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:
neg_mul,

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: case mpr.left
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
ι : Type u_4
inst✝ : DecidableEq ι
v : ι → E
h : ∀ (i j : ι), ⟪v i, v j⟫ = if i = j then 1 else 0
i : ι
h' : ‖v i‖ ^ 2 = 1 ^ 2
h₁ : 0 ≤ ‖v i‖
h₂ : 0 ≤ 1
⊢ ‖v i‖ = 1

Tactic:
sq_eq_sq₀ h₁ h₂

Goal state after:
case mpr.left
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
ι : Type u_4
inst✝ : DecidableEq ι
v : ι → E
h : ∀ (i j : ι), ⟪v i, v j⟫ = if i = j then 1 else 0
i : ι
h' : ‖v i‖ = 1
h₁ : 0 ≤ ‖v i‖
h₂ : 0 ≤ 1
⊢ ‖v i‖ = 1
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
h : Convex ℝ K
u v : F
hv : v ∈ K
this✝ : Nonempty ↑K := Nonempty.intro ⟨v, hv⟩
eq : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
w : F
hw : w ∈ K
δ : ℝ := ⨅ w, ‖u - ↑w‖
p : ℝ := ⟪u - v, w - v⟫_ℝ
q : ℝ := ‖w - v‖ ^ 2
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
this : ∀ (θ : ℝ), 0 < θ → θ ≤ 1 → 2 * p ≤ θ * 0
hq : q = 0
⊢ 0 < 1

Tactic:
by simp

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

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
h : Convex ℝ K
u v : F
hv : v ∈ K
this✝ : Nonempty ↑K := Nonempty.intro ⟨v, hv⟩
eq : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
w : F
hw : w ∈ K
δ : ℝ := ⨅ w, ‖u - ↑w‖
p : ℝ := ⟪u - v, w - v⟫_ℝ
q : ℝ := ‖w - v‖ ^ 2
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
this : ∀ (θ : ℝ), 0 < θ → θ ≤ 1 → 2 * p ≤ θ * 0
hq : q = 0
⊢ 1 ≤ 1

Tactic:
by simp

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

Goal state before:
pp: E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
x : E
⊢ 0 ≤ ((innerₗ E) x) x

Tactic:
by simp

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

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
h : Convex ℝ K
u v : F
hv : v ∈ K
this✝ : Nonempty ↑K := Nonempty.intro ⟨v, hv⟩
eq : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
w : F
hw : w ∈ K
δ : ℝ := ⨅ w, ‖u - ↑w‖
p : ℝ := ⟪u - v, w - v⟫_ℝ
q : ℝ := ‖w - v‖ ^ 2
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
this : ∀ (θ : ℝ), 0 < θ → θ ≤ 1 → 2 * p ≤ θ * q
hq : ¬q = 0
q_pos : 0 < q
hp : 0 < ⟪u - v, w - v⟫_ℝ
θ : ℝ := min 1 (p / q)
eq₁ : θ * q ≤ p
⊢ 0 < 1

Tactic:
by simp

Goal state after:
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
η : Type u_5
s : η → Set E
hs : Directed (fun x1 x2 ↦ x1 ⊆ x2) s
h : ∀ (i : η), Orthonormal 𝕜 fun x ↦ ↑x
⊢ ∀ v ∈ ⋃ i, s i, ∀ w ∈ ⋃ i, s i, ⟪v, w⟫ = if v = w then 1 else 0

Tactic:
rintro x ⟨_, ⟨i, rfl⟩, hxi⟩ y ⟨_, ⟨j, rfl⟩, hyj⟩

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
η : Type u_5
s : η → Set E
hs : Directed (fun x1 x2 ↦ x1 ⊆ x2) s
h : ∀ (i : η), Orthonormal 𝕜 fun x ↦ ↑x
x : E
i : η
hxi : x ∈ (fun i ↦ s i) i
y : E
j : η
hyj : y ∈ (fun i ↦ s i) j
⊢ ⟪x, y⟫ = if x = y then 1 else 0
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Submodule ℝ F
u v : F
hv : v ∈ K
⊢ ‖u - v‖ = ⨅ w, ‖u - ↑w‖ → ∀ w ∈ K, ⟪u - v, w⟫_ℝ = 0

Tactic:
intro h
      have h : ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0 := by
        rwa [norm_eq_iInf_iff_real_inner_le_zero] at h
        exacts [K.convex, hv]
      intro w hw
      have le : ⟪u - v, w⟫_ℝ ≤ 0 := by
        let w' := w + v
        have : w' ∈ K := Submodule.add_mem _ hw hv
        have h₁ := h w' this
        have h₂ : w' - v = w := by
          simp only [w', add_neg_cancel_right, sub_eq_add_neg]
        rw [h₂] at h₁
        exact h₁
      have ge : ⟪u - v, w⟫_ℝ ≥ 0 := by
        let w'' := -w + v
        have : w'' ∈ K := Submodule.add_mem _ (Submodule.neg_mem _ hw) hv
        have h₁ := h w'' this
        have h₂ : w'' - v = -w := by
          simp only [w'', add_neg_cancel_right, sub_eq_add_neg]
        rw [h₂, inner_neg_right] at h₁
        linarith
      exact le_antisymm le ge

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
⊢ ∀ (x : E), ‖(innerSL 𝕜) x‖ ≤ 1 * ‖x‖

Tactic:
by simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
u v : E
hv : v ∈ K
this : InnerProductSpace ℝ E := rclikeToReal 𝕜 E
K' : Submodule ℝ E := restrictScalars ℝ K
H : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
A : ∀ w ∈ K, re ⟪u - v, w⟫_𝕜 = 0
w : E
hw : w ∈ K
⊢ re (-I * ⟪u - v, w⟫_𝕜) = im ⟪u - v, w⟫_𝕜

Tactic:
by simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : CompleteSpace E
f : (i : ι) → G i
hf : ∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ⊆ m → ∀ (n : Finset ι), N ⊆ n → ‖∑ i ∈ m, (V i) (f i) - ∑ i ∈ n, (V i) (f i)‖ < ε
ε : ℝ
hε : ε > 0
a : Finset ι
H : ∀ (m : Finset ι), a ⊆ m → ∀ (n : Finset ι), a ⊆ n → ‖∑ i ∈ m, (V i) (f i) - ∑ i ∈ n, (V i) (f i)‖ < √ε
s₁ : Finset ι
hs₁ : a ⊆ s₁
s₂ : Finset ι
hs₂ : a ⊆ s₂
this : ∀ (i : ι), 0 ≤ ‖f i‖ ^ 2
⊢ |‖∑ i ∈ s₁, (V i) (f i) - ∑ i ∈ s₂, (V i) (f i)‖| < √ε

Tactic:
abs_of_nonneg (norm_nonneg _)

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : CompleteSpace E
f : (i : ι) → G i
hf : ∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ⊆ m → ∀ (n : Finset ι), N ⊆ n → ‖∑ i ∈ m, (V i) (f i) - ∑ i ∈ n, (V i) (f i)‖ < ε
ε : ℝ
hε : ε > 0
a : Finset ι
H : ∀ (m : Finset ι), a ⊆ m → ∀ (n : Finset ι), a ⊆ n → ‖∑ i ∈ m, (V i) (f i) - ∑ i ∈ n, (V i) (f i)‖ < √ε
s₁ : Finset ι
hs₁ : a ⊆ s₁
s₂ : Finset ι
hs₂ : a ⊆ s₂
this : ∀ (i : ι), 0 ≤ ‖f i‖ ^ 2
⊢ ‖∑ i ∈ s₁, (V i) (f i) - ∑ i ∈ s₂, (V i) (f i)‖ < √ε
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 y : E
⊢ ⟪(T - S) x, y⟫ = ⟪x, (T - S) y⟫

Tactic:
rw [sub_apply, inner_sub_left, hT x y, hS x y, ← inner_sub_right, sub_apply]

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

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
v w : F
h : ‖v‖ = ‖w‖
R : F ≃ₗᵢ[ℝ] F := (ℝ ∙ (v - w))ᗮ.reflection
this : R v + R v = w + w
⊢ 2 ≠ 0

Tactic:
by simp

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)
⊢ ⟪y - T x, y - T x⟫ = 0

Tactic:
have hlhs : ∀ k : ℕ, ⟪T (u k) - T x, y - T x⟫ = ⟪u k - x, T (y - T x)⟫ := by
    intro k
    rw [← T.map_sub, hT]

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)⟫
⊢ ⟪y - T x, y - T x⟫ = 0
Module: Mathlib.Analysis.InnerProductSpace.Symmetric

Goal state before:
pp: case refine_2
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
U V : Submodule 𝕜 E
hUV : IsCompl U V
h : ∀ u ∈ U, ∀ v ∈ V, ⟪u, v⟫ = 0
x y : E
⊢ ⟪(U.subtype ∘ₗ U.projectionOnto V hUV) x, (U.projection V hUV) y + (V.projection U ⋯) y⟫ =
    ⟪(U.projection V hUV) x + (V.projection U ⋯) x, (U.subtype ∘ₗ U.projectionOnto V hUV) y⟫

Tactic:
simp [inner_add_right, inner_add_left, h, inner_eq_zero_symm]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : InnerProductSpace ℝ F
K : Submodule 𝕜 E
v : E
⊢ v - v ∈ ⊤ᗮ

Tactic:
by simp

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

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
K : Submodule 𝕜 E
inst✝ : K.HasOrthogonalProjection
x : E
⊢ ↑K.starProjection x ∈ K

Tactic:
by simp

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

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
K : Submodule 𝕜 E
inst✝ : K.HasOrthogonalProjection
v : E
⊢ 1 * ‖v‖ = ‖v‖

Tactic:
by simp

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

Goal state before:
pp: case hvo
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
v w : E
⊢ ∀ w_1 ∈ 𝕜 ∙ v, ⟪↑‖v‖ ^ 2 • w - ⟪v, w⟫ • v, w_1⟫ = 0

Tactic:
rw [← Submodule.mem_orthogonal', Submodule.mem_orthogonal_singleton_iff_inner_left]

Goal state after:
case hvo
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
v w : E
⊢ ⟪↑‖v‖ ^ 2 • w - ⟪v, w⟫ • v, v⟫ = 0
Module: Mathlib.Analysis.InnerProductSpace.Projection.Basic

Goal state before:
pp: case neg
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
v w : E
hv : ¬v = 0
hv' : ‖v‖ ≠ 0
key : ((↑(‖v‖ ^ 2))⁻¹ * ↑(‖v‖ ^ 2)) • (𝕜 ∙ v).starProjection w = ((↑(‖v‖ ^ 2))⁻¹ * ⟪v, w⟫) • v
⊢ (𝕜 ∙ v).starProjection w = (⟪v, w⟫ / ↑(‖v‖ ^ 2)) • v

Tactic:
convert! key using 1 <;> match_scalars <;> field_simp [hv']

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

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
K : Submodule 𝕜 E
inst✝ : K.HasOrthogonalProjection
v : E
⊢ v = ↑(K.orthogonalProjectionOnto v) + (v - ↑(K.orthogonalProjectionOnto v))

Tactic:
by simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
p : E →ₗ[𝕜] E
hp : p.IsSymmetricProjection
this : p.range.HasOrthogonalProjection
x : E
⊢ p x ∈ p.range

Tactic:
by simp

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
f : E →SL[σ] F
σ' : 𝕜' →+* 𝕜
inst✝⁴ : RingHomInvPair σ σ'
inst✝³ : RingHomIsometric σ
inst✝² : RingHomIsometric σ'
inst✝¹ : CompleteSpace F
inst✝ : CompleteSpace E
surj : Surjective ⇑f
C : ℝ
C0 : C ≥ 0
g : F → E
hg : ∀ (y : F), dist (f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖
h : F → F := fun y ↦ y - f (g y)
hle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖
y : F
hnle : ∀ (n : ℕ), ‖h^[n] y‖ ≤ (1 / 2) ^ n * ‖y‖
u : ℕ → E := fun n ↦ g (h^[n] y)
ule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)
⊢ 0 ≤ 1 / 2

Tactic:
by simp

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

Goal state before:
pp: n : Type u_2
R : Type u_4
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommRing R
p : ℝ≥0∞
⊢ (toLpLin p p) ((toLpLin p p).symm LinearMap.id) = (toLpLin p p) 1

Tactic:
by simp

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

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
inst✝³ : Semiring 𝕜
η : Type u_5
ιs : η → Type u_6
Ms : η → Type u_7
inst✝² : (i : η) → AddCommGroup (Ms i)
inst✝¹ : (i : η) → Module 𝕜 (Ms i)
inst✝ : DecidableEq η
v : (j : η) → ιs j → Ms j
hs : ∀ (i : η), LinearIndependent 𝕜 (v i)
⊢ InjOn ⇑↑(WithLp.linearEquiv p 𝕜 ((j : η) → Ms j)).symm
    ↑(Submodule.span 𝕜 (range fun ji ↦ Pi.single ji.fst (v ji.fst ji.snd)))

Tactic:
by simp

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

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
inst✝¹ : DecidableEq ι
inst✝ : Ring 𝕜
⊢ InjOn ⇑↑(WithLp.linearEquiv p 𝕜 (ι → 𝕜)).symm ↑(Submodule.span 𝕜 (range fun i ↦ Pi.single i 1))

Tactic:
by simp

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

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : Ring 𝕜
inst✝³ : IsDomain 𝕜
M : Type u_5
inst✝² : AddCommGroup M
inst✝¹ : Module 𝕜 M
inst✝ : IsTorsionFree 𝕜 M
v : ι → M
hv : ∀ (i : ι), v i ≠ 0
⊢ InjOn ⇑↑(WithLp.linearEquiv p 𝕜 (ι → M)).symm ↑(Submodule.span 𝕜 (range fun i ↦ Pi.single i (v i)))

Tactic:
by simp

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
f✝ : E →SL[σ] 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
f : E →SL[σ] F
hinj : Injective ⇑f
hclo : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Set.range ⇑f)
x : E
⊢ f x ∈ (↑f).range

Tactic:
by simp

Goal state after:
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
p : ℝ≥0∞
⊢ toLp p (WithLp.fst 0, 0) = 0

Tactic:
by simp

Goal state after:
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
p : ℝ≥0∞
⊢ toLp p (0, WithLp.snd 0) = 0

Tactic:
by simp

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

Goal state before:
pp: case neg
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
hp : ¬p = 0
h✝ : ¬p = ∞
⊢ 0 ≤ (∑ i, dist (f.ofLp i) (g.ofLp i) ^ p.toReal) ^ (1 / p.toReal)

Tactic:
· exact rpow_nonneg (Fintype.sum_nonneg fun i ↦ by positivity) (1 / p.toReal)

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
h : 1 ≤ p.toReal
this : 0 < p.toReal
⊢ 0 ≤ 1 / p.toReal

Tactic:
by simp

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

Goal state before:
pp: ι : Type u_2
inst✝¹ : Fintype ι
β : ι → Type u_5
inst✝ : (i : ι) → SeminormedAddCommGroup (β i)
x : PiLp 2 β
⊢ ‖x‖ ^ 2 = ∑ i, ‖x.ofLp i‖ ^ 2

Tactic:
suffices ‖x‖₊ ^ 2 = ∑ i : ι, ‖x i‖₊ ^ 2 by
    simpa only [NNReal.coe_sum] using! congr_arg ((↑) : ℝ≥0 → ℝ) this

Goal state after:
ι : Type u_2
inst✝¹ : Fintype ι
β : ι → Type u_5
inst✝ : (i : ι) → SeminormedAddCommGroup (β i)
x : PiLp 2 β
⊢ ‖x‖₊ ^ 2 = ∑ i, ‖x.ofLp i‖₊ ^ 2
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Basic

Goal state before:
pp: a : ℝ
⊢ volume {a} = 0

Tactic:
by simp

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

Goal state before:
pp: r : ℝ≥0
⊢ ↑r = volume (Icc 0 ↑r)

Tactic:
by simp

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

Goal state before:
pp: a : ℝ
n : ℕ
⊢ ↑n = volume (Ioo a (a + ↑n))

Tactic:
by simp

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

Goal state before:
pp: a : ℝ
n : ℕ
⊢ ↑n = volume (Ioo (a - ↑n) a)

Tactic:
by simp

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

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
x y : ℝ
⊢ (volume.restrict (Icc x y)) univ < ∞

Tactic:
by simp

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

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
x y : ℝ
⊢ (volume.restrict (Ico x y)) univ < ∞

Tactic:
by simp

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

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
x y : ℝ
⊢ (volume.restrict (Ioc x y)) univ < ∞

Tactic:
by simp

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

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
x y : ℝ
⊢ (volume.restrict (Ioo x y)) univ < ∞

Tactic:
by simp

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

Goal state before:
pp: a : ℝ
h : a ≠ 0
s : Set ℝ
⊢ (Measure.map (fun x ↦ x * a) volume) s = ofReal |a⁻¹| * volume s

Tactic:
rw [map_volume_mul_right h]; rfl

Goal state after:
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
v : ι → E
t : ι → ℝ
ht : ∀ (i : ι), t i ∈ Icc (0 i) (1 i)
i : ι
⊢ (fun a ↦ a • v i) (t i) = (t • v) i

Tactic:
by simp

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

Goal state before:
pp: case h.right.snd
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.2 = (∑ i, Sum.elim t s i • (v.prod w) i).2

Tactic:
simp [hs2, Prod.snd_sum]

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

Goal state before:
pp: p : ℝ≥0∞
ι : Type u_2
hp✝ : Fact (1 ≤ p)
inst✝¹ : Fintype ι
β : Type u_5
inst✝ : SeminormedAddCommGroup β
hp : p ≠ ∞
b : β
⊢ ↑(↑(Fintype.card ι) ^ (1 / p).toReal * ‖b‖₊) = ↑↑(Fintype.card ι) ^ (1 / p).toReal * ‖b‖

Tactic:
by simp

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

Goal state before:
pp: p : ℝ≥0∞
ι : Type u_2
hp : Fact (1 ≤ p)
inst✝² : Fintype ι
β : Type u_5
inst✝¹ : SeminormedAddCommGroup β
inst✝ : Nonempty ι
b : β
⊢ ↑(↑(Fintype.card ι) ^ (1 / p).toReal * ‖b‖₊) = ↑↑(Fintype.card ι) ^ (1 / p).toReal * ‖b‖

Tactic:
by simp

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

Goal state before:
pp: 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
s : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s
⊢ 0 < 1 / 2

Tactic:
by simp

Goal state after:
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: case refine_1
E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁶ : NormedField K
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace K E
b : Basis ι K E
inst✝³ : LinearOrder K
inst✝² : IsStrictOrderedRing K
inst✝¹ : FloorRing K
inst✝ : Fintype ι
q : E ⧸ span ℤ (Set.range ⇑b)
x✝¹ x✝ : E
h : x✝¹ ≈ x✝
⊢ ↑(fractRestrict b x✝¹) = ↑(fractRestrict b x✝)

Tactic:
fractRestrict_apply,

Goal state after:
case refine_1
E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁶ : NormedField K
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace K E
b : Basis ι K E
inst✝³ : LinearOrder K
inst✝² : IsStrictOrderedRing K
inst✝¹ : FloorRing K
inst✝ : Fintype ι
q : E ⧸ span ℤ (Set.range ⇑b)
x✝¹ x✝ : E
h : x✝¹ ≈ x✝
⊢ fract b x✝¹ = ↑(fractRestrict b x✝)
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: case refine_1
E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁶ : NormedField K
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace K E
b : Basis ι K E
inst✝³ : LinearOrder K
inst✝² : IsStrictOrderedRing K
inst✝¹ : FloorRing K
inst✝ : Fintype ι
q : E ⧸ span ℤ (Set.range ⇑b)
x✝¹ x✝ : E
h : x✝¹ ≈ x✝
⊢ fract b x✝¹ = ↑(fractRestrict b x✝)

Tactic:
fractRestrict_apply,

Goal state after:
case refine_1
E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁶ : NormedField K
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace K E
b : Basis ι K E
inst✝³ : LinearOrder K
inst✝² : IsStrictOrderedRing K
inst✝¹ : FloorRing K
inst✝ : Fintype ι
q : E ⧸ span ℤ (Set.range ⇑b)
x✝¹ x✝ : E
h : x✝¹ ≈ x✝
⊢ fract b x✝¹ = fract b x✝
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: case refine_1
E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁶ : NormedField K
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace K E
b : Basis ι K E
inst✝³ : LinearOrder K
inst✝² : IsStrictOrderedRing K
inst✝¹ : FloorRing K
inst✝ : Fintype ι
q : E ⧸ span ℤ (Set.range ⇑b)
x✝¹ x✝ : E
h : x✝¹ ≈ x✝
⊢ fract b x✝¹ = fract b x✝

Tactic:
fract_eq_fract

Goal state after:
case refine_1
E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁶ : NormedField K
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace K E
b : Basis ι K E
inst✝³ : LinearOrder K
inst✝² : IsStrictOrderedRing K
inst✝¹ : FloorRing K
inst✝ : Fintype ι
q : E ⧸ span ℤ (Set.range ⇑b)
x✝¹ x✝ : E
h : x✝¹ ≈ x✝
⊢ -x✝¹ + x✝ ∈ span ℤ (Set.range ⇑b)
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: case refine_2.h
E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁶ : NormedField K
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace K E
b : Basis ι K E
inst✝³ : LinearOrder K
inst✝² : IsStrictOrderedRing K
inst✝¹ : FloorRing K
inst✝ : Fintype ι
a✝ b✝ : E
⊢ ⟦a✝⟧.liftOn (fractRestrict b) ⋯ = ⟦b✝⟧.liftOn (fractRestrict b) ⋯ → ⟦a✝⟧ = ⟦b✝⟧

Tactic:
intro hxy

Goal state after:
case refine_2.h
E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁶ : NormedField K
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace K E
b : Basis ι K E
inst✝³ : LinearOrder K
inst✝² : IsStrictOrderedRing K
inst✝¹ : FloorRing K
inst✝ : Fintype ι
a✝ b✝ : E
hxy : ⟦a✝⟧.liftOn (fractRestrict b) ⋯ = ⟦b✝⟧.liftOn (fractRestrict b) ⋯
⊢ ⟦a✝⟧ = ⟦b✝⟧
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
x : E
r : ℝ
this : ball 0 r = (fun x_1 ↦ x + x_1) ⁻¹' ball x r
⊢ μ (ball x r) = μ ((fun x_1 ↦ x + x_1) ⁻¹' ball x r)

Tactic:
measure_preimage_add

Goal state after:
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
x : E
r : ℝ
this : ball 0 r = (fun x_1 ↦ x + x_1) ⁻¹' ball x r
⊢ μ (ball x r) = μ (ball x r)
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
x : E
r : ℝ
⊢ μ.real (ball x r) = μ.real (ball 0 r)

Tactic:
simp [measureReal_def, addHaar_ball_center]

Goal state after: