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

Goal state before:
pp: x y : Circle
hne : x ≠ y
⊢ (↑x).arg ≤ x.angleDiff y + (↑x).arg

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.Circle

Goal state before:
pp: x y : Circle
⊢ (↑x).arg ≤ x.angleDiff y + (↑x).arg

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.Circle

Goal state before:
pp: x y : Circle
h : x ≠ y
hdisj : Disjoint (Ioc (↑x).arg (x.angleDiff y + (↑x).arg)) (Ioc (↑y).arg (y.angleDiff x + (↑y).arg))
a : ℝ
ha : a ∈ Ioc (↑x).arg (x.angleDiff y + (↑x).arg)
b : ℝ
hb : b ∈ Ioc (↑y).arg (y.angleDiff x + (↑y).arg)
⊢ min (↑x).arg (↑y).arg + 2 * π - min (↑x).arg (↑y).arg ≤ 2 * π

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.Circle

Goal state before:
pp: x y : Circle
h : x ≠ y
⊢ (↑x).arg ≤ x.angleDiff y + (↑x).arg

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.Circle

Goal state before:
pp: x y : Circle
h : x ≠ y
⊢ 0 < 1

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.Circle

Goal state before:
pp: x y : Circle
h : x ≠ y
⊢ x ∉ {x}ᶜ

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.Circle

Goal state before:
pp: x y : Circle
h : x ≠ y
⊢ 0 ≤ 1

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
u : A
hu : u ∈ unitary A
x : ℝ
hz : IsLeast (re '' spectrum ℂ u) x
h✝ : Nontrivial A
h_eqOn : Set.EqOn (fun z ↦ ‖z - 1‖ ^ 2) (fun z ↦ 2 * (1 - z.re)) (spectrum ℂ u)
h₂ : IsGreatest ((fun z ↦ 2 * (1 - z.re)) '' spectrum ℂ u) (2 * (1 - x))
⊢ (fun x ↦ ‖x - 1‖) '' spectrum ℂ u ⊆ {x | 0 ≤ x}

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: case inr
A : Type u_1
inst✝ : CStarAlgebra A
u : A
hu : u ∈ unitary A
x : ℝ
hz : IsLeast (re '' spectrum ℂ u) x
h✝ : Nontrivial A
h_eqOn : Set.EqOn (fun z ↦ ‖z - 1‖ ^ 2) (fun z ↦ 2 * (1 - z.re)) (spectrum ℂ u)
h₂ : IsGreatest ((fun z ↦ 2 * (1 - z.re)) '' spectrum ℂ u) (2 * (1 - x))
h₃ : IsGreatest ((fun x ↦ ‖x - 1‖ ^ 2) '' spectrum ℂ u) (‖cfc (fun x ↦ x - 1) u‖ ^ 2)
⊢ ‖cfc (fun x ↦ x - 1 x) u‖ ^ 2 = 2 * (1 - x)

Tactic:
exact h₃.unique (h_eqOn.image_eq ▸ h₂)

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
x : ↥(selfAdjoint A)
hx : ‖x‖ ≤ π
a✝ : Nontrivial A
h : -‖↑x‖ ∈ spectrum ℝ ↑x
⊢ (fun a ↦ Real.cos a) (-‖↑x‖) = Real.cos ‖x‖

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.Circle

Goal state before:
pp: ⊢ ∀ᶠ (i : ℕ) in Filter.atTop, True

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Maps

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CStarRing A
inst✝² : Ring R
inst✝¹ : Module R A
inst✝ : SMulCommClass R A A
u v : ↥(unitary A)
x : A
⊢ ((mulLeft R A) (u * v)) x = ((mulLeft R A) u) (((mulLeft R A) v) x)

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Span

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
x✝¹ : (A : Type ?u.39) → [NonUnitalCStarAlgebra A] → PartialOrder A := spectralOrder
x✝ : ∀ (A : Type ?u.41) [inst : NonUnitalCStarAlgebra A], StarOrderedRing A := spectralOrderedRing
⊢ 0 = ∑ i, 0 i • ↑(![1, -1, 1, -1] i) ∧ ∀ (i : Fin 4), ‖0 i‖ ≤ ‖0‖ / 2

Tactic:
by simp

Goal state after:
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✝ : CompleteSpace E
⊢ adjoint (ContinuousLinearMap.id 𝕜 E) = ContinuousLinearMap.id 𝕜 E

Tactic:
by simp

Goal state after:
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✝ : CompleteSpace E
⊢ adjoint 1 = 1

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
u v : ↥(unitary A)
⊢ ‖(↑u * ↑(star v) - 1) * ↑v‖ = ‖↑(u * star v) - 1‖

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
x : ↥(selfAdjoint A)
⊢ expUnitary (↑0 • x) = 1

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
x : ↥(selfAdjoint A)
⊢ expUnitary (↑1 • x) = expUnitary x

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
⊢ (fun y ↦ (List.map expUnitary y).prod) [] = 1

Tactic:
by simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace F
A : E →L[𝕜] F
y : F
x : E
⊢ A x ∈ (↑A).range

Tactic:
by simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace F
A : E →L[𝕜] F
⊢ (innerₛₗ 𝕜).flip.IsAdjointPair (innerₛₗ 𝕜).flip ⇑A ⇑(adjoint A)

Tactic:
by
  intro x y
  simp [adjoint_inner_left]

Goal state after:
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
T : E →L[𝕜] E
inst✝ : CompleteSpace E
hT : IsIdempotentElem T
h : IsStarNormal T
⊢ (∀ (x : E), ‖(1 - T) (T x)‖ = 0) ↔ ∀ (x : E), ‖(T - T * T) x‖ = 0

Tactic:
by simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : FiniteDimensional 𝕜 E
inst✝ : FiniteDimensional 𝕜 F
A : E →ₗ[𝕜] F
x : E
y : F
this : CompleteSpace E
⊢ ⟪(adjoint A) y, x⟫_𝕜 = ⟪y, A x⟫_𝕜

Tactic:
have := FiniteDimensional.complete 𝕜 F

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : FiniteDimensional 𝕜 E
inst✝ : FiniteDimensional 𝕜 F
A : E →ₗ[𝕜] F
x : E
y : F
this✝ : CompleteSpace E
this : CompleteSpace F
⊢ ⟪(adjoint A) y, x⟫_𝕜 = ⟪y, A x⟫_𝕜
Module: Mathlib.Analysis.InnerProductSpace.Adjoint

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : FiniteDimensional 𝕜 E
inst✝ : FiniteDimensional 𝕜 F
A : E →ₗ[𝕜] F
x : E
y : F
this : CompleteSpace E
⊢ ⟪x, (adjoint A) y⟫_𝕜 = ⟪A x, y⟫_𝕜

Tactic:
have := FiniteDimensional.complete 𝕜 F

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : FiniteDimensional 𝕜 E
inst✝ : FiniteDimensional 𝕜 F
A : E →ₗ[𝕜] F
x : E
y : F
this✝ : CompleteSpace E
this : CompleteSpace F
⊢ ⟪x, (adjoint A) y⟫_𝕜 = ⟪A x, y⟫_𝕜
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
⊢ adjoint id = id

Tactic:
by simp

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✝⁴ : DivisionRing k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
n : ℕ
inst✝ : CharZero k
s : Simplex k P n
t : Set (Fin (n + 1))
ht : t ≠ Set.univ
hssubset : t ⊂ Set.univ
i : Fin (n + 1)
hi : i ∈ Set.univ ∧ i ∉ t
w : Fin (n + 1) → k := centroidWeights k univ
h : (affineCombination k univ s.points) w ∈ affineSpan k (s.points '' t)
wdef : w = centroidWeights k univ
⊢ univ.Nonempty

Tactic:
by simp

Goal state after:
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
⊢ adjoint 1 = 1

Tactic:
by simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : FiniteDimensional 𝕜 E
inst✝ : FiniteDimensional 𝕜 F
A : E →ₗ[𝕜] F
this : CompleteSpace E
⊢ A.kerᗮ = (adjoint A).range

Tactic:
have := FiniteDimensional.complete 𝕜 F

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : FiniteDimensional 𝕜 E
inst✝ : FiniteDimensional 𝕜 F
A : E →ₗ[𝕜] F
this✝ : CompleteSpace E
this : CompleteSpace F
⊢ A.kerᗮ = (adjoint A).range
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
n : ℕ
inst✝ : CharZero k
s : Simplex k P n
t : Set (Fin (n + 1))
ht : t ≠ Set.univ
hssubset : t ⊂ Set.univ
i : Fin (n + 1)
hi : i ∈ Set.univ ∧ i ∉ t
w : Fin (n + 1) → k := centroidWeights k univ
h : (affineCombination k univ s.points) w ∈ affineSpan k (s.points '' t)
wdef : w = centroidWeights k univ
hw : ∑ i, w i = 1
⊢ i ∈ univ

Tactic:
by simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : FiniteDimensional 𝕜 E
inst✝ : FiniteDimensional 𝕜 F
A : E →ₗ[𝕜] F
this : CompleteSpace E
⊢ A.rangeᗮ = (adjoint A).ker

Tactic:
have := FiniteDimensional.complete 𝕜 F

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : FiniteDimensional 𝕜 E
inst✝ : FiniteDimensional 𝕜 F
A : E →ₗ[𝕜] F
this✝ : CompleteSpace E
this : CompleteSpace F
⊢ A.rangeᗮ = (adjoint A).ker
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
n : ℕ
inst✝ : CharZero k
s : Simplex k P n
p : P
⊢ univ.Nonempty

Tactic:
by simp

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✝⁴ : DivisionRing k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
n : ℕ
inst✝ : CharZero k
s : Simplex k P n
p : P
p✝ : Fin (n + 1) → V
⊢ univ.Nonempty

Tactic:
by simp

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✝⁴ : DivisionRing k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
n : ℕ
inst✝ : CharZero k
s : Simplex k P n
i : Fin (n + 1)
⊢ s.centroid = (↑n + 1)⁻¹ • ∑ x, (s.points x -ᵥ s.points i) +ᵥ s.points i

Tactic:
by
  rw [← s.centroid_vsub_eq, vsub_vadd]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : FiniteDimensional 𝕜 E
inst✝ : FiniteDimensional 𝕜 F
A : E →ₗ[𝕜] F
this : CompleteSpace E
⊢ (adjoint A ∘ₗ A).ker = A.ker

Tactic:
have := FiniteDimensional.complete 𝕜 F

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : FiniteDimensional 𝕜 E
inst✝ : FiniteDimensional 𝕜 F
A : E →ₗ[𝕜] F
this✝ : CompleteSpace E
this : CompleteSpace F
⊢ (adjoint A ∘ₗ A).ker = A.ker
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

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
n : ℕ
s₁ s₂ : Simplex k P n
h : Set.range s₁.points = Set.range s₂.points
⊢ Finset.centroid k univ s₁.points = Finset.centroid k univ s₂.points

Tactic:
← Set.image_univ,

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
n : ℕ
s₁ s₂ : Simplex k P n
h : s₁.points '' Set.univ = Set.range s₂.points
⊢ Finset.centroid k univ s₁.points = Finset.centroid k univ s₂.points
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

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
n : ℕ
s₁ s₂ : Simplex k P n
h : s₁.points '' Set.univ = Set.range s₂.points
⊢ Finset.centroid k univ s₁.points = Finset.centroid k univ s₂.points

Tactic:
← Set.image_univ,

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
n : ℕ
s₁ s₂ : Simplex k P n
h : s₁.points '' Set.univ = s₂.points '' Set.univ
⊢ Finset.centroid k univ s₁.points = Finset.centroid k univ s₂.points
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
n : ℕ
inst✝ : CharZero k
s : Simplex k P n
i : Fin (n + 1)
⊢ {i}ᶜ ≠ Set.univ

Tactic:
by simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : FiniteDimensional 𝕜 E
inst✝ : FiniteDimensional 𝕜 F
A : E →ₗ[𝕜] F
⊢ (innerₛₗ 𝕜).flip.IsAdjointPair (innerₛₗ 𝕜).flip ⇑A ⇑(adjoint A)

Tactic:
by
  intro x y
  simp [adjoint_inner_left]

Goal state after:
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
⊢ im ⟪x, (adjoint T) (T x)⟫_𝕜 = 0

Tactic:
simp only [adjoint_inner_right, inner_self_eq_norm_sq_to_K]
  norm_cast

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
H : Type u_5
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : CompleteSpace H
K : Type u_6
inst✝² : NormedAddCommGroup K
inst✝¹ : InnerProductSpace 𝕜 K
inst✝ : CompleteSpace K
e : H ≃ₗᵢ[𝕜] K
⊢ adjoint ↑↑e = adjoint ↑↑e ∘SL ↑↑e ∘SL ↑↑e.symm

Tactic:
by simp

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

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝⁴ : DivisionRing k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
inst✝ : Fintype ι
p : ι → P
hi : AffineIndependent k p
n : ℕ
hc : #Finset.univ = n + 1
⊢ finrank k ↥(vectorSpan k (Set.range p)) = n

Tactic:
← Set.image_univ,

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝⁴ : DivisionRing k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
inst✝ : Fintype ι
p : ι → P
hi : AffineIndependent k p
n : ℕ
hc : #Finset.univ = n + 1
⊢ finrank k ↥(vectorSpan k (p '' Set.univ)) = n
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝⁴ : DivisionRing k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
inst✝ : Fintype ι
p : ι → P
n : ℕ
hc : Fintype.card ι = n + 1
⊢ finrank k ↥(vectorSpan k (Set.range p)) ≤ n

Tactic:
← Set.image_univ,

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝⁴ : DivisionRing k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
inst✝ : Fintype ι
p : ι → P
n : ℕ
hc : Fintype.card ι = n + 1
⊢ finrank k ↥(vectorSpan k (p '' Set.univ)) ≤ n
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)
⊢ ↑((s.restrict S hS).faceOppositeCentroid i) = s.faceOppositeCentroid i

Tactic:
by
  rw [eq_comm]
  have := Nonempty.map (AffineSubspace.inclusion hS) inferInstance
  have hf : Function.Injective (S.subtype) := by
    simp only [coe_subtype, Subtype.val_injective]
  exact (s.restrict S hS).faceOppositeCentroid_map S.subtype hf (i := i)

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

Goal state before:
pp: case inr.inr
k : Type u_1
V : Type u_2
inst✝² : DivisionRing k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
s t : Finset V
hs : AffineIndependent k Subtype.val
hst : ↑s ⊆ ↑(affineSpan k ↑t)
hs' : s.Nonempty
ht' : t.Nonempty
this✝ : Nonempty ↥s
this : Nonempty ↑↑t
direction_le : vectorSpan k (Set.range Subtype.val) ≤ vectorSpan k ↑t
⊢ #s ≤ #t

Tactic:
← @Subtype.range_coe _ (t : Set V)

Goal state after:
case inr.inr
k : Type u_1
V : Type u_2
inst✝² : DivisionRing k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
s t : Finset V
hs : AffineIndependent k Subtype.val
hst : ↑s ⊆ ↑(affineSpan k ↑t)
hs' : s.Nonempty
ht' : t.Nonempty
this✝ : Nonempty ↥s
this : Nonempty ↑↑t
direction_le : vectorSpan k (Set.range Subtype.val) ≤ vectorSpan k (Set.range Subtype.val)
⊢ #s ≤ #t
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: case inr.inr
k : Type u_1
V : Type u_2
inst✝² : DivisionRing k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
s t : Finset V
hs : AffineIndependent k Subtype.val
hst : affineSpan k ↑s < affineSpan k ↑t
hs' : s.Nonempty
ht' : t.Nonempty
this✝ : Nonempty ↥s
this : Nonempty ↑↑t
dir_lt : vectorSpan k (Set.range Subtype.val) < vectorSpan k ↑t
⊢ #s < #t

Tactic:
← @Subtype.range_coe _ (t : Set V)

Goal state after:
case inr.inr
k : Type u_1
V : Type u_2
inst✝² : DivisionRing k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
s t : Finset V
hs : AffineIndependent k Subtype.val
hst : affineSpan k ↑s < affineSpan k ↑t
hs' : s.Nonempty
ht' : t.Nonempty
this✝ : Nonempty ↥s
this : Nonempty ↑↑t
dir_lt : vectorSpan k (Set.range Subtype.val) < vectorSpan k (Set.range Subtype.val)
⊢ #s < #t
Module: Mathlib.Analysis.InnerProductSpace.Adjoint

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
m : Type u_5
n : Type u_6
inst✝⁴ : Fintype m
inst✝³ : DecidableEq m
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : FiniteDimensional 𝕜 E
v₁ : OrthonormalBasis n 𝕜 E
e : n ≃ m
f : E →ₗ[𝕜] E
i j : m
⊢ ((v₁.reindex e).repr (f ((v₁.reindex e) j))).ofLp i = (v₁.repr (f (v₁ (e.symm j)))).ofLp (e.symm i)

Tactic:
by simp

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

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
inst✝¹ : Fintype ι
p : ι → P
hi : AffineIndependent k p
sp : AffineSubspace k P
inst✝ : FiniteDimensional k ↥sp.direction
hle : affineSpan k (Set.range p) ≤ sp
hc : #Finset.univ = finrank k ↥sp.direction + 1
⊢ affineSpan k (Set.range p) = sp

Tactic:
← Set.image_univ,

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
inst✝¹ : Fintype ι
p : ι → P
hi : AffineIndependent k p
sp : AffineSubspace k P
inst✝ : FiniteDimensional k ↥sp.direction
hle : affineSpan k (p '' Set.univ) ≤ sp
hc : #Finset.univ = finrank k ↥sp.direction + 1
⊢ affineSpan k (p '' Set.univ) = sp
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

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
s : Set P
inst✝ : FiniteDimensional k ↥(vectorSpan k s)
h : Collinear k s ↔ Module.rank k ↥(vectorSpan k s) ≤ 1
⊢ Collinear k s ↔ finrank k ↥(vectorSpan k s) ≤ 1

Tactic:
rw [← finrank_eq_rank] at h

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
s : Set P
inst✝ : FiniteDimensional k ↥(vectorSpan k s)
h : Collinear k s ↔ ↑(finrank k ↥(vectorSpan k s)) ≤ 1
⊢ Collinear k s ↔ finrank k ↥(vectorSpan k s) ≤ 1
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.medial.points 0 ∈ Set.range s.medial.points

Tactic:
by simp

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
hmem1 : s.medial.points 0 ∈ affineSpan k (Set.range s.medial.points)
⊢ Set.range (s.faceOpposite 0).points ⊆ Set.range s.points

Tactic:
by simp

Goal state after:
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 (Set.range p) ↔ ¬Collinear k {p i₁, p i₂, p i₃}

Tactic:
← Set.image_univ,

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 '' Set.univ) ↔ ¬Collinear k {p i₁, p i₂, p i₃}
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
p₁ p₂ p₃ p : P
ha : AffineIndependent k ![p₁, p₂, p₃]
hcol : Collinear k {p₂, p₃, p}
hne : p₂ ≠ p
h : Collinear k {p₁, p₂, p}
⊢ p₂ ∈ {p₁, p₂, p}

Tactic:
by simp

Goal state after:
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
p₁ p₂ p₃ p : P
ha : AffineIndependent k ![p₁, p₂, p₃]
hcol : Collinear k {p₂, p₃, p}
hne : p₂ ≠ p
h : Collinear k {p₁, p₂, p}
⊢ p ∈ {p₁, p₂, p}

Tactic:
by simp

Goal state after:
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
p₁ p₂ p₃ p : P
ha : AffineIndependent k ![p₁, p₂, p₃]
hcol : Collinear k {p₂, p₃, p}
hne : p₂ ≠ p
h : Collinear k {p₁, p₂, p}
⊢ p₁ ∈ {p₁, p₂, p}

Tactic:
by simp

Goal state after: