - (1 + x)⁻¹
g : ℝ≥0 → ℝ≥0 := fun x ↦ x * (1 - x)⁻¹
a b : A
ha₁ : 0 ≤ a
ha₂ : ‖a‖ < 1
x✝ : ℝ≥0
a✝ : x✝ ∈ σ ℝ≥0 ↑?a
⊢ x✝ ≠ 0
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ≥0 → ℝ≥0 := fun x ↦ 1 - (1 + x)⁻¹
g : ℝ≥0 → ℝ≥0 := fun x ↦ x * (1 - x)⁻¹
a b : A
ha₁ : 0 ≤ a
ha₂ : ‖a‖ < 1
x✝ : ℝ≥0
⊢ x✝ ≠ 0

Tactic:
_

Goal state after:
case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ≥0 → ℝ≥0 := fun x ↦ 1 - (1 + x)⁻¹
g : ℝ≥0 → ℝ≥0 := fun x ↦ x * (1 - x)⁻¹
a b : A
ha₁ : 0 ≤ a
ha₂ : ‖a‖ < 1
x✝ : ℝ≥0
a✝ : x✝ = 0
⊢ False
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ≥0 → ℝ≥0 := fun x ↦ 1 - (1 + x)⁻¹
g : ℝ≥0 → ℝ≥0 := fun x ↦ x * (1 - x)⁻¹
a b : A
ha₁ : 0 ≤ a
ha₂ : ‖a‖ < 1
x✝ : ℝ≥0
⊢ x✝ ∈ g '' σₙ ℝ≥0 a → 1 + x✝ ≠ 0

Tactic:
_

Goal state after:
case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ≥0 → ℝ≥0 := fun x ↦ 1 - (1 + x)⁻¹
g : ℝ≥0 → ℝ≥0 := fun x ↦ x * (1 - x)⁻¹
a b : A
ha₁ : 0 ≤ a
ha₂ : ‖a‖ < 1
x✝ : ℝ≥0
a✝ : x✝ ∈ g '' σₙ ℝ≥0 a
⊢ 1 + x✝ ≠ 0
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: case pos
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : Nontrivial R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : MetricSpace A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : IsScalarTower R A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus R A p
a : A
ha : p a
f : R →ᵤ[{quasispectrum R a}] R
hf : ContinuousOn ((toFun {quasispectrum R a}) f) (quasispectrum R a)
hf0 : f 0 = 0
g : R →ᵤ[{quasispectrum R a}] R
hg : ContinuousOn ((toFun {quasispectrum R a}) g) (quasispectrum R a)
hg0 : g 0 = 0
⊢ edist
      ((cfcₙHom ha)
        { toFun := (quasispectrum R a).domRestrict ((toFun {quasispectrum R a}) f), continuous_toFun := ⋯,
          map_zero' := hf0 })
      ((cfcₙHom ha)
        { toFun := (quasispectrum R a).domRestrict ((toFun {quasispectrum R a}) g), continuous_toFun := ⋯,
          map_zero' := hg0 }) ≤
    ↑1 * edist f g

Tactic:
isometry_cfcₙHom (R := R) a ha |>.edist_eq

Goal state after:
case pos
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : Nontrivial R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : MetricSpace A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : IsScalarTower R A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus R A p
a : A
ha : p a
f : R →ᵤ[{quasispectrum R a}] R
hf : ContinuousOn ((toFun {quasispectrum R a}) f) (quasispectrum R a)
hf0 : f 0 = 0
g : R →ᵤ[{quasispectrum R a}] R
hg : ContinuousOn ((toFun {quasispectrum R a}) g) (quasispectrum R a)
hg0 : g 0 = 0
⊢ edist
      { toFun := (quasispectrum R a).domRestrict ((toFun {quasispectrum R a}) f), continuous_toFun := ⋯,
        map_zero' := hf0 }
      { toFun := (quasispectrum R a).domRestrict ((toFun {quasispectrum R a}) g), continuous_toFun := ⋯,
        map_zero' := hg0 } ≤
    ↑1 * edist f g
Module: Mathlib.Topology.UniformSpace.Matrix

Goal state before:
pp: m : Type u_1
n : Type u_2
𝕜 : Type u_3
inst✝ : UniformSpace 𝕜
⊢ 𝓤 (Matrix m n 𝕜) = ⨅ i, ⨅ j, Filter.comap (fun a ↦ (a.1 i j, a.2 i j)) (𝓤 𝕜)

Tactic:
erw [Pi.uniformity]

Goal state after:
m : Type u_1
n : Type u_2
𝕜 : Type u_3
inst✝ : UniformSpace 𝕜
⊢ ⨅ i, Filter.comap (fun a ↦ (a.1 i, a.2 i)) (𝓤 (n → 𝕜)) = ⨅ i, ⨅ j, Filter.comap (fun a ↦ (a.1 i j, a.2 i j)) (𝓤 𝕜)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: case mul
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : ContinuousStar A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
inst✝ : TopologicalSpace X
s : Set 𝕜
hs : IsCompact s
hs0 : Fact (0 ∈ s)
a : X → A
ha_cont : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a
ha : ∀ (x : X), quasispectrum 𝕜 (a x) ⊆ s
ha' : ∀ (x : X), p (a x)
this : CompactSpace ↑s
f g : C(↑s, 𝕜)₀
hf : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦ (cfcₙHomSuperset ⋯ ⋯) f
hg : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦ (cfcₙHomSuperset ⋯ ⋯) g
⊢ Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦ (cfcₙHomSuperset ⋯ ⋯) (f * g)

Tactic:
| mul f g hf hg =>

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

Goal state before:
pp: case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
m✝¹ m✝ : A
ε : ℝ≥0
hε : 0 < ε
x✝¹ : A
x m : Unitization ℂ A
hm₁ : 0 ≤ m
hm₂ : ‖m‖ < 1
hx₂ : ‖x‖ ≤ 1
hx₀ : 0 ≤ x
g : ℝ≥0 → ℝ≥0 := fun y ↦ 1 - (1 + y)⁻¹
hx₁ : cfc g (ε⁻¹ ^ 2 • m) ≤ x
x✝ : ℝ≥0
⊢ x✝ ∈ σ ℝ≥0 ↑?a → x✝ ≠ 0

Tactic:
_

Goal state after:
case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
m✝¹ m✝ : A
ε : ℝ≥0
hε : 0 < ε
x✝¹ : A
x m : Unitization ℂ A
hm₁ : 0 ≤ m
hm₂ : ‖m‖ < 1
hx₂ : ‖x‖ ≤ 1
hx₀ : 0 ≤ x
g : ℝ≥0 → ℝ≥0 := fun y ↦ 1 - (1 + y)⁻¹
hx₁ : cfc g (ε⁻¹ ^ 2 • m) ≤ x
x✝ : ℝ≥0
a✝ : x✝ ∈ σ ℝ≥0 ↑?a
⊢ x✝ ≠ 0
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
m✝¹ m✝ : A
ε : ℝ≥0
hε : 0 < ε
x✝¹ : A
x m : Unitization ℂ A
hm₁ : 0 ≤ m
hm₂ : ‖m‖ < 1
hx₂ : ‖x‖ ≤ 1
hx₀ : 0 ≤ x
g : ℝ≥0 → ℝ≥0 := fun y ↦ 1 - (1 + y)⁻¹
hx₁ : cfc g (ε⁻¹ ^ 2 • m) ≤ x
x✝ : ℝ≥0
⊢ x✝ ≠ 0

Tactic:
_

Goal state after:
case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
m✝¹ m✝ : A
ε : ℝ≥0
hε : 0 < ε
x✝¹ : A
x m : Unitization ℂ A
hm₁ : 0 ≤ m
hm₂ : ‖m‖ < 1
hx₂ : ‖x‖ ≤ 1
hx₀ : 0 ≤ x
g : ℝ≥0 → ℝ≥0 := fun y ↦ 1 - (1 + y)⁻¹
hx₁ : cfc g (ε⁻¹ ^ 2 • m) ≤ x
x✝ : ℝ≥0
a✝ : x✝ = 0
⊢ False
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
m✝¹ m✝ : A
ε : ℝ≥0
hε : 0 < ε
x✝¹ : A
x m : Unitization ℂ A
hm₁ : 0 ≤ m
hm₂ : ‖m‖ < 1
hx₂ : ‖x‖ ≤ 1
hx₀ : 0 ≤ x
g : ℝ≥0 → ℝ≥0 := fun y ↦ 1 - (1 + y)⁻¹
hx₁ : cfc g (ε⁻¹ ^ 2 • m) ≤ x
x✝ : ℝ≥0
⊢ x✝ ∈ Set.univ → 1 + x✝ ≠ 0

Tactic:
_

Goal state after:
case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
m✝¹ m✝ : A
ε : ℝ≥0
hε : 0 < ε
x✝¹ : A
x m : Unitization ℂ A
hm₁ : 0 ≤ m
hm₂ : ‖m‖ < 1
hx₂ : ‖x‖ ≤ 1
hx₀ : 0 ≤ x
g : ℝ≥0 → ℝ≥0 := fun y ↦ 1 - (1 + y)⁻¹
hx₁ : cfc g (ε⁻¹ ^ 2 • m) ≤ x
x✝ : ℝ≥0
a✝ : x✝ ∈ Set.univ
⊢ 1 + x✝ ≠ 0
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Commute

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁷ : RCLike 𝕜
inst✝⁶ : Ring A
inst✝⁵ : StarRing A
inst✝⁴ : Algebra 𝕜 A
inst✝³ : TopologicalSpace A
inst✝² : ContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a b : A
ha : p a
hb₁ : Commute a b
hb₂ : Commute (star a) b
r : 𝕜
| Commute ((cfcHom ha) (ContinuousMap.const (↑(spectrum 𝕜 a)) r)) b

Tactic:
1,

Goal state after:
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁷ : RCLike 𝕜
inst✝⁶ : Ring A
inst✝⁵ : StarRing A
inst✝⁴ : Algebra 𝕜 A
inst✝³ : TopologicalSpace A
inst✝² : ContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a b : A
ha : p a
hb₁ : Commute a b
hb₂ : Commute (star a) b
r : 𝕜
| (cfcHom ha) (ContinuousMap.const (↑(spectrum 𝕜 a)) r)
Module: Mathlib.MeasureTheory.SpecificCodomains.ContinuousMap

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝³ : MeasurableSpace X
μ : Measure X
inst✝² : TopologicalSpace Y
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : CompactSpace Y
f : X → C(Y, E)
bound : X → ℝ
bound_int : HasFiniteIntegral bound μ
bound_ge : ∀ᵐ (x : X) ∂μ, ∀ (y : Y), ‖(f x) y‖ ≤ bound x
h : Nonempty Y
bound_nonneg : 0 ≤ᵐ[μ] bound
x : X
⊢ (∀ (y : Y), ‖(f x) y‖ ≤ bound x) → 0 x ≤ bound x → ‖f x‖ ≤ bound x

Tactic:
bound_ge_x

Goal state after:
X : Type u_1
Y : Type u_2
inst✝³ : MeasurableSpace X
μ : Measure X
inst✝² : TopologicalSpace Y
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : CompactSpace Y
f : X → C(Y, E)
bound : X → ℝ
bound_int : HasFiniteIntegral bound μ
bound_ge : ∀ᵐ (x : X) ∂μ, ∀ (y : Y), ‖(f x) y‖ ≤ bound x
h : Nonempty Y
bound_nonneg : 0 ≤ᵐ[μ] bound
x : X
bound_ge_x : ∀ (y : Y), ‖(f x) y‖ ≤ bound x
⊢ 0 x ≤ bound x → ‖f x‖ ≤ bound x
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Commute

Goal state before:
pp: case mul
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁷ : RCLike 𝕜
inst✝⁶ : Ring A
inst✝⁵ : StarRing A
inst✝⁴ : Algebra 𝕜 A
inst✝³ : TopologicalSpace A
inst✝² : ContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a b : A
ha : p a
hb₁ : Commute a b
hb₂ : Commute (star a) b
f g : C(↑(spectrum 𝕜 a), 𝕜)
hf : Commute ((cfcHom ha) f) b
hg : Commute ((cfcHom ha) g) b
⊢ Commute ((cfcHom ha) (f * g)) b

Tactic:
| mul f g hf hg =>

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝³ : MeasurableSpace X
μ : Measure X
inst✝² : TopologicalSpace Y
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : CompactSpace Y
f : X → Y → E
g : C(Y, E)
f_ae_cont : ∀ᵐ (x : X) ∂μ, Continuous (f x)
bound : X → ℝ
bound_int : HasFiniteIntegral bound μ
bound_ge : ∀ᵐ (x : X) ∂μ, ∀ (y : Y), ‖f x y‖ ≤ bound x
x : X
⊢ (∀ (y : Y), ‖f x y‖ ≤ bound x) → Continuous (f x) → ∀ (y : Y), ‖(mkD (f x) g) y‖ ≤ bound x

Tactic:
bound_ge_x

Goal state after:
X : Type u_1
Y : Type u_2
inst✝³ : MeasurableSpace X
μ : Measure X
inst✝² : TopologicalSpace Y
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : CompactSpace Y
f : X → Y → E
g : C(Y, E)
f_ae_cont : ∀ᵐ (x : X) ∂μ, Continuous (f x)
bound : X → ℝ
bound_int : HasFiniteIntegral bound μ
bound_ge : ∀ᵐ (x : X) ∂μ, ∀ (y : Y), ‖f x y‖ ≤ bound x
x : X
bound_ge_x : ∀ (y : Y), ‖f x y‖ ≤ bound x
⊢ Continuous (f x) → ∀ (y : Y), ‖(mkD (f x) g) y‖ ≤ bound x
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Commute

Goal state before:
pp: case mul
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalRing A
inst✝⁷ : StarRing A
inst✝⁶ : Module 𝕜 A
inst✝⁵ : IsScalarTower 𝕜 A A
inst✝⁴ : SMulCommClass 𝕜 A A
inst✝³ : TopologicalSpace A
inst✝² : NonUnitalContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : IsTopologicalRing A
inst✝ : T2Space A
a b : A
ha : p a
hb₁ : Commute a b
hb₂ : Commute (star a) b
f g : C(↑(quasispectrum 𝕜 a), 𝕜)₀
hf : Commute ((cfcₙHom ha) f) b
hg : Commute ((cfcₙHom ha) g) b
⊢ Commute ((cfcₙHom ha) (f * g)) b

Tactic:
| mul f g hf hg =>

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
inst✝¹ : CompactSpace Y
inst✝ : Zero Y
f : X → C(Y, E)₀
bound : X → ℝ
bound_int : HasFiniteIntegral bound μ
bound_ge : ∀ᵐ (x : X) ∂μ, ∀ (y : Y), ‖(f x) y‖ ≤ bound x
bound_nonneg : 0 ≤ᵐ[μ] bound
x : X
⊢ (∀ (y : Y), ‖(f x) y‖ ≤ bound x) → 0 x ≤ bound x → ‖f x‖ ≤ bound x

Tactic:
bound_ge_x

Goal state after:
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
inst✝¹ : CompactSpace Y
inst✝ : Zero Y
f : X → C(Y, E)₀
bound : X → ℝ
bound_int : HasFiniteIntegral bound μ
bound_ge : ∀ᵐ (x : X) ∂μ, ∀ (y : Y), ‖(f x) y‖ ≤ bound x
bound_nonneg : 0 ≤ᵐ[μ] bound
x : X
bound_ge_x : ∀ (y : Y), ‖(f x) y‖ ≤ bound x
⊢ 0 x ≤ bound x → ‖f x‖ ≤ bound x
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
inst✝¹ : CompactSpace Y
inst✝ : Zero Y
f : X → Y → E
g : C(Y, E)₀
f_ae_cont : ∀ᵐ (x : X) ∂μ, Continuous (f x)
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 : Y), ‖f x y‖ ≤ bound x
x : X
⊢ (∀ (y : Y), ‖f x y‖ ≤ bound x) → Continuous (f x) → f x 0 = 0 → ∀ (y : Y), ‖(mkD (f x) g) y‖ ≤ bound x

Tactic:
bound_ge_x

Goal state after:
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
inst✝¹ : CompactSpace Y
inst✝ : Zero Y
f : X → Y → E
g : C(Y, E)₀
f_ae_cont : ∀ᵐ (x : X) ∂μ, Continuous (f x)
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 : Y), ‖f x y‖ ≤ bound x
x : X
bound_ge_x : ∀ (y : Y), ‖f x y‖ ≤ bound x
⊢ Continuous (f x) → f x 0 = 0 → ∀ (y : Y), ‖(mkD (f x) g) y‖ ≤ bound x
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart

Goal state before:
pp: A : Type u_1
inst✝⁷ : TopologicalSpace A
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : Module ℂ A
inst✝³ : IsScalarTower ℂ A A
inst✝² : SMulCommClass ℂ A A
inst✝¹ : StarModule ℂ A
inst✝ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
a : A
ha : IsStarNormal a
| 2⁻¹ • (a + star a)

Tactic:
← cfcₙ_id' ℂ a,

Goal state after:
A : Type u_1
inst✝⁷ : TopologicalSpace A
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : Module ℂ A
inst✝³ : IsScalarTower ℂ A A
inst✝² : SMulCommClass ℂ A A
inst✝¹ : StarModule ℂ A
inst✝ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
a : A
ha : IsStarNormal a
| 2⁻¹ • (cfcₙ (fun x ↦ x) a + star (cfcₙ (fun x ↦ x) a))
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Range

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : RCLike 𝕜
inst✝⁷ : Ring A
inst✝⁶ : StarRing A
inst✝⁵ : Algebra 𝕜 A
inst✝⁴ : TopologicalSpace A
inst✝³ : StarModule 𝕜 A
inst✝² : ContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : IsTopologicalRing A
inst✝ : ContinuousStar A
a : A
ha : p a
⊢ (cfcHom ha).range ≤ elemental 𝕜 a

Tactic:
by
  grw [StarAlgHom.range_eq_map_top, ← ContinuousMap.elemental_id_eq_top, StarAlgebra.elemental,
    StarSubalgebra.map_topologicalClosure_le _ _ (cfcHom_continuous ha (R := 𝕜)),
    StarAlgHom.map_adjoin]
  simp [cfcHom_id ha, elemental]

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

Goal state before:
pp: X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : MeasurableSpace X
μ : Measure X
inst✝⁸ : NormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedAlgebra 𝕜 A
inst✝⁵ : ContinuousFunctionalCalculus 𝕜 A p
inst✝⁴ : CompleteSpace A
inst✝³ : NormedSpace ℝ A
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
s : Set X
hs : MeasurableSet s
f : X → 𝕜 → 𝕜
bound : X → ℝ
a : A
inst✝ : SecondCountableTopologyEither X C(↑(spectrum 𝕜 a), 𝕜)
hf : ContinuousOn (Function.uncurry f) (s ×ˢ spectrum 𝕜 a)
bound_ge : ∀ᵐ (x : X) ∂μ.restrict s, ∀ z ∈ spectrum 𝕜 a, ‖f x z‖ ≤ bound x
bound_int : HasFiniteIntegral bound (μ.restrict s)
ha : p a
⊢ cfc (fun r ↦ ∫ (x : X) in s, f x r ∂μ) a = ∫ (x : X) in s, cfc (f x) a ∂μ

Tactic:
by
  have : ∀ᵐ (x : X) ∂(μ.restrict s), ContinuousOn (f x) (spectrum 𝕜 a) :=
    ae_restrict_of_forall_mem hs fun x hx ↦
      hf.comp (Continuous.prodMk_right x).continuousOn fun _ hz ↦ ⟨hx, hz⟩
  refine cfc_setIntegral' _ _ this ⟨?_, ?_⟩ ha
  · exact aeStronglyMeasurable_restrict_mkD_restrict_of_uncurry hs _ _ hf
  · exact hasFiniteIntegral_mkD_restrict_of_bound f _ this bound bound_int bound_ge

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
| 2⁻¹ • cfc (fun x ↦ x + star x) a

Tactic:
← cfc_smul ..

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
| cfc (fun x ↦ 2⁻¹ • (x + star x)) a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unitary

Goal state before:
pp: A : Type u_1
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra ℂ A
inst✝ : ContinuousFunctionalCalculus ℂ A IsStarNormal
u : A
hu : IsStarNormal u
⊢ cfc id u ∈ unitary A ↔ spectrum ℂ u ⊆ ↑(unitary ℂ)

Tactic:
cfc_unitary_iff id u,

Goal state after:
A : Type u_1
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra ℂ A
inst✝ : ContinuousFunctionalCalculus ℂ A IsStarNormal
u : A
hu : IsStarNormal u
⊢ (∀ x ∈ spectrum ℂ u, star (id x) * id x = 1) ↔ spectrum ℂ u ⊆ ↑(unitary ℂ)
Module: Mathlib.Analysis.CStarAlgebra.Matrix

Goal state before:
pp: case e_2
𝕜 : Type u_1
m : Type u_2
n : Type u_3
l : Type u_4
E : Type u_5
inst✝⁵ : RCLike 𝕜
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : Fintype l
inst✝ : DecidableEq l
normed_add_comm_group : NormedAddCommGroup (Matrix m n 𝕜) :=
  let __src := l2OpNormedAddCommGroupAux.replaceTopology ⋯;
  { toNorm := l2OpNormedAddCommGroupAux.toNorm, toAddCommGroup := addCommGroup, toMetricSpace := __src, dist_eq := ⋯ }
⊢ IsTopologicalAddGroup.rightUniformSpace (Matrix m n 𝕜) = PseudoEMetricSpace.toUniformSpace

Tactic:
@IsUniformAddGroup.rightUniformSpace_eq _ PseudoEMetricSpace.toUniformSpace _ _

Goal state after:
case e_2
𝕜 : Type u_1
m : Type u_2
n : Type u_3
l : Type u_4
E : Type u_5
inst✝⁵ : RCLike 𝕜
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : Fintype l
inst✝ : DecidableEq l
normed_add_comm_group : NormedAddCommGroup (Matrix m n 𝕜) :=
  let __src := l2OpNormedAddCommGroupAux.replaceTopology ⋯;
  { toNorm := l2OpNormedAddCommGroupAux.toNorm, toAddCommGroup := addCommGroup, toMetricSpace := __src, dist_eq := ⋯ }
⊢ PseudoEMetricSpace.toUniformSpace = PseudoEMetricSpace.toUniformSpace
Module: Mathlib.Analysis.Complex.Circle

Goal state before:
pp: ⊢ 𝐞 ≠ 1

Tactic:
by
  rw [DFunLike.ne_iff]
  use 2⁻¹
  simp only [fourierChar_apply', AddChar.one_apply]
  rw [mul_comm, ← mul_assoc, inv_mul_cancel₀ (by positivity), one_mul]
  exact Circle.exp_pi_ne_one

Goal state after:
Module: Mathlib.Topology.OpenPartialHomeomorph.Composition

Goal state before:
pp: case refine_1
X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : OpenPartialHomeomorph X Y
s : Set X
e' : OpenPartialHomeomorph X Y
hs : IsOpen[inst✝¹] s
hs' : IsOpen[inst✝] (↑e '' s)
hs'' : s ⊆ e.source
⊢ ((e.restr s).symm.trans e').source = ((e.symm.trans e').restr (↑e '' s)).source

Tactic:
simp only [trans_toPartialEquiv, symm_toPartialEquiv, restr_toPartialEquiv,
      PartialEquiv.trans_source, PartialEquiv.symm_source, PartialEquiv.restr_target,
      coe_toPartialEquiv_symm, PartialEquiv.restr_coe_symm, PartialEquiv.restr_source]

Goal state after:
case refine_1
X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : OpenPartialHomeomorph X Y
s : Set X
e' : OpenPartialHomeomorph X Y
hs : IsOpen[inst✝¹] s
hs' : IsOpen[inst✝] (↑e '' s)
hs'' : s ⊆ e.source
⊢ e.target ∩ ↑e.symm ⁻¹' interior s ∩ ↑e.symm ⁻¹' e'.source = e.target ∩ ↑e.symm ⁻¹' e'.source ∩ interior (↑e '' s)