mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x : F
r : ℝ
hx : x ≠ 0
hr : r < 0
⊢ r * (‖x‖ * ‖x‖) / -(r * (‖x‖ * ‖x‖)) = -1

Tactic:
div_neg_eq_neg_div,

Goal state after:
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x : F
r : ℝ
hx : x ≠ 0
hr : r < 0
⊢ -(r * (‖x‖ * ‖x‖) / (r * (‖x‖ * ‖x‖))) = -1
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
ι' : Type u_5
v : ι → E
hv : Orthonormal 𝕜 v
f : ι' → ι
hf : Function.Injective f
⊢ Orthonormal 𝕜 (v ∘ f)

Tactic:
rw [orthonormal_iff_ite] at hv ⊢
  intro i j
  convert! hv (f i) (f j) using 1
  simp [hf.eq_iff]

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

Goal state before:
pp: V : Type u_4
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℂ V
S T : V →ₗ[ℂ] V
⊢ (∀ (x : V), ⟪S x, x⟫_ℂ = ⟪T x, x⟫_ℂ) ↔ S = T

Tactic:
rw [← sub_eq_zero, ← inner_map_self_eq_zero]
  refine forall_congr' fun x => ?_
  rw [LinearMap.sub_apply, inner_sub_left, sub_eq_zero]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Reflection

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : InnerProductSpace ℝ F
K : Submodule 𝕜 E
inst✝ : K.HasOrthogonalProjection
x : E
w : ↥K := K.orthogonalProjectionOnto x
v : E := x - ↑w
this : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ ‖K.reflectionLinearEquiv x‖ = ‖x‖

Tactic:
convert norm_sub_eq_norm_add this

Goal state after:
case e'_2
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : InnerProductSpace ℝ F
K : Submodule 𝕜 E
inst✝ : K.HasOrthogonalProjection
x : E
w : ↥K := K.orthogonalProjectionOnto x
v : E := x - ↑w
this : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ K.reflectionLinearEquiv x = ↑w - v

case e'_3
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : InnerProductSpace ℝ F
K : Submodule 𝕜 E
inst✝ : K.HasOrthogonalProjection
x : E
w : ↥K := K.orthogonalProjectionOnto x
v : E := x - ↑w
this : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ x = ↑w + v
Module: Mathlib.Analysis.InnerProductSpace.Projection.Submodule

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
K₀ K₁ : Submodule 𝕜 E
inst✝¹ : K₀.HasOrthogonalProjection
inst✝ : K₁.HasOrthogonalProjection
⊢ K₁ᗮ ≤ K₀ᗮᗮ ↔ K₁ᗮ ≤ K₀

Tactic:
orthogonal_orthogonal

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
K₀ K₁ : Submodule 𝕜 E
inst✝¹ : K₀.HasOrthogonalProjection
inst✝ : K₁.HasOrthogonalProjection
⊢ K₁ᗮ ≤ K₀ ↔ K₁ᗮ ≤ K₀
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: case inr
𝕜 : Type u_1
𝕜' : Type u_2
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜' F
f : E →SL[σ] F
σ' : 𝕜' →+* 𝕜
inst✝³ : RingHomInvPair σ σ'
inst✝² : RingHomIsometric σ
inst✝¹ : RingHomIsometric σ'
inst✝ : CompleteSpace F
surj : Surjective ⇑f
A : ⋃ n, closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (⇑f '' ball 0 ↑n) = Set.univ
n : ℕ
a : F
ε : ℝ
εpos : ε > 0
H : ball a ε ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (⇑f '' ball 0 ↑n)
c : 𝕜
hc : 1 < ‖c‖
y : F
hy : y ≠ 0
hc' : 1 < ‖σ c‖
d : 𝕜'
hd : d ≠ 0
ydlt : ‖d • y‖ < ε / 2
dinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖σ c‖ * ‖y‖
δ : ℝ := ‖d‖ * ‖y‖ / 4
δpos : 0 < δ
this : a + d • y ∈ ball a ε
z₁ : F
z₁im : z₁ ∈ ⇑f '' ball 0 ↑n
x₁ : E
h₁ : dist (a + d • y) (f x₁) < δ
hx₁ : x₁ ∈ ball 0 ↑n
xz₁ : f x₁ = z₁
⊢ ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖

Tactic:
rw [mem_ball, dist_eq_norm, sub_zero] at hx₁

Goal state after:
case inr
𝕜 : Type u_1
𝕜' : Type u_2
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜' F
f : E →SL[σ] F
σ' : 𝕜' →+* 𝕜
inst✝³ : RingHomInvPair σ σ'
inst✝² : RingHomIsometric σ
inst✝¹ : RingHomIsometric σ'
inst✝ : CompleteSpace F
surj : Surjective ⇑f
A : ⋃ n, closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (⇑f '' ball 0 ↑n) = Set.univ
n : ℕ
a : F
ε : ℝ
εpos : ε > 0
H : ball a ε ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (⇑f '' ball 0 ↑n)
c : 𝕜
hc : 1 < ‖c‖
y : F
hy : y ≠ 0
hc' : 1 < ‖σ c‖
d : 𝕜'
hd : d ≠ 0
ydlt : ‖d • y‖ < ε / 2
dinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖σ c‖ * ‖y‖
δ : ℝ := ‖d‖ * ‖y‖ / 4
δpos : 0 < δ
this : a + d • y ∈ ball a ε
z₁ : F
z₁im : z₁ ∈ ⇑f '' ball 0 ↑n
x₁ : E
h₁ : dist (a + d • y) (f x₁) < δ
hx₁ : ‖x₁‖ < ↑n
xz₁ : f x₁ = z₁
⊢ ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖
Module: Mathlib.Analysis.InnerProductSpace.Projection.Submodule

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
K₀ K₁ : Submodule 𝕜 E
inst✝¹ : K₀.HasOrthogonalProjection
inst✝ : K₁.HasOrthogonalProjection
⊢ K₁ᗮᗮ ≤ K₀ᗮ ↔ K₁ ≤ K₀ᗮ

Tactic:
orthogonal_orthogonal

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
K₀ K₁ : Submodule 𝕜 E
inst✝¹ : K₀.HasOrthogonalProjection
inst✝ : K₁.HasOrthogonalProjection
⊢ K₁ ≤ K₀ᗮ ↔ K₁ ≤ K₀ᗮ
Module: Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional

Goal state before:
pp: case pos
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : FiniteDimensional ℝ F
n : ℕ
IH :
  ∀ (φ : F ≃ₗᵢ[ℝ] F),
    finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ n →
      ∃ l, l.length ≤ n ∧ φ = (List.map (fun v ↦ (ℝ ∙ v)ᗮ.reflection) l).prod
φ : F ≃ₗᵢ[ℝ] F
hn : finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ n + 1
W : Submodule ℝ F := (↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).ker
hW : ∀ w ∈ W, φ w = w
hn' : finrank ℝ ↥Wᗮ ≤ n
⊢ ∃ l, l.length ≤ n + 1 ∧ φ = (List.map (fun v ↦ (ℝ ∙ v)ᗮ.reflection) l).prod

Tactic:
obtain ⟨V, hV₁, hV₂⟩ := IH φ hn'

Goal state after:
case pos
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : FiniteDimensional ℝ F
n : ℕ
IH :
  ∀ (φ : F ≃ₗᵢ[ℝ] F),
    finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ n →
      ∃ l, l.length ≤ n ∧ φ = (List.map (fun v ↦ (ℝ ∙ v)ᗮ.reflection) l).prod
φ : F ≃ₗᵢ[ℝ] F
hn : finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ n + 1
W : Submodule ℝ F := (↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).ker
hW : ∀ w ∈ W, φ w = w
hn' : finrank ℝ ↥Wᗮ ≤ n
V : List F
hV₁ : V.length ≤ n
hV₂ : φ = (List.map (fun v ↦ (ℝ ∙ v)ᗮ.reflection) V).prod
⊢ ∃ l, l.length ≤ n + 1 ∧ φ = (List.map (fun v ↦ (ℝ ∙ v)ᗮ.reflection) l).prod
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
α : Type u_2
β : Type u_3
hp : Fact (1 ≤ p)
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f g : WithLp p (α × β)
⊢ 0 ≤ dist f g

Tactic:
rcases p.dichotomy with (rfl | h)

Goal state after:
case inl
𝕜 : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
hp : Fact (1 ≤ ∞)
f g : WithLp ∞ (α × β)
⊢ 0 ≤ dist f g

case inr
p : ℝ≥0∞
𝕜 : Type u_1
α : Type u_2
β : Type u_3
hp : Fact (1 ≤ p)
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f g : WithLp p (α × β)
h : 1 ≤ p.toReal
⊢ 0 ≤ dist f g
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
α : Type u_2
β : Type u_3
hp : Fact (1 ≤ p)
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f g : WithLp p (α × β)
⊢ edist f g = ENNReal.ofReal (dist f g)

Tactic:
rcases p.dichotomy with (rfl | h)

Goal state after:
case inl
𝕜 : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
hp : Fact (1 ≤ ∞)
f g : WithLp ∞ (α × β)
⊢ edist f g = ENNReal.ofReal (dist f g)

case inr
p : ℝ≥0∞
𝕜 : Type u_1
α : Type u_2
β : Type u_3
hp : Fact (1 ≤ p)
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f g : WithLp p (α × β)
h : 1 ≤ p.toReal
⊢ edist f g = ENNReal.ofReal (dist f g)
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: p : ℝ≥0∞
α : Type u_2
β : Type u_3
hp : Fact (1 ≤ p)
inst✝¹ : PseudoEMetricSpace α
inst✝ : PseudoEMetricSpace β
x y : WithLp p (α × β)
⊢ edist x.fst y.fst ≤ edist x y ∧ edist x.snd y.snd ≤ edist x y

Tactic:
rcases p.dichotomy with (rfl | h)

Goal state after:
case inl
α : Type u_2
β : Type u_3
inst✝¹ : PseudoEMetricSpace α
inst✝ : PseudoEMetricSpace β
hp : Fact (1 ≤ ∞)
x y : WithLp ∞ (α × β)
⊢ edist x.fst y.fst ≤ edist x y ∧ edist x.snd y.snd ≤ edist x y

case inr
p : ℝ≥0∞
α : Type u_2
β : Type u_3
hp : Fact (1 ≤ p)
inst✝¹ : PseudoEMetricSpace α
inst✝ : PseudoEMetricSpace β
x y : WithLp p (α × β)
h : 1 ≤ p.toReal
⊢ edist x.fst y.fst ≤ edist x y ∧ edist x.snd y.snd ≤ edist x y
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: p : ℝ≥0∞
α : Type u_2
β : Type u_3
hp : Fact (1 ≤ p)
inst✝¹ : PseudoEMetricSpace α
inst✝ : PseudoEMetricSpace β
x y : WithLp p (α × β)
⊢ edist x y ≤ ↑(2 ^ (1 / p).toReal) * edist x.ofLp y.ofLp

Tactic:
rcases p.dichotomy with (rfl | h)

Goal state after:
case inl
α : Type u_2
β : Type u_3
inst✝¹ : PseudoEMetricSpace α
inst✝ : PseudoEMetricSpace β
hp : Fact (1 ≤ ∞)
x y : WithLp ∞ (α × β)
⊢ edist x y ≤ ↑(2 ^ (1 / ∞).toReal) * edist x.ofLp y.ofLp

case inr
p : ℝ≥0∞
α : Type u_2
β : Type u_3
hp : Fact (1 ≤ p)
inst✝¹ : PseudoEMetricSpace α
inst✝ : PseudoEMetricSpace β
x y : WithLp p (α × β)
h : 1 ≤ p.toReal
⊢ edist x y ≤ ↑(2 ^ (1 / p).toReal) * edist x.ofLp y.ofLp
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
α : Type u_2
β : Type u_3
hp : Fact (1 ≤ p)
inst✝¹ : SeminormedAddCommGroup α
inst✝ : SeminormedAddCommGroup β
x y : WithLp p (α × β)
⊢ dist x y = ‖-x + y‖

Tactic:
rcases p.dichotomy with (rfl | h)

Goal state after:
case inl
𝕜 : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : SeminormedAddCommGroup α
inst✝ : SeminormedAddCommGroup β
hp : Fact (1 ≤ ∞)
x y : WithLp ∞ (α × β)
⊢ dist x y = ‖-x + y‖

case inr
p : ℝ≥0∞
𝕜 : Type u_1
α : Type u_2
β : Type u_3
hp : Fact (1 ≤ p)
inst✝¹ : SeminormedAddCommGroup α
inst✝ : SeminormedAddCommGroup β
x y : WithLp p (α × β)
h : 1 ≤ p.toReal
⊢ dist x y = ‖-x + y‖
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : SeminormedAddCommGroup α
inst✝ : SeminormedAddCommGroup β
f : WithLp ∞ (α × β)
⊢ max ‖f.ofLp.1‖₊ ‖f.ofLp.2‖₊ = max ‖f.fst‖₊ ‖f.snd‖₊

Tactic:
ofLp_fst,

Goal state after:
α : Type u_2
β : Type u_3
inst✝¹ : SeminormedAddCommGroup α
inst✝ : SeminormedAddCommGroup β
f : WithLp ∞ (α × β)
⊢ max ‖f.fst‖₊ ‖f.ofLp.2‖₊ = max ‖f.fst‖₊ ‖f.snd‖₊
Module: Mathlib.Data.Matrix.Reflection

Goal state before:
pp: α : Type u_1
m n : ℕ
A : Matrix (Fin m) (Fin (n + 1)) α
i : Fin (n + 1)
j : Fin m
⊢ A.transposeᵣ i j = Aᵀ i j

Tactic:
simp_rw [transposeᵣ, transposeᵣ_eq]
      refine i.cases ?_ fun i => ?_
      · dsimp
        rw [FinVec.map_eq, Function.comp_apply]
      · simp only [of_apply, Matrix.cons_val_succ]
        rfl

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
ι : Type u_2
β : ι → Type u_4
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → Zero (β i)
i : ι
a : β i
⊢ Pi.single i a i = a

Tactic:
Pi.single_eq_same

Goal state after:
p : ℝ≥0∞
ι : Type u_2
β : ι → Type u_4
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → Zero (β i)
i : ι
a : β i
⊢ a = a
Module: Mathlib.LinearAlgebra.UnitaryGroup

Goal state before:
pp: n : Type u
inst✝³ : DecidableEq n
inst✝² : Fintype n
α : Type v
inst✝¹ : CommRing α
inst✝ : StarRing α
A✝ : Matrix n n α
A : ↥(unitaryGroup n α)
x : n → α
⊢ (toLin' (A⁻¹ * A)) x = x

Tactic:
rw [inv_mul_cancel, toLin'_one, id_apply]

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜' F
σ' : 𝕜' →+* 𝕜
inst✝⁵ : RingHomInvPair σ σ'
inst✝⁴ : RingHomIsometric σ
inst✝³ : RingHomIsometric σ'
inst✝² : CompleteSpace F
inst✝¹ : CompleteSpace E
inst✝ : RingHomInvPair σ' σ
e : E ≃ₛₗ[σ] F
h :
  Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
    ⇑e
⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
    ⇑e.symm

Tactic:
rw [continuous_def]
  intro s hs
  rw [← e.image_eq_preimage_symm]
  rw [← e.coe_coe] at h ⊢
  exact ContinuousLinearMap.isOpenMap (σ := σ) ⟨_, h⟩ e.surjective s hs

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
inst✝³ : Fact (1 ≤ p)
inst✝² : (i : ι) → PseudoMetricSpace (α i)
inst✝¹ : (i : ι) → PseudoEMetricSpace (β i)
inst✝ : Fintype ι
f g : PiLp p α
⊢ 0 ≤ dist f g

Tactic:
rcases p.dichotomy with (rfl | h)

Goal state after:
case inl
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
inst✝³ : (i : ι) → PseudoMetricSpace (α i)
inst✝² : (i : ι) → PseudoEMetricSpace (β i)
inst✝¹ : Fintype ι
inst✝ : Fact (1 ≤ ∞)
f g : PiLp ∞ α
⊢ 0 ≤ dist f g

case inr
p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
inst✝³ : Fact (1 ≤ p)
inst✝² : (i : ι) → PseudoMetricSpace (α i)
inst✝¹ : (i : ι) → PseudoEMetricSpace (β i)
inst✝ : Fintype ι
f g : PiLp p α
h : 1 ≤ p.toReal
⊢ 0 ≤ dist f g
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
inst✝³ : Fact (1 ≤ p)
inst✝² : (i : ι) → PseudoMetricSpace (α i)
inst✝¹ : (i : ι) → PseudoEMetricSpace (β i)
inst✝ : Fintype ι
f g : PiLp p α
⊢ edist f g = ENNReal.ofReal (dist f g)

Tactic:
rcases p.dichotomy with (rfl | h)

Goal state after:
case inl
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
inst✝³ : (i : ι) → PseudoMetricSpace (α i)
inst✝² : (i : ι) → PseudoEMetricSpace (β i)
inst✝¹ : Fintype ι
inst✝ : Fact (1 ≤ ∞)
f g : PiLp ∞ α
⊢ edist f g = ENNReal.ofReal (dist f g)

case inr
p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
inst✝³ : Fact (1 ≤ p)
inst✝² : (i : ι) → PseudoMetricSpace (α i)
inst✝¹ : (i : ι) → PseudoEMetricSpace (β i)
inst✝ : Fintype ι
f g : PiLp p α
h : 1 ≤ p.toReal
⊢ edist f g = ENNReal.ofReal (dist f g)
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
ι : Type u_2
β : ι → Type u_4
inst✝² : Fact (1 ≤ p)
inst✝¹ : (i : ι) → PseudoEMetricSpace (β i)
inst✝ : Fintype ι
x y : PiLp p β
i : ι
⊢ edist (x.ofLp i) (y.ofLp i) ≤ edist x y

Tactic:
rcases p.dichotomy with (rfl | h)

Goal state after:
case inl
ι : Type u_2
β : ι → Type u_4
inst✝² : (i : ι) → PseudoEMetricSpace (β i)
inst✝¹ : Fintype ι
i : ι
inst✝ : Fact (1 ≤ ∞)
x y : PiLp ∞ β
⊢ edist (x.ofLp i) (y.ofLp i) ≤ edist x y

case inr
p : ℝ≥0∞
ι : Type u_2
β : ι → Type u_4
inst✝² : Fact (1 ≤ p)
inst✝¹ : (i : ι) → PseudoEMetricSpace (β i)
inst✝ : Fintype ι
x y : PiLp p β
i : ι
h : 1 ≤ p.toReal
⊢ edist (x.ofLp i) (y.ofLp i) ≤ edist x y
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace E
f : E →L[𝕜] E
⊢ spectrum 𝕜 f = spectrum 𝕜 ↑f

Tactic:
ext μ
  rw [spectrum.mem_iff, spectrum.mem_iff, ContinuousLinearMap.isUnit_iff_isUnit_toLinearMap]
  rfl

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
ι : Type u_2
β : ι → Type u_4
inst✝² : Fact (1 ≤ p)
inst✝¹ : (i : ι) → PseudoEMetricSpace (β i)
inst✝ : Fintype ι
x y : WithLp p ((i : ι) → β i)
⊢ edist x y ≤ ↑(↑(Fintype.card ι) ^ (1 / p).toReal) * edist x.ofLp y.ofLp

Tactic:
rcases p.dichotomy with (rfl | h)

Goal state after:
case inl
ι : Type u_2
β : ι → Type u_4
inst✝² : (i : ι) → PseudoEMetricSpace (β i)
inst✝¹ : Fintype ι
inst✝ : Fact (1 ≤ ∞)
x y : WithLp ∞ ((i : ι) → β i)
⊢ edist x y ≤ ↑(↑(Fintype.card ι) ^ (1 / ∞).toReal) * edist x.ofLp y.ofLp

case inr
p : ℝ≥0∞
ι : Type u_2
β : ι → Type u_4
inst✝² : Fact (1 ≤ p)
inst✝¹ : (i : ι) → PseudoEMetricSpace (β i)
inst✝ : Fintype ι
x y : WithLp p ((i : ι) → β i)
h : 1 ≤ p.toReal
⊢ edist x y ≤ ↑(↑(Fintype.card ι) ^ (1 / p).toReal) * edist x.ofLp y.ofLp
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
hp : Fact (1 ≤ p)
inst✝¹ : Fintype ι
inst✝ : (i : ι) → SeminormedAddCommGroup (β i)
x y : PiLp p β
⊢ dist x y = ‖-x + y‖

Tactic:
rcases p.dichotomy with (rfl | h)

Goal state after:
case inl
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
inst✝¹ : Fintype ι
inst✝ : (i : ι) → SeminormedAddCommGroup (β i)
hp : Fact (1 ≤ ∞)
x y : PiLp ∞ β
⊢ dist x y = ‖-x + y‖

case inr
p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
hp : Fact (1 ≤ p)
inst✝¹ : Fintype ι
inst✝ : (i : ι) → SeminormedAddCommGroup (β i)
x y : PiLp p β
h : 1 ≤ p.toReal
⊢ dist x y = ‖-x + y‖
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
hp✝ : Fact (1 ≤ p)
inst✝¹ : Fintype ι
inst✝ : (i : ι) → SeminormedAddCommGroup (β i)
x y : PiLp p β
h : 1 ≤ ∞.toReal
hp : p = ∞
⊢ False

Tactic:
ENNReal.toReal_top

Goal state after:
p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
hp✝ : Fact (1 ≤ p)
inst✝¹ : Fintype ι
inst✝ : (i : ι) → SeminormedAddCommGroup (β i)
x y : PiLp p β
h : 1 ≤ 0
hp : p = ∞
⊢ False
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Basic

Goal state before:
pp: a r : ℝ
⊢ volume (closedBall a r) = ofReal (2 * r)

Tactic:
by
  rw [closedBall_eq_Icc, volume_Icc, ← sub_add, add_sub_cancel_left, two_mul]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Basic

Goal state before:
pp: case inl
a : ℝ
⊢ volume univ = 2 * ∞

Tactic:
volume_univ,

Goal state after:
case inl
a : ℝ
⊢ ∞ = 2 * ∞
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
hp : Fact (1 ≤ p)
inst✝⁸ : Fintype ι
inst✝⁷ : Semiring 𝕜
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (α i)
inst✝⁵ : (i : ι) → SeminormedAddCommGroup (β i)
inst✝⁴ : (i : ι) → Module 𝕜 (α i)
inst✝³ : (i : ι) → Module 𝕜 (β i)
c : 𝕜
ι' : Type u_5
inst✝² : Fintype ι'
E : Type u_6
inst✝¹ : SeminormedAddCommGroup E
inst✝ : Module 𝕜 E
e : ι ≃ ι'
x' : PiLp p fun x ↦ E
⊢ ‖(WithLp.linearEquiv p 𝕜 (ι → E) ≪≫ₗ
          (LinearEquiv.piCongrLeft' 𝕜 (fun x ↦ E) e ≪≫ₗ (WithLp.linearEquiv p 𝕜 (ι' → E)).symm))
        x'‖ =
    ‖x'‖

Tactic:
rcases p.dichotomy with (rfl | h)

Goal state after:
case inl
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
inst✝⁸ : Fintype ι
inst✝⁷ : Semiring 𝕜
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (α i)
inst✝⁵ : (i : ι) → SeminormedAddCommGroup (β i)
inst✝⁴ : (i : ι) → Module 𝕜 (α i)
inst✝³ : (i : ι) → Module 𝕜 (β i)
c : 𝕜
ι' : Type u_5
inst✝² : Fintype ι'
E : Type u_6
inst✝¹ : SeminormedAddCommGroup E
inst✝ : Module 𝕜 E
e : ι ≃ ι'
hp : Fact (1 ≤ ∞)
__LinearEquiv✝ : (PiLp ∞ fun x ↦ E) ≃ₗ[𝕜] PiLp ∞ fun x ↦ E :=
  WithLp.linearEquiv ∞ 𝕜 (ι → E) ≪≫ₗ
    (LinearEquiv.piCongrLeft' 𝕜 (fun x ↦ E) e ≪≫ₗ (WithLp.linearEquiv ∞ 𝕜 (ι' → E)).symm)
x' : PiLp ∞ fun x ↦ E
⊢ ‖(WithLp.linearEquiv ∞ 𝕜 (ι → E) ≪≫ₗ
          (LinearEquiv.piCongrLeft' 𝕜 (fun x ↦ E) e ≪≫ₗ (WithLp.linearEquiv ∞ 𝕜 (ι' → E)).symm))
        x'‖ =
    ‖x'‖

case inr
p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
hp : Fact (1 ≤ p)
inst✝⁸ : Fintype ι
inst✝⁷ : Semiring 𝕜
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (α i)
inst✝⁵ : (i : ι) → SeminormedAddCommGroup (β i)
inst✝⁴ : (i : ι) → Module 𝕜 (α i)
inst✝³ : (i : ι) → Module 𝕜 (β i)
c : 𝕜
ι' : Type u_5
inst✝² : Fintype ι'
E : Type u_6
inst✝¹ : SeminormedAddCommGroup E
inst✝ : Module 𝕜 E
e : ι ≃ ι'
x' : PiLp p fun x ↦ E
h : 1 ≤ p.toReal
⊢ ‖(WithLp.linearEquiv p 𝕜 (ι → E) ≪≫ₗ
          (LinearEquiv.piCongrLeft' 𝕜 (fun x ↦ E) e ≪≫ₗ (WithLp.linearEquiv p 𝕜 (ι' → E)).symm))
        x'‖ =
    ‖x'‖
Module: Mathlib.MeasureTheory.Measure.Haar.OfBasis

Goal state before:
pp: ι : Type u_1
E : Type u_3
inst✝² : Fintype ι
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
b : Basis ι ℝ E
x : E
⊢ (∃ t, ∀ (x_1 : ι), (0 ≤ t x_1 ∧ t x_1 ≤ 1) ∧ (b.repr x) x_1 = t x_1) ↔ ∀ (i : ι), 0 ≤ (b.repr x) i ∧ (b.repr x) i ≤ 1

Tactic:
aesop

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Basic

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
a b : ι → ℝ
h : a ≤ b
⊢ (volume (univ.pi fun i ↦ Ioc (a i) (b i))).toReal = ∏ i, (b i - a i)

Tactic:
simp only [volume_pi_Ioc, ENNReal.toReal_prod, ENNReal.toReal_ofReal (sub_nonneg.2 (h _))]

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: case top
ι : Type u_2
β : ι → Type u_4
inst✝² : Fintype ι
inst✝¹ : (i : ι) → SeminormedAddCommGroup (β i)
inst✝ : DecidableEq ι
i : ι
b : β i
this : Nonempty ι
hp : Fact (1 ≤ ∞)
⊢ ‖single ∞ i b‖₊ = ‖b‖₊

Tactic:
simp_rw [nnnorm_eq_ciSup]

Goal state after:
case top
ι : Type u_2
β : ι → Type u_4
inst✝² : Fintype ι
inst✝¹ : (i : ι) → SeminormedAddCommGroup (β i)
inst✝ : DecidableEq ι
i : ι
b : β i
this : Nonempty ι
hp : Fact (1 ≤ ∞)
⊢ ⨆ i_1, ‖(single ∞ i b).ofLp i_1‖₊ = ‖b‖₊
Module: Mathlib.MeasureTheory.Measure.Haar.OfBasis

Goal state before:
pp: case h.left.left
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁵ : Fintype ι
inst✝⁴ : Fintype ι'
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
v : Basis ι ℝ E
w : Basis ι' ℝ F
x : E × F
t : ι → ℝ
ht1 : t ∈ Icc 0 1
ht2 : x.1 = ∑ i, t i • v i
s : ι' → ℝ
hs1 : s ∈ Icc 0 1
hs2 : x.2 = ∑ i, s i • w i
⊢ ∀ (x : ι ⊕ ι'), 0 ≤ Sum.elim t s x

Tactic:
aesop

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.OfBasis

Goal state before:
pp: case h.left.right
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁵ : Fintype ι
inst✝⁴ : Fintype ι'
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
v : Basis ι ℝ E
w : Basis ι' ℝ F
x : E × F
t : ι → ℝ
ht1 : t ∈ Icc 0 1
ht2 : x.1 = ∑ i, t i • v i
s : ι' → ℝ
hs1 : s ∈ Icc 0 1
hs2 : x.2 = ∑ i, s i • w i
⊢ ∀ (x : ι ⊕ ι'), Sum.elim t s x ≤ 1

Tactic:
aesop

Goal state after:
Module: Mathlib.Analysis.BoxIntegral.Box.Basic

Goal state before:
pp: ι : Type u_1
I✝ J✝ : Box ι
x y : ι → ℝ
I J : WithBot (Box ι)
⊢ I ⊓ J ≤ I

Tactic:
rw [← withBotCoe_subset_iff, coe_inf]

Goal state after:
ι : Type u_1
I✝ J✝ : Box ι
x y : ι → ℝ
I J : WithBot (Box ι)
⊢ ↑I ∩ ↑J ⊆ ↑I