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

Goal state before:
pp: case h
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
ne : K.Nonempty
h₁ : IsComplete K
h₂ : Convex ℝ K
u : F
δ : ℝ := ⋯
this✝ : Nonempty ↑K := ⋯
zero_le_δ : 0 ≤ δ
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
w : ℕ → ↑K
hw : ∀ (n : ℕ), ‖u - ↑(w n)‖ < δ + 1 / (↑n + 1)
norm_tendsto : Tendsto (fun n ↦ ‖u - ↑(w n)‖) atTop (𝓝 δ)
b✝ : ℕ → ℝ := ⋯
p q N : ℕ
hp : N ≤ p
hq : N ≤ q
wp : F := ⋯
wq : F := ⋯
a : F := ⋯
b : F := ⋯
half : ℝ := ⋯
div : ℝ := ⋯
this : 4 * ‖u - half • (wq + wp)‖ * ‖u - half • (wq + wp)‖ + ‖wp - wq‖ * ‖wp - wq‖ = 2 * (‖a‖ * ‖a‖ + ‖b‖ * ‖b‖)
eq : δ ≤ ‖u - half • (wq + wp)‖
eq₂ : ‖a‖ ≤ δ + div
eq₂' : ‖b‖ ≤ δ + div
⊢ ‖↑(w p) - ↑(w q)‖ * ‖↑(w p) - ↑(w q)‖ ≤ √(b✝ N) * √(b✝ N)

Tactic:
rw [mul_self_sqrt]

Goal state after:
case h
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
ne : K.Nonempty
h₁ : IsComplete K
h₂ : Convex ℝ K
u : F
δ : ℝ := ⨅ w, ‖u - ↑w‖
this✝ : Nonempty ↑K := Set.Nonempty.to_subtype ne
zero_le_δ : 0 ≤ δ
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
w : ℕ → ↑K
hw : ∀ (n : ℕ), ‖u - ↑(w n)‖ < δ + 1 / (↑n + 1)
norm_tendsto : Tendsto (fun n ↦ ‖u - ↑(w n)‖) atTop (𝓝 δ)
b✝ : ℕ → ℝ := fun n ↦ 8 * δ * (1 / (↑n + 1)) + 4 * (1 / (↑n + 1)) * (1 / (↑n + 1))
p q N : ℕ
hp : N ≤ p
hq : N ≤ q
wp : F := ↑(w p)
wq : F := ↑(w q)
a : F := u - wq
b : F := u - wp
half : ℝ := 1 / 2
div : ℝ := 1 / (↑N + 1)
this : 4 * ‖u - half • (wq + wp)‖ * ‖u - half • (wq + wp)‖ + ‖wp - wq‖ * ‖wp - wq‖ = 2 * (‖a‖ * ‖a‖ + ‖b‖ * ‖b‖)
eq : δ ≤ ‖u - half • (wq + wp)‖
eq₂ : ‖a‖ ≤ δ + div
eq₂' : ‖b‖ ≤ δ + div
⊢ ‖↑(w p) - ↑(w q)‖ * ‖↑(w p) - ↑(w q)‖ ≤ b✝ N

case h
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
ne : K.Nonempty
h₁ : IsComplete K
h₂ : Convex ℝ K
u : F
δ : ℝ := ⨅ w, ‖u - ↑w‖
this✝ : Nonempty ↑K := Set.Nonempty.to_subtype ne
zero_le_δ : 0 ≤ δ
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
w : ℕ → ↑K
hw : ∀ (n : ℕ), ‖u - ↑(w n)‖ < δ + 1 / (↑n + 1)
norm_tendsto : Tendsto (fun n ↦ ‖u - ↑(w n)‖) atTop (𝓝 δ)
b✝ : ℕ → ℝ := fun n ↦ 8 * δ * (1 / (↑n + 1)) + 4 * (1 / (↑n + 1)) * (1 / (↑n + 1))
p q N : ℕ
hp : N ≤ p
hq : N ≤ q
wp : F := ↑(w p)
wq : F := ↑(w q)
a : F := u - wq
b : F := u - wp
half : ℝ := 1 / 2
div : ℝ := 1 / (↑N + 1)
this : 4 * ‖u - half • (wq + wp)‖ * ‖u - half • (wq + wp)‖ + ‖wp - wq‖ * ‖wp - wq‖ = 2 * (‖a‖ * ‖a‖ + ‖b‖ * ‖b‖)
eq : δ ≤ ‖u - half • (wq + wp)‖
eq₂ : ‖a‖ ≤ δ + div
eq₂' : ‖b‖ ≤ δ + div
⊢ 0 ≤ b✝ N
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: case mp
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
h₀ : x ≠ 0
h₀' : ¬↑‖x‖ = 0
h : ⟪x, y⟫ = ↑‖x‖ * ↑‖y‖
this : x = 0 ∨ y = (⟪x, y⟫ / ⟪x, x⟫) • x
⊢ (↑‖y‖ / ↑‖x‖) • x = y

Tactic:
rw [this.resolve_left h₀, h]

Goal state after:
case mp
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
h₀ : x ≠ 0
h₀' : ¬↑‖x‖ = 0
h : ⟪x, y⟫ = ↑‖x‖ * ↑‖y‖
this : x = 0 ∨ y = (⟪x, y⟫ / ⟪x, x⟫) • x
⊢ (↑‖(↑‖x‖ * ↑‖y‖ / ⟪x, x⟫) • x‖ / ↑‖x‖) • x = (↑‖x‖ * ↑‖y‖ / ⟪x, x⟫) • x
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
v : ι → E
hv : Orthonormal 𝕜 v
l : ι →₀ 𝕜
i : ι
⊢ (starRingEnd 𝕜) ⟪v i, (linearCombination 𝕜 v) l⟫ = (starRingEnd 𝕜) (l i)

Tactic:
hv.inner_right_finsupp

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
hv : Orthonormal 𝕜 v
l : ι →₀ 𝕜
i : ι
⊢ (starRingEnd 𝕜) (l i) = (starRingEnd 𝕜) (l i)
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
v : ι → E
hv : Function.Injective v
f : ι ≃ ↑(Set.range v) := Equiv.ofInjective v hv
h : Orthonormal 𝕜 v
⊢ Orthonormal 𝕜 Subtype.val

Tactic:
← Equiv.self_comp_ofInjective_symm hv

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
hv : Function.Injective v
f : ι ≃ ↑(Set.range v) := Equiv.ofInjective v hv
h : Orthonormal 𝕜 v
⊢ Orthonormal 𝕜 (v ∘ ⇑(Equiv.ofInjective v hv).symm)
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
ι : Type u_4
E' : Type u_7
inst✝¹ : SeminormedAddCommGroup E'
inst✝ : InnerProductSpace 𝕜 E'
v : ι → E
f : E →ₗᵢ[𝕜] E'
⊢ (∀ (i j : ι), ⟪(⇑f ∘ v) i, (⇑f ∘ v) j⟫ = if i = j then 1 else 0) ↔ ∀ (i j : ι), ⟪v i, v j⟫ = if i = j then 1 else 0

Tactic:
Function.comp_apply,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
ι : Type u_4
E' : Type u_7
inst✝¹ : SeminormedAddCommGroup E'
inst✝ : InnerProductSpace 𝕜 E'
v : ι → E
f : E →ₗᵢ[𝕜] E'
⊢ (∀ (i j : ι), ⟪f (v i), f (v j)⟫ = if i = j then 1 else 0) ↔ ∀ (i j : ι), ⟪v i, v j⟫ = if i = j then 1 else 0
Module: Mathlib.Topology.Algebra.Module.ClosedSubmodule

Goal state before:
pp: R : Type u_2
M : Type u_3
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : TopologicalSpace M
inst✝² : Module R M
inst✝¹ : ContinuousAdd M
inst✝ : ContinuousConstSMul R M
s : ClosedSubmodule R M
x✝ : M
⊢ x✝ ∈ closure ↑s ↔ x✝ ∈ s

Tactic:
closure_eq_iff_isClosed.mpr

Goal state after:
R : Type u_2
M : Type u_3
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : TopologicalSpace M
inst✝² : Module R M
inst✝¹ : ContinuousAdd M
inst✝ : ContinuousConstSMul R M
s : ClosedSubmodule R M
x✝ : M
⊢ x✝ ∈ ↑s ↔ x✝ ∈ s

R : Type u_2
M : Type u_3
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : TopologicalSpace M
inst✝² : Module R M
inst✝¹ : ContinuousAdd M
inst✝ : ContinuousConstSMul R M
s : ClosedSubmodule R M
x✝ : M
⊢ IsClosed ↑s
Module: Mathlib.Topology.Algebra.Module.ClosedSubmodule

Goal state before:
pp: R : Type u_2
M : Type u_3
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : TopologicalSpace M
inst✝² : Module R M
inst✝¹ : ContinuousAdd M
inst✝ : ContinuousConstSMul R M
s : ClosedSubmodule R M
x✝ : M
⊢ IsClosed ↑s

Tactic:
· exact s.isClosed'

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T S : E →ₗ[𝕜] E
hT : T.IsSymmetric
hS : S.IsSymmetric
x : E
⊢ ∀ (y : E), ⟪(T + S) x, y⟫ = ⟪x, (T + S) y⟫

Tactic:
y

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T S : E →ₗ[𝕜] E
hT : T.IsSymmetric
hS : S.IsSymmetric
x y : E
⊢ ⟪(T + S) x, y⟫ = ⟪x, (T + S) y⟫
Module: Mathlib.Analysis.InnerProductSpace.Symmetric

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T S : E →ₗ[𝕜] E
hT : T.IsSymmetric
hS : S.IsSymmetric
x : E
⊢ ∀ (y : E), ⟪(T - S) x, y⟫ = ⟪x, (T - S) y⟫

Tactic:
y

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T S : E →ₗ[𝕜] E
hT : T.IsSymmetric
hS : S.IsSymmetric
x y : E
⊢ ⟪(T - S) x, y⟫ = ⟪x, (T - S) y⟫
Module: Mathlib.Analysis.InnerProductSpace.Symmetric

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
c : 𝕜
hc : (starRingEnd 𝕜) c = c
T : E →ₗ[𝕜] E
hT : T.IsSymmetric
x : E
⊢ ∀ (y : E), ⟪(c • T) x, y⟫ = ⟪x, (c • T) y⟫

Tactic:
y

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
c : 𝕜
hc : (starRingEnd 𝕜) c = c
T : E →ₗ[𝕜] E
hT : T.IsSymmetric
x y : E
⊢ ⟪(c • T) x, y⟫ = ⟪x, (c • T) y⟫
Module: Mathlib.Analysis.InnerProductSpace.Symmetric

Goal state before:
pp: case mpr
V : Type u_3
inst✝¹ : SeminormedAddCommGroup V
inst✝ : InnerProductSpace ℂ V
T : V →ₗ[ℂ] V
h : ∀ (v : V), (starRingEnd ℂ) ⟪T v, v⟫_ℂ = ⟪T v, v⟫_ℂ
x : V
⊢ ∀ (y : V), ⟪T x, y⟫_ℂ = ⟪x, T y⟫_ℂ

Tactic:
y

Goal state after:
case mpr
V : Type u_3
inst✝¹ : SeminormedAddCommGroup V
inst✝ : InnerProductSpace ℂ V
T : V →ₗ[ℂ] V
h : ∀ (v : V), (starRingEnd ℂ) ⟪T v, v⟫_ℂ = ⟪T v, v⟫_ℂ
x y : V
⊢ ⟪T x, y⟫_ℂ = ⟪x, T y⟫_ℂ
Module: Mathlib.Analysis.InnerProductSpace.Orthogonal

Goal state before:
pp: 𝕜 : Type u_4
E : Type u_5
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
⊢ (↑K.closure)ᗮ = K.topologicalClosureᗮ

Tactic:
congr

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

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
T : E →ₗ[𝕜] E
hT : T.IsSymmetric
u : ℕ → E
x y : E
hu : Filter.Tendsto u Filter.atTop (nhds x)
hTu : Filter.Tendsto (⇑T ∘ u) Filter.atTop (nhds y)
hlhs : ∀ (k : ℕ), ⟪T (u k) - T x, y - T x⟫ = ⟪u k - x, T (y - T x)⟫
⊢ Filter.Tendsto (fun t ↦ ⟪(⇑T ∘ u) t - T x, y - T x⟫) Filter.atTop (nhds 0)

Tactic:
Function.comp_apply,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
T : E →ₗ[𝕜] E
hT : T.IsSymmetric
u : ℕ → E
x y : E
hu : Filter.Tendsto u Filter.atTop (nhds x)
hTu : Filter.Tendsto (⇑T ∘ u) Filter.atTop (nhds y)
hlhs : ∀ (k : ℕ), ⟪T (u k) - T x, y - T x⟫ = ⟪u k - x, T (y - T x)⟫
⊢ Filter.Tendsto (fun t ↦ ⟪T (u t) - T x, y - T x⟫) Filter.atTop (nhds 0)
Module: Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional

Goal state before:
pp: case zero
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : FiniteDimensional ℝ F
φ : F ≃ₗᵢ[ℝ] F
hn : finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ 0
this✝ : (↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).ker = ⊤
x : F
this : ↑(ContinuousLinearMap.id ℝ F - ↑↑φ) x = 0 x
⊢ 1 x = φ x

Tactic:
simpa only [sub_eq_zero, ContinuousLinearMap.toLinearMap_sub, LinearMap.sub_apply,
      LinearMap.zero_apply] using! this

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

Goal state before:
pp: case inl
α : Type u_2
β : Type u_3
inst✝¹ : PseudoEMetricSpace α
inst✝ : PseudoEMetricSpace β
f g : WithLp 0 (α × β)
⊢ edist f g = edist g f

pp: case inr.inl
α : Type u_2
β : Type u_3
inst✝¹ : PseudoEMetricSpace α
inst✝ : PseudoEMetricSpace β
f g : WithLp ∞ (α × β)
⊢ edist f g = edist g f

pp: case inr.inr
p : ℝ≥0∞
α : Type u_2
β : Type u_3
inst✝¹ : PseudoEMetricSpace α
inst✝ : PseudoEMetricSpace β
f g : WithLp p (α × β)
h : 0 < p.toReal
⊢ edist f g = edist g f

Tactic:
· simp only [prod_edist_eq_card, edist_comm]

Goal state after:
case inr.inl
α : Type u_2
β : Type u_3
inst✝¹ : PseudoEMetricSpace α
inst✝ : PseudoEMetricSpace β
f g : WithLp ∞ (α × β)
⊢ edist f g = edist g f

case inr.inr
p : ℝ≥0∞
α : Type u_2
β : Type u_3
inst✝¹ : PseudoEMetricSpace α
inst✝ : PseudoEMetricSpace β
f g : WithLp p (α × β)
h : 0 < p.toReal
⊢ edist f g = edist g f
Module: Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional

Goal state before:
pp: F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : FiniteDimensional ℝ F
n : ℕ
φ : F ≃ₗᵢ[ℝ] F
hn : finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ n
⊢ ∃ l, l.length ≤ n ∧ φ = (List.map (fun v ↦ (ℝ ∙ v)ᗮ.reflection) l).prod

Tactic:
induction n generalizing φ with

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : InnerProductSpace ℝ F
K : Submodule 𝕜 E
ι : Type u_4
inst✝² : DecidableEq ι
inst✝¹ : Fintype ι
V : ι → Submodule 𝕜 E
inst✝ : ∀ (i : ι), CompleteSpace ↥(V i)
hV : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ
h : iSup V = ⊤
x : E
this : (i : ι) → DecidableEq ↥(V i) := fun i ↦ Classical.decEq ↥(V i)
⊢ ∑ x_1, (AddSubmonoidClass.subtype (V x_1)) ((V x_1).orthogonalProjectionOnto x) = x

Tactic:
AddSubmonoidClass.coe_subtype

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : InnerProductSpace ℝ F
K : Submodule 𝕜 E
ι : Type u_4
inst✝² : DecidableEq ι
inst✝¹ : Fintype ι
V : ι → Submodule 𝕜 E
inst✝ : ∀ (i : ι), CompleteSpace ↥(V i)
hV : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ
h : iSup V = ⊤
x : E
this : (i : ι) → DecidableEq ↥(V i) := fun i ↦ Classical.decEq ↥(V i)
⊢ ∑ x_1, ↑((V x_1).orthogonalProjectionOnto x) = x
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 : WithLp p (α × β)
⊢ ∀ (y : WithLp p (α × β)), edist x.ofLp y.ofLp ≤ ↑1 * edist x y

Tactic:
y

Goal state after:
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.ofLp y.ofLp ≤ ↑1 * 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 : WithLp p (α × β)
⊢ ∀ (y : WithLp p (α × β)), edist x y ≤ ↑(2 ^ (1 / p).toReal) * edist x.ofLp y.ofLp

Tactic:
y

Goal state after:
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
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 : SeminormedAddCommGroup α
inst✝ : SeminormedAddCommGroup β
x : WithLp p (α × β)
⊢ ‖x.snd‖ₑ ≤ ‖x‖ₑ

Tactic:
by
  simpa using edist_snd_le x 0

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

Goal state before:
pp: case mpr
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
v : Set E
hv : Orthonormal 𝕜 Subtype.val
h : ∀ x ∈ (span 𝕜 v)ᗮ, x = 0
u : Set E
huv : v ⊆ u
hu : Orthonormal 𝕜 Subtype.val
x : E
hxu : x ∈ u
hxv : x ∉ v
⊢ x ∈ (span 𝕜 v)ᗮ

Tactic:
y

Goal state after:
case mpr
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
v : Set E
hv : Orthonormal 𝕜 Subtype.val
h : ∀ x ∈ (span 𝕜 v)ᗮ, x = 0
u : Set E
huv : v ⊆ u
hu : Orthonormal 𝕜 Subtype.val
x : E
hxu : x ∈ u
hxv : x ∉ v
y : E
⊢ y ∈ span 𝕜 v → ⟪y, x⟫_𝕜 = 0
Module: Mathlib.LinearAlgebra.UnitaryGroup

Goal state before:
pp: R : Type v
inst✝ : CommRing R
M : Matrix (Fin 2) (Fin 2) R
⊢ M ∈ specialOrthogonalGroup (Fin 2) R ↔ M 0 0 = M 1 1 ∧ M 0 1 = -M 1 0 ∧ M 0 0 ^ 2 + M 0 1 ^ 2 = 1

Tactic:
rw [← M.etaExpand_eq]
  exact of_mem_specialOrthogonalGroup_fin_two_iff

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

Goal state before:
pp: 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

Tactic:
simp only [dist]

Goal state after:
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 ≤
    if p = 0 then ↑⋯.toFinset.card
    else
      if p = ∞ then ⨆ i, dist (f.ofLp i) (g.ofLp i) else (∑ i, dist (f.ofLp i) (g.ofLp i) ^ p.toReal) ^ (1 / p.toReal)
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: 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
cancel : p.toReal * (1 / p.toReal) = 1
⊢ edist (x.ofLp i) (y.ofLp i) ≤ edist x y

Tactic:
rw [edist_eq_sum (zero_lt_one.trans_le h)]

Goal state after:
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
cancel : p.toReal * (1 / p.toReal) = 1
⊢ edist (x.ofLp i) (y.ofLp i) ≤ (∑ i, edist (x.ofLp i) (y.ofLp i) ^ p.toReal) ^ (1 / p.toReal)
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 : WithLp p ((i : ι) → β i)
⊢ ∀ (y : WithLp p ((i : ι) → β i)), edist x y ≤ ↑(↑(Fintype.card ι) ^ (1 / p).toReal) * edist x.ofLp y.ofLp

Tactic:
y

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

Goal state before:
pp: ι : Type u_2
β : ι → Type u_4
inst✝¹ : Fintype ι
inst✝ : (i : ι) → SeminormedAddCommGroup (β i)
x : PiLp 1 β
⊢ ↑‖x‖₊ = ↑(∑ i, ‖x.ofLp i‖₊)

Tactic:
by push_cast; exact norm_eq_of_L1 x

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

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
E : Type u_3
inst✝³ : Fintype ι
inst✝² : Fintype ι'
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
v : ι → E
e : ι' ≃ ι
K : (ι' → ℝ) ≃ (ι → ℝ) := Equiv.piCongrLeft' (fun _a ↦ ℝ) e
this✝ : Icc 0 1 = ⇑K '' Icc 0 1
x : E
this : ∀ (z : ι' → ℝ), ∑ x, z (e.symm x) • v x = ∑ i, z i • v (e i)
⊢ x ∈ (fun t ↦ ∑ i, t i • (v ∘ ⇑e) i) '' Icc 0 1 ↔ x ∈ (fun t ↦ ∑ i, t i • v i) ∘ ⇑K '' Icc 0 1

Tactic:
Function.comp_apply,

Goal state after:
ι : Type u_1
ι' : Type u_2
E : Type u_3
inst✝³ : Fintype ι
inst✝² : Fintype ι'
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
v : ι → E
e : ι' ≃ ι
K : (ι' → ℝ) ≃ (ι → ℝ) := Equiv.piCongrLeft' (fun _a ↦ ℝ) e
this✝ : Icc 0 1 = ⇑K '' Icc 0 1
x : E
this : ∀ (z : ι' → ℝ), ∑ x, z (e.symm x) • v x = ∑ i, z i • v (e i)
⊢ x ∈ (fun a ↦ ∑ x, a x • v (e x)) '' Icc 0 1 ↔ x ∈ (fun a ↦ ∑ i, K a i • v i) '' Icc 0 1
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: case inl
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
f : E →ₗ[ℝ] E
s : Set E
hf : LinearMap.det f ≠ 0
g : E ≃L[ℝ] E := (f.equivOfDetNeZero hf).toContinuousLinearEquiv
⊢ ENNReal.ofReal |LinearMap.det ↑↑g.symm.symm| * μ s = ENNReal.ofReal |LinearMap.det f| * μ s

Tactic:
congr

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

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : Nontrivial E
x : E
r : ℝ
hr : 0 ≤ r
⊢ μ (ball x r) = ENNReal.ofReal (r ^ finrank ℝ E) * μ (ball 0 1)

Tactic:
← addHaar_ball_mul μ x hr,

Goal state after:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : Nontrivial E
x : E
r : ℝ
hr : 0 ≤ r
⊢ μ (ball x r) = μ (ball x (r * 1))
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Basic

Goal state before:
pp: μ : Measure ℝ
inst✝ : NullSingletonClass μ
s : Set ℝ
p : ℝ → Prop
h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x
T : ↑s × ↑s → Set ℝ := fun p ↦ Ioo ↑p.1 ↑p.2
u : Set ℝ := ⋃ i, T i
hfinite : (s \ u).Finite
A : Set (↑s × ↑s)
A_count : A.Countable
hA : ⋃ i ∈ A, T i = ⋃ i, T i
M : ∀ᵐ (x : ℝ) ∂μ, x ∉ s \ u
⊢ ∀ᵐ (x : ℝ) ∂μ, ∀ i ∈ A, x ∈ s ∩ T i → p x

Tactic:
ae_ball_iff A_count

Goal state after:
μ : Measure ℝ
inst✝ : NullSingletonClass μ
s : Set ℝ
p : ℝ → Prop
h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x
T : ↑s × ↑s → Set ℝ := fun p ↦ Ioo ↑p.1 ↑p.2
u : Set ℝ := ⋃ i, T i
hfinite : (s \ u).Finite
A : Set (↑s × ↑s)
A_count : A.Countable
hA : ⋃ i ∈ A, T i = ⋃ i, T i
M : ∀ᵐ (x : ℝ) ∂μ, x ∉ s \ u
⊢ ∀ i ∈ A, ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ T i → p x
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: ι : Type u_2
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι ℝ (ι → ℝ)
⊢ ENNReal.ofReal |(Pi.basisFun ℝ ι).det ⇑b| * (Measure.pi fun x ↦ volume) (Set.univ.pi fun x ↦ Set.Ico 0 1) =
    ENNReal.ofReal |(Matrix.of ⇑b).det|

Tactic:
Measure.pi_pi,

Goal state after:
ι : Type u_2
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι ℝ (ι → ℝ)
⊢ ENNReal.ofReal |(Pi.basisFun ℝ ι).det ⇑b| * ∏ i, volume (Set.Ico 0 1) = ENNReal.ofReal |(Matrix.of ⇑b).det|
Module: Mathlib.Analysis.BoxIntegral.Box.Basic

Goal state before:
pp: ι : Type u_1
l₁ u₁ : ι → ℝ
h₁ : ∀ (i : ι), l₁ i < u₁ i
l₂ u₂ : ι → ℝ
h₂ : ∀ (i : ι), l₂ i < u₂ i
h : (l₂ ≤ l₁ ∧ u₁ ≤ u₂) ∧ l₁ ≤ l₂ ∧ u₂ ≤ u₁
⊢ { lower := l₁, upper := u₁, lower_lt_upper := h₁ } = { lower := l₂, upper := u₂, lower_lt_upper := h₂ }

Tactic:
congr

Goal state after:
case e_lower
ι : Type u_1
l₁ u₁ : ι → ℝ
h₁ : ∀ (i : ι), l₁ i < u₁ i
l₂ u₂ : ι → ℝ
h₂ : ∀ (i : ι), l₂ i < u₂ i
h : (l₂ ≤ l₁ ∧ u₁ ≤ u₂) ∧ l₁ ≤ l₂ ∧ u₂ ≤ u₁
⊢ l₁ = l₂

case e_upper
ι : Type u_1
l₁ u₁ : ι → ℝ
h₁ : ∀ (i : ι), l₁ i < u₁ i
l₂ u₂ : ι → ℝ
h₂ : ∀ (i : ι), l₂ i < u₂ i
h : (l₂ ≤ l₁ ∧ u₁ ≤ u₂) ∧ l₁ ≤ l₂ ∧ u₂ ≤ u₁
⊢ u₁ = u₂
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: case refine_1
K : Type u_1
inst✝⁹ : NormedField K
inst✝⁸ : LinearOrder K
inst✝⁷ : IsStrictOrderedRing K
inst✝⁶ : HasSolidNorm K
inst✝⁵ : FloorRing K
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace K E
inst✝² : FiniteDimensional K E
inst✝¹ : ProperSpace E
L : Submodule ℤ E
inst✝ : DiscreteTopology ↥L
hs : IsZLattice K L
s : Set E
h_incl : s ⊆ ↑L
h_span : span K s = span K ↑L
h_lind : LinearIndependent K Subtype.val
b : Basis { x // x ∈ s } K E := Basis.mk h_lind ⋯
this✝ : Fintype ↑s
this : (fundamentalDomain b ∩ ↑L).Finite
x : E
h_mem : x ∈ ↑L
⊢ (Subtype.val ∘ ⇑(quotientEquiv b)) ((span ℤ (Set.range ⇑b)).mkQ x) ∈ fundamentalDomain b ∩ ↑L

Tactic:
Function.comp_apply,

Goal state after:
case refine_1
K : Type u_1
inst✝⁹ : NormedField K
inst✝⁸ : LinearOrder K
inst✝⁷ : IsStrictOrderedRing K
inst✝⁶ : HasSolidNorm K
inst✝⁵ : FloorRing K
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace K E
inst✝² : FiniteDimensional K E
inst✝¹ : ProperSpace E
L : Submodule ℤ E
inst✝ : DiscreteTopology ↥L
hs : IsZLattice K L
s : Set E
h_incl : s ⊆ ↑L
h_span : span K s = span K ↑L
h_lind : LinearIndependent K Subtype.val
b : Basis { x // x ∈ s } K E := Basis.mk h_lind ⋯
this✝ : Fintype ↑s
this : (fundamentalDomain b ∩ ↑L).Finite
x : E
h_mem : x ∈ ↑L
⊢ ↑((quotientEquiv b) ((span ℤ (Set.range ⇑b)).mkQ x)) ∈ fundamentalDomain b ∩ ↑L