mathlib-initiative/mathlib-tactics huggingface dataset:
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]

Goal state after:
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‖
⊢ 0 ≤ ↑C * ‖v‖
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Integral

Goal state before:
pp: case refine_1
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
this : ∀ᵐ (x : X) ∂μ.restrict s, ContinuousOn (f x) (spectrum 𝕜 a)
⊢ AEStronglyMeasurable (fun x ↦ mkD ((spectrum 𝕜 a).domRestrict (f x)) 0) (μ.restrict s)

Tactic:
exact aeStronglyMeasurable_restrict_mkD_restrict_of_uncurry hs _ _ hf

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

Goal state before:
pp: case mp
A : Type u_1
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra ℝ A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : IsTopologicalRing A
inst✝⁴ : T2Space A
inst✝³ : PartialOrder A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : StarOrderedRing A
inst✝ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : 0 ≤ a
f : ℝ≥0 → ℝ≥0
⊢ (fun x ↦ cfc x a) f ∈ (fun x ↦ cfc x a) '' {f | ∀ x ∈ spectrum ℝ a, 0 ≤ f x}

Tactic:
simp only [cfc_nnreal_eq_real f a ha]

Goal state after:
case mp
A : Type u_1
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra ℝ A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : IsTopologicalRing A
inst✝⁴ : T2Space A
inst✝³ : PartialOrder A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : StarOrderedRing A
inst✝ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : 0 ≤ a
f : ℝ≥0 → ℝ≥0
⊢ cfc (fun x ↦ ↑(f x.toNNReal)) a ∈ (fun x ↦ cfc x a) '' {f | ∀ x ∈ spectrum ℝ a, 0 ≤ f x}
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✝⁹ : RCLike 𝕜
inst✝⁸ : MeasurableSpace X
μ : Measure X
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 : NonUnitalContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : CompleteSpace A
inst✝ : NormedSpace ℝ A
f : X → 𝕜 → 𝕜
a : A
hf₁ : ∀ᵐ (x : X) ∂μ, ContinuousOn (f x) (quasispectrum 𝕜 a)
hf₂ : ∀ᵐ (x : X) ∂μ, f x 0 = 0
hf₃ : Integrable (fun x ↦ mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0) μ
ha : p a
z : ↑(quasispectrum 𝕜 a)
x : X
⊢ ContinuousOn (f x) (quasispectrum 𝕜 a) → f x 0 = 0 → f x ↑z = (mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0) z

Tactic:
cont_x

Goal state after:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁹ : RCLike 𝕜
inst✝⁸ : MeasurableSpace X
μ : Measure X
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 : NonUnitalContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : CompleteSpace A
inst✝ : NormedSpace ℝ A
f : X → 𝕜 → 𝕜
a : A
hf₁ : ∀ᵐ (x : X) ∂μ, ContinuousOn (f x) (quasispectrum 𝕜 a)
hf₂ : ∀ᵐ (x : X) ∂μ, f x 0 = 0
hf₃ : Integrable (fun x ↦ mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0) μ
ha : p a
z : ↑(quasispectrum 𝕜 a)
x : X
cont_x : ContinuousOn (f x) (quasispectrum 𝕜 a)
⊢ f x 0 = 0 → f x ↑z = (mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0) z
Module: Mathlib.Analysis.Normed.Group.NullSubmodule

Goal state before:
pp: M : Type u_1
inst✝ : SeminormedCommGroup M
x : M
hx : ‖x‖ = 0
⊢ x⁻¹ ∈ {x | ‖x‖ = 0}

Tactic:
simpa only [Set.mem_ofPred_eq, norm_inv'] using hx

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Dual

Goal state before:
pp: 𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
x : E
y : F
hx : x ≠ 0
hy : y ≠ 0
⊢ Module.rank 𝕜 ↥(↑(((innerSL 𝕜) y).smulRight x)).range = 1

Tactic:
range_smulRight_apply,

Goal state after:
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
x : E
y : F
hx : x ≠ 0
hy : y ≠ 0
⊢ Module.rank 𝕜 ↥(𝕜 ∙ x) = 1

case hf
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
x : E
y : F
hx : x ≠ 0
hy : y ≠ 0
⊢ (innerSL 𝕜) y ≠ 0
Module: Mathlib.Analysis.CStarAlgebra.Matrix

Goal state before:
pp: 𝕜 : Type u_1
m : Type u_2
n : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : DecidableEq m
A : Matrix m n 𝕜
⊢ ‖(toLin (EuclideanSpace.basisFun m 𝕜).toBasis (EuclideanSpace.basisFun n 𝕜).toBasis ≪≫ₗ toContinuousLinearMap) Aᴴ‖ =
    ‖A‖

Tactic:
LinearEquiv.trans_apply,

Goal state after:
𝕜 : Type u_1
m : Type u_2
n : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : DecidableEq m
A : Matrix m n 𝕜
⊢ ‖toContinuousLinearMap ((toLin (EuclideanSpace.basisFun m 𝕜).toBasis (EuclideanSpace.basisFun n 𝕜).toBasis) Aᴴ)‖ = ‖A‖
Module: Mathlib.Analysis.Complex.Circle

Goal state before:
pp: x : Circle
⊢ ↑x = 1 ↔ x = 1

Tactic:
← coe_inj,

Goal state after:
x : Circle
⊢ ↑x = 1 ↔ ↑x = ↑1
Module: Mathlib.Analysis.CStarAlgebra.Matrix

Goal state before:
pp: 𝕜 : Type u_1
m : Type u_2
n : Type u_3
inst✝³ : RCLike 𝕜
inst✝² : Fintype m
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix m n 𝕜
⊢ ‖(toLin (EuclideanSpace.basisFun n 𝕜).toBasis (EuclideanSpace.basisFun n 𝕜).toBasis ≪≫ₗ toContinuousLinearMap)
        (Aᴴ * A)‖ =
    ‖A‖ * ‖A‖

Tactic:
LinearEquiv.trans_apply,

Goal state after:
𝕜 : Type u_1
m : Type u_2
n : Type u_3
inst✝³ : RCLike 𝕜
inst✝² : Fintype m
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix m n 𝕜
⊢ ‖toContinuousLinearMap
        ((toLin (EuclideanSpace.basisFun n 𝕜).toBasis (EuclideanSpace.basisFun n 𝕜).toBasis) (Aᴴ * A))‖ =
    ‖A‖ * ‖A‖
Module: Mathlib.Analysis.CStarAlgebra.Matrix

Goal state before:
pp: 𝕜 : Type u_1
m : Type u_2
n : Type u_3
l : Type u_4
inst✝⁵ : RCLike 𝕜
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : Fintype l
inst✝ : DecidableEq l
A : Matrix m n 𝕜
B : Matrix n l 𝕜
this :
  ‖(toEuclideanLin ≪≫ₗ toContinuousLinearMap) A ∘SL (toEuclideanLin ≪≫ₗ toContinuousLinearMap) B‖ ≤
    ‖(toEuclideanLin ≪≫ₗ toContinuousLinearMap) A‖ * ‖(toEuclideanLin ≪≫ₗ toContinuousLinearMap) B‖
⊢ (toEuclideanLin ≪≫ₗ toContinuousLinearMap) (A * B) =
    (toEuclideanLin ≪≫ₗ toContinuousLinearMap) A ∘SL (toEuclideanLin ≪≫ₗ toContinuousLinearMap) B

Tactic:
ext1 x

Goal state after:
𝕜 : Type u_1
m : Type u_2
n : Type u_3
l : Type u_4
inst✝⁵ : RCLike 𝕜
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : Fintype l
inst✝ : DecidableEq l
A : Matrix m n 𝕜
B : Matrix n l 𝕜
this :
  ‖(toEuclideanLin ≪≫ₗ toContinuousLinearMap) A ∘SL (toEuclideanLin ≪≫ₗ toContinuousLinearMap) B‖ ≤
    ‖(toEuclideanLin ≪≫ₗ toContinuousLinearMap) A‖ * ‖(toEuclideanLin ≪≫ₗ toContinuousLinearMap) B‖
x : EuclideanSpace 𝕜 l
⊢ ((toEuclideanLin ≪≫ₗ toContinuousLinearMap) (A * B)) x =
    ((toEuclideanLin ≪≫ₗ toContinuousLinearMap) A ∘SL (toEuclideanLin ≪≫ₗ toContinuousLinearMap) B) x
Module: Mathlib.Data.Bundle

Goal state before:
pp: B : Type u_1
F : Type u_2
E : B → Type u_3
b : B
y y' : E b
⊢ mk' F b y = mk' F b y' ↔ y = y'

Tactic:
simp [TotalSpace.ext_iff]

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

Goal state before:
pp: 𝕜 : Type u
A : Type v
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : IsScalarTower 𝕜 A A
a b : 𝓜(𝕜, A)
h : a.toProd = b.toProd
⊢ a = b

Tactic:
cases a

Goal state after:
case mk
𝕜 : Type u
A : Type v
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : IsScalarTower 𝕜 A A
b : 𝓜(𝕜, A)
toProd✝ : (A →L[𝕜] A) × (A →L[𝕜] A)
central✝ : ∀ (x y : A), toProd✝.2 x * y = x * toProd✝.1 y
h : { toProd := toProd✝, central := central✝ }.toProd = b.toProd
⊢ { toProd := toProd✝, central := central✝ } = b
Module: Mathlib.Topology.FiberBundle.Trivialization

Goal state before:
pp: B : Type u_1
F : Type u_2
E : B → Type u_3
Z : Type u_4
inst✝³ : TopologicalSpace B
inst✝² : TopologicalSpace F
proj : Z → B
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace (TotalSpace F E)
e✝ : Trivialization F proj
x : Z
e'✝ : Trivialization F TotalSpace.proj
b : B
y : E b
e e' : Trivialization F proj
s : Set B
Hs : e.baseSet ∩ frontier s = e'.baseSet ∩ frontier s
Heq : EqOn (↑e) (↑e') (proj ⁻¹' (e.baseSet ∩ frontier s))
⊢ proj ⁻¹' (e.baseSet ∩ frontier s) = proj ⁻¹' (e'.baseSet ∩ frontier s)

Tactic:
Hs

Goal state after:
B : Type u_1
F : Type u_2
E : B → Type u_3
Z : Type u_4
inst✝³ : TopologicalSpace B
inst✝² : TopologicalSpace F
proj : Z → B
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace (TotalSpace F E)
e✝ : Trivialization F proj
x : Z
e'✝ : Trivialization F TotalSpace.proj
b : B
y : E b
e e' : Trivialization F proj
s : Set B
Hs : e.baseSet ∩ frontier s = e'.baseSet ∩ frontier s
Heq : EqOn (↑e) (↑e') (proj ⁻¹' (e.baseSet ∩ frontier s))
⊢ proj ⁻¹' (e'.baseSet ∩ frontier s) = proj ⁻¹' (e'.baseSet ∩ frontier s)
Module: Mathlib.Topology.IsLocalHomeomorph

Goal state before:
pp: case refine_1
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : IsLocalHomeomorph f
U : Set Y
hU : IsOpen[inst✝] U
s : C(↑U, X)
hs : f ∘ ⇑s = Subtype.val
⊢ IsOpen[inst✝] (Set.range Subtype.val)

Tactic:
rwa [Subtype.range_val]

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : StarRing 𝕜
inst✝⁶ : NonUnitalNormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : CStarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : StarModule 𝕜 A
a : 𝓜(𝕜, A)
hball : (Metric.closedBall 0 1).Nonempty
key : ∀ (x y : A), ‖x‖₊ ≤ 1 → ‖y‖₊ ≤ 1 → ‖a.toProd.2 (star (a.toProd.1 (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊
⊢ ‖star a * a‖₊ = ‖a‖₊ * ‖a‖₊

Tactic:
← nnnorm_snd

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : StarRing 𝕜
inst✝⁶ : NonUnitalNormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : CStarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : StarModule 𝕜 A
a : 𝓜(𝕜, A)
hball : (Metric.closedBall 0 1).Nonempty
key : ∀ (x y : A), ‖x‖₊ ≤ 1 → ‖y‖₊ ≤ 1 → ‖a.toProd.2 (star (a.toProd.1 (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊
⊢ ‖(star a * a).toProd.2‖₊ = ‖a‖₊ * ‖a‖₊
Module: Mathlib.Analysis.SpecialFunctions.Complex.Circle

Goal state before:
pp: x y : Circle
h : x ≠ y
⊢ range ⇑(x.path y) ∪ range ⇑(y.path x) = univ

Tactic:
rw [range_path, range_path, ← image_union, Icc_union_Icc_angleDiff_add_arg h,
    periodic_exp.image_Icc two_pi_pos]

Goal state after:
x y : Circle
h : x ≠ y
⊢ range ⇑exp = univ
Module: Mathlib.Analysis.SpecialFunctions.Complex.Circle

Goal state before:
pp: case H
T : ℝ
hT : T ≠ 0
z✝ : ℝ
⊢ (homeomorphCircle hT) ↑z✝ = toCircle ↑z✝

Tactic:
homeomorphCircle,

Goal state after:
case H
T : ℝ
hT : T ≠ 0
z✝ : ℝ
⊢ ((homeomorphAddCircle T (2 * π) hT homeomorphCircle._proof_1).trans homeomorphCircle') ↑z✝ = toCircle ↑z✝
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
hu : dist u 1 < 2
ε : ℝ
huε : dist u 1 ^ 2 < ε
hε2 : ε < 2 ^ 2
hε : 0 < ε
huε' : dist u 1 < √ε
⊢ ContinuousAt (fun x ↦ cfc (fun x ↦ ↑x.arg) ↑x) u

Tactic:
apply ContinuousOn.continuousAt ?_ (closedBall_mem_nhds_of_mem huε')

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
hu : dist u 1 < 2
ε : ℝ
huε : dist u 1 ^ 2 < ε
hε2 : ε < 2 ^ 2
hε : 0 < ε
huε' : dist u 1 < √ε
⊢ ContinuousOn (fun x ↦ cfc (fun x ↦ ↑x.arg) ↑x) (closedBall 1 √ε)
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : DivisionRing k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
inst✝ : CharZero k
n : ℕ
s : Simplex k P n
S : AffineSubspace k P
hS : affineSpan k (Set.range s.points) ≤ S
this : Nonempty ↥S
⊢ s.centroid = ↑(s.restrict S hS).centroid

Tactic:
have hf : Function.Injective (S.subtype) := by
    simp only [coe_subtype, Subtype.val_injective]

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : DivisionRing k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
inst✝ : CharZero k
n : ℕ
s : Simplex k P n
S : AffineSubspace k P
hS : affineSpan k (Set.range s.points) ≤ S
this : Nonempty ↥S
hf : Function.Injective ⇑S.subtype
⊢ s.centroid = ↑(s.restrict S hS).centroid
Module: Mathlib.Analysis.InnerProductSpace.Adjoint

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
T : E →ₗ[𝕜] E
x : E
⊢ 0 ≤ re ⟪x, (adjoint T * T) x⟫_𝕜

Tactic:
by
  simp only [Module.End.mul_apply, adjoint_inner_right, inner_self_eq_norm_sq_to_K]
  norm_cast
  exact sq_nonneg _

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
S : AffineSubspace k P
hS : affineSpan k (Set.range s.points) ≤ S
i : Fin (n + 1)
this : Nonempty ↥S
⊢ s.faceOppositeCentroid i = ↑((s.restrict S hS).faceOppositeCentroid i)

Tactic:
have hf : Function.Injective (S.subtype) := by
    simp only [coe_subtype, Subtype.val_injective]

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
S : AffineSubspace k P
hS : affineSpan k (Set.range s.points) ≤ S
i : Fin (n + 1)
this : Nonempty ↥S
hf : Function.Injective ⇑S.subtype
⊢ s.faceOppositeCentroid i = ↑((s.restrict S hS).faceOppositeCentroid i)
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p : Fin 3 → P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
hu : Finset.univ = {i₁, i₂, i₃}
⊢ ¬Collinear k (p '' insert i₁ (insert i₂ ↑{i₃})) ↔ ¬Collinear k {p i₁, p i₂, p i₃}

Tactic:
Finset.coe_singleton,

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p : Fin 3 → P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
hu : Finset.univ = {i₁, i₂, i₃}
⊢ ¬Collinear k (p '' {i₁, i₂, i₃}) ↔ ¬Collinear k {p i₁, p i₂, p i₃}
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: R : Type u_1
S : Type u_2
V : Type u_3
W : Type u_4
P : Type u_5
inst✝¹⁴ : Ring R
inst✝¹³ : Ring S
inst✝¹² : AddCommGroup V
inst✝¹¹ : Module R V
inst✝¹⁰ : Module.Finite R V
inst✝⁹ : Module.Free R V
inst✝⁸ : AffineSpace V P
inst✝⁷ : AddCommGroup W
inst✝⁶ : Module R W
inst✝⁵ : Module S W
inst✝⁴ : Module.Finite S W
inst✝³ : SMulCommClass R S W
inst✝² : Module.Free S W
inst✝¹ : StrongRankCondition R
inst✝ : StrongRankCondition S
⊢ Module.finrank S (W × (V →ₗ[R] W)) = (Module.finrank R V + 1) * Module.finrank S W

Tactic:
rw [Module.finrank_prod, Module.finrank_linearMap]

Goal state after:
R : Type u_1
S : Type u_2
V : Type u_3
W : Type u_4
P : Type u_5
inst✝¹⁴ : Ring R
inst✝¹³ : Ring S
inst✝¹² : AddCommGroup V
inst✝¹¹ : Module R V
inst✝¹⁰ : Module.Finite R V
inst✝⁹ : Module.Free R V
inst✝⁸ : AffineSpace V P
inst✝⁷ : AddCommGroup W
inst✝⁶ : Module R W
inst✝⁵ : Module S W
inst✝⁴ : Module.Finite S W
inst✝³ : SMulCommClass R S W
inst✝² : Module.Free S W
inst✝¹ : StrongRankCondition R
inst✝ : StrongRankCondition S
⊢ Module.finrank S W + Module.finrank R V * Module.finrank S W = (Module.finrank R V + 1) * Module.finrank S W
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: case inl
𝕜 : Type u_1
α : Type u_3
E : α → Type u_4
inst✝³ : (i : α) → NormedAddCommGroup (E i)
inst✝² : NormedRing 𝕜
inst✝¹ : (i : α) → Module 𝕜 (E i)
inst✝ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)
f : (i : α) → E i
c : 𝕜
hf : Memℓp f 0
⊢ Memℓp (c • f) 0

pp: case inr.inl
𝕜 : Type u_1
α : Type u_3
E : α → Type u_4
inst✝³ : (i : α) → NormedAddCommGroup (E i)
inst✝² : NormedRing 𝕜
inst✝¹ : (i : α) → Module 𝕜 (E i)
inst✝ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)
f : (i : α) → E i
c : 𝕜
hf : Memℓp f ∞
⊢ Memℓp (c • f) ∞

pp: case inr.inr
𝕜 : Type u_1
α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝³ : (i : α) → NormedAddCommGroup (E i)
inst✝² : NormedRing 𝕜
inst✝¹ : (i : α) → Module 𝕜 (E i)
inst✝ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)
f : (i : α) → E i
hf : Memℓp f p
c : 𝕜
hp : 0 < p.toReal
⊢ Memℓp (c • f) p

Tactic:
· apply memℓp_zero
    refine hf.finite_dsupport.subset fun i => (?_ : ¬c • f i = 0 → ¬f i = 0)
    exact not_imp_not.mpr fun hf' => hf'.symm ▸ smul_zero c

Goal state after:
case inr.inl
𝕜 : Type u_1
α : Type u_3
E : α → Type u_4
inst✝³ : (i : α) → NormedAddCommGroup (E i)
inst✝² : NormedRing 𝕜
inst✝¹ : (i : α) → Module 𝕜 (E i)
inst✝ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)
f : (i : α) → E i
c : 𝕜
hf : Memℓp f ∞
⊢ Memℓp (c • f) ∞

case inr.inr
𝕜 : Type u_1
α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝³ : (i : α) → NormedAddCommGroup (E i)
inst✝² : NormedRing 𝕜
inst✝¹ : (i : α) → Module 𝕜 (E i)
inst✝ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)
f : (i : α) → E i
hf : Memℓp f p
c : 𝕜
hp : 0 < p.toReal
⊢ Memℓp (c • f) p
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
hp : 0 < p.toReal
f : ↥(lp E p)
⊢ ‖f‖ = (∑' (i : α), ‖↑f i‖ ^ p.toReal) ^ (1 / p.toReal)

Tactic:
by
  dsimp [norm]
  rw [ENNReal.toReal_pos_iff] at hp
  rw [dif_neg hp.1.ne', if_neg hp.2.ne]

Goal state after:
Module: Mathlib.Analysis.Calculus.SmoothSeries

Goal state before:
pp: α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : IsRCLikeNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
hu : Summable u
hf : ∀ (n : α) (x : E), HasFDerivAt (f n) (f' n x) x
hf' : ∀ (n : α) (x : E), ‖f' n x‖ ≤ u n
h : ∀ (x₀ : E), ¬Summable fun n ↦ f n x₀
⊢ (fun x ↦ ∑' (n : α), f n x) = 0

Tactic:
ext1 x

Goal state after:
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : IsRCLikeNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
hu : Summable u
hf : ∀ (n : α) (x : E), HasFDerivAt (f n) (f' n x) x
hf' : ∀ (n : α) (x : E), ‖f' n x‖ ≤ u n
h : ∀ (x₀ : E), ¬Summable fun n ↦ f n x₀
x : E
⊢ ∑' (n : α), f n x = 0 x
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: case inl.inr
𝕜 : Type u_1
𝕜' : Type u_2
α : Type u_3
E : α → Type u_4
q : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
hp : Fact (1 ≤ ∞)
f g : ↥(lp E ∞)
h✝ : Nonempty α
i : α
⊢ (fun i ↦ ‖↑(f + g) i‖) i ≤ (fun x1 x2 ↦ x1 + x2) ((fun i ↦ ‖↑f i‖) i) ((fun i ↦ ‖↑g i‖) i)

Tactic:
exact norm_add_le (f i) (g i)

Goal state after:
Module: Mathlib.Analysis.Calculus.SmoothSeries

Goal state before:
pp: α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : IsRCLikeNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
x₀ : E
hu : Summable u
hf : ∀ (n : α), Differentiable 𝕜 (f n)
hf' : ∀ (n : α) (x : E), ‖fderiv 𝕜 (f n) x‖ ≤ u n
hf0 : Summable fun n ↦ f n x₀
⊢ (fderiv 𝕜 fun y ↦ ∑' (n : α), f n y) = fun x ↦ ∑' (n : α), fderiv 𝕜 (f n) x

Tactic:
ext1 x

Goal state after:
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : IsRCLikeNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
x₀ : E
hu : Summable u
hf : ∀ (n : α), Differentiable 𝕜 (f n)
hf' : ∀ (n : α) (x : E), ‖fderiv 𝕜 (f n) x‖ ≤ u n
hf0 : Summable fun n ↦ f n x₀
x : E
⊢ fderiv 𝕜 (fun y ↦ ∑' (n : α), f n y) x = ∑' (n : α), fderiv 𝕜 (f n) x
Module: Mathlib.Analysis.Calculus.SmoothSeries

Goal state before:
pp: α : Type u_1
𝕜 : Type u_3
F : Type u_5
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
g : α → 𝕜 → F
y₀ : 𝕜
hu : Summable u
hg : ∀ (n : α), Differentiable 𝕜 (g n)
hg' : ∀ (n : α) (y : 𝕜), ‖deriv (g n) y‖ ≤ u n
hg0 : Summable fun n ↦ g n y₀
⊢ (deriv fun y ↦ ∑' (n : α), g n y) = fun y ↦ ∑' (n : α), deriv (g n) y

Tactic:
ext1 x

Goal state after:
α : Type u_1
𝕜 : Type u_3
F : Type u_5
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
g : α → 𝕜 → F
y₀ : 𝕜
hu : Summable u
hg : ∀ (n : α), Differentiable 𝕜 (g n)
hg' : ∀ (n : α) (y : 𝕜), ‖deriv (g n) y‖ ≤ u n
hg0 : Summable fun n ↦ g n y₀
x : 𝕜
⊢ deriv (fun y ↦ ∑' (n : α), g n y) x = ∑' (n : α), deriv (g n) x
Module: Mathlib.Analysis.Calculus.SmoothSeries

Goal state before:
pp: case zero
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : IsRCLikeNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
inst✝ : NormedSpace 𝕜 F
f : α → E → F
v : ℕ → α → ℝ
N : ℕ∞
hf : ∀ (i : α), ContDiff 𝕜 (↑N) (f i)
hv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)
h'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i
hk : ↑0 ≤ N
⊢ (iteratedFDeriv 𝕜 0 fun y ↦ ∑' (n : α), f n y) = fun x ↦ ∑' (n : α), iteratedFDeriv 𝕜 0 (f n) x

Tactic:
ext1 x

Goal state after:
case zero
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : IsRCLikeNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
inst✝ : NormedSpace 𝕜 F
f : α → E → F
v : ℕ → α → ℝ
N : ℕ∞
hf : ∀ (i : α), ContDiff 𝕜 (↑N) (f i)
hv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)
h'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i
hk : ↑0 ≤ N
x : E
⊢ iteratedFDeriv 𝕜 0 (fun y ↦ ∑' (n : α), f n y) x = ∑' (n : α), iteratedFDeriv 𝕜 0 (f n) x