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

Goal state before:
pp: case of.inr
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
ι : Type u_4
inst✝¹ : DecidableEq ι
V : ι → Submodule 𝕜 E
hV : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ
i : ι
inst✝ : CompleteSpace ↥(V i)
j : ι
x : ↥(V j)
hij : i ≠ j
⊢ (V i).orthogonalProjectionOnto ↑x = (DFinsupp.single j x) i

Tactic:
· rw [orthogonalProjectionOnto_apply_of_mem_orthogonal, DFinsupp.single_eq_of_ne hij]
      exact hV.isOrtho hij.symm x.prop

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‖)
sNu : Summable fun n ↦ ‖u n‖
su : Summable u
x : E := tsum u
x_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖
fsumeq : ∀ (n : ℕ), f (∑ i ∈ Finset.range n, u i) = y - h^[n] y
this : Tendsto (fun n ↦ ∑ i ∈ Finset.range n, u i) atTop (𝓝 x)
L₁ : Tendsto (fun n ↦ y - h^[n] y) atTop (𝓝 (f x))
⊢ Tendsto (fun e ↦ ‖h^[e] y‖) atTop (𝓝 0)

Tactic:
refine squeeze_zero (fun _ => norm_nonneg _) hnle ?_

Goal state after:
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‖)
sNu : Summable fun n ↦ ‖u n‖
su : Summable u
x : E := tsum u
x_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖
fsumeq : ∀ (n : ℕ), f (∑ i ∈ Finset.range n, u i) = y - h^[n] y
this : Tendsto (fun n ↦ ∑ i ∈ Finset.range n, u i) atTop (𝓝 x)
L₁ : Tendsto (fun n ↦ y - h^[n] y) atTop (𝓝 (f x))
⊢ Tendsto (fun t ↦ (1 / 2) ^ t * ‖y‖) atTop (𝓝 0)
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: case inr.right
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
cancel : p.toReal * (1 / p.toReal) = 1
⊢ edist x.snd y.snd ≤ (edist x.fst y.fst ^ p.toReal + edist x.snd y.snd ^ p.toReal) ^ (1 / p.toReal)

Tactic:
calc
        edist x.snd y.snd ≤ (edist x.snd y.snd ^ p.toReal) ^ (1 / p.toReal) := by
          simp only [← ENNReal.rpow_mul, cancel, ENNReal.rpow_one, le_refl]
        _ ≤ (edist x.fst y.fst ^ p.toReal + edist x.snd y.snd ^ p.toReal) ^ (1 / p.toReal) := by
          gcongr
          simp only [self_le_add_left]

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

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

Tactic:
toLp_add

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)
⊢ LinearIndependent 𝕜 (⇑↑(WithLp.linearEquiv p 𝕜 ((j : η) → Ms j)).symm ∘ fun ji ↦ Pi.single ji.fst (v ji.fst ji.snd))

Tactic:
rw [LinearMap.linearIndependent_iff_of_injOn _ (by simp)]

Goal state after:
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)
⊢ LinearIndependent 𝕜 fun ji ↦ Pi.single ji.fst (v ji.fst ji.snd)
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
⊢ LinearIndependent 𝕜 (⇑↑(WithLp.linearEquiv p 𝕜 (ι → 𝕜)).symm ∘ fun i ↦ Pi.single i 1)

Tactic:
rw [LinearMap.linearIndependent_iff_of_injOn _ (by simp)]

Goal state after:
p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
inst✝¹ : DecidableEq ι
inst✝ : Ring 𝕜
⊢ LinearIndependent 𝕜 fun i ↦ Pi.single i 1
Module: Mathlib.LinearAlgebra.UnitaryGroup

Goal state before:
pp: n : Type u
inst✝¹³ : DecidableEq n
inst✝¹² : Fintype n
R : Type u_1
A : Type u_2
B : Type u_3
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Semiring A
inst✝⁹ : Semiring B
inst✝⁸ : Algebra R A
inst✝⁷ : Algebra R B
inst✝⁶ : StarRing A
inst✝⁵ : StarRing B
inst✝⁴ : StarRing R
inst✝³ : StarModule R A
inst✝² : StarModule R B
m : Type u_4
inst✝¹ : Fintype m
inst✝ : DecidableEq m
U : Matrix m m A
V : Matrix n n B
hU : star U * U = 1 ∧ U * star U = 1
hV : star V * V = 1 ∧ V * star V = 1
⊢ star (U ⊗ₖₜ[R] V) * U ⊗ₖₜ[R] V = 1 ∧ U ⊗ₖₜ[R] V * star (U ⊗ₖₜ[R] V) = 1

Tactic:
star_eq_conjTranspose

Goal state after:
n : Type u
inst✝¹³ : DecidableEq n
inst✝¹² : Fintype n
R : Type u_1
A : Type u_2
B : Type u_3
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Semiring A
inst✝⁹ : Semiring B
inst✝⁸ : Algebra R A
inst✝⁷ : Algebra R B
inst✝⁶ : StarRing A
inst✝⁵ : StarRing B
inst✝⁴ : StarRing R
inst✝³ : StarModule R A
inst✝² : StarModule R B
m : Type u_4
inst✝¹ : Fintype m
inst✝ : DecidableEq m
U : Matrix m m A
V : Matrix n n B
hU : Uᴴ * U = 1 ∧ U * Uᴴ = 1
hV : Vᴴ * V = 1 ∧ V * Vᴴ = 1
⊢ (U ⊗ₖₜ[R] V)ᴴ * U ⊗ₖₜ[R] V = 1 ∧ U ⊗ₖₜ[R] V * (U ⊗ₖₜ[R] V)ᴴ = 1
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
⊢ LinearIndependent 𝕜 (⇑↑(WithLp.linearEquiv p 𝕜 (ι → M)).symm ∘ fun i ↦ Pi.single i (v i))

Tactic:
rw [LinearMap.linearIndependent_iff_of_injOn _ (by simp)]

Goal state after:
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
⊢ LinearIndependent 𝕜 fun i ↦ Pi.single i (v i)
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
⊢ ↑(toGL 1) = ↑1

Tactic:
simp only [coe_toGL, toLin'_one]

Goal state after:
n : Type u
inst✝³ : DecidableEq n
inst✝² : Fintype n
α : Type v
inst✝¹ : CommRing α
inst✝ : StarRing α
⊢ LinearMap.id = ↑1
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
this : 0 < p.toReal
⊢ (∑ x, ENNReal.ofReal (dist (f.ofLp x) (g.ofLp x)) ^ p.toReal) ^ (1 / p.toReal) =
    ENNReal.ofReal ((∑ i, dist (f.ofLp i) (g.ofLp i) ^ p.toReal) ^ (1 / p.toReal))

Tactic:
← ENNReal.ofReal_rpow_of_nonneg (by simp [Finset.sum_nonneg, Real.rpow_nonneg]) (by simp)

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
this : 0 < p.toReal
⊢ (∑ x, ENNReal.ofReal (dist (f.ofLp x) (g.ofLp x)) ^ p.toReal) ^ (1 / p.toReal) =
    ENNReal.ofReal (∑ i, dist (f.ofLp i) (g.ofLp i) ^ p.toReal) ^ (1 / p.toReal)
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 y : PiLp 2 β
⊢ ‖x - y‖ = √(∑ x_1, ‖x.ofLp x_1 - y.ofLp x_1‖ ^ 2)

Tactic:
norm_eq_of_L2,

Goal state after:
ι : Type u_2
β : ι → Type u_4
inst✝¹ : Fintype ι
inst✝ : (i : ι) → SeminormedAddCommGroup (β i)
x y : PiLp 2 β
⊢ √(∑ i, ‖(x - y).ofLp i‖ ^ 2) = √(∑ x_1, ‖x.ofLp x_1 - y.ofLp x_1‖ ^ 2)
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: case inl
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
inst✝⁵ : Fintype ι
inst✝⁴ : Semiring 𝕜
inst✝³ : (i : ι) → SeminormedAddCommGroup (α i)
inst✝² : (i : ι) → SeminormedAddCommGroup (β i)
inst✝¹ : (i : ι) → Module 𝕜 (α i)
inst✝ : (i : ι) → Module 𝕜 (β i)
c : 𝕜
e : (i : ι) → α i ≃ₗᵢ[𝕜] β i
x : (i : ι) → α i
hp : Fact (1 ≤ ∞)
__LinearEquiv✝ : PiLp ∞ α ≃ₗ[𝕜] PiLp ∞ β :=
  (WithLp.linearEquiv ∞ 𝕜 ((i : ι) → α i) ≪≫ₗ LinearEquiv.piCongrRight fun i ↦ (e i).toLinearEquiv) ≪≫ₗ
    (WithLp.linearEquiv ∞ 𝕜 ((i : ι) → β i)).symm
⊢ ↑(Finset.univ.sup fun b ↦ ‖(e b).toLinearEquiv (x b)‖₊) = ↑(Finset.univ.sup fun b ↦ ‖x b‖₊)

Tactic:
LinearIsometryEquiv.coe_toLinearEquiv,

Goal state after:
case inl
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
inst✝⁵ : Fintype ι
inst✝⁴ : Semiring 𝕜
inst✝³ : (i : ι) → SeminormedAddCommGroup (α i)
inst✝² : (i : ι) → SeminormedAddCommGroup (β i)
inst✝¹ : (i : ι) → Module 𝕜 (α i)
inst✝ : (i : ι) → Module 𝕜 (β i)
c : 𝕜
e : (i : ι) → α i ≃ₗᵢ[𝕜] β i
x : (i : ι) → α i
hp : Fact (1 ≤ ∞)
__LinearEquiv✝ : PiLp ∞ α ≃ₗ[𝕜] PiLp ∞ β :=
  (WithLp.linearEquiv ∞ 𝕜 ((i : ι) → α i) ≪≫ₗ LinearEquiv.piCongrRight fun i ↦ (e i).toLinearEquiv) ≪≫ₗ
    (WithLp.linearEquiv ∞ 𝕜 ((i : ι) → β i)).symm
⊢ ↑(Finset.univ.sup fun b ↦ ‖(e b) (x b)‖₊) = ↑(Finset.univ.sup fun b ↦ ‖x b‖₊)
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
hp✝ : Fact (1 ≤ p✝)
inst✝¹⁰ : Fintype ι✝
inst✝⁹ : Semiring 𝕜
inst✝⁸ : (i : ι✝) → SeminormedAddCommGroup (α✝ i)
inst✝⁷ : (i : ι✝) → SeminormedAddCommGroup (β i)
inst✝⁶ : (i : ι✝) → Module 𝕜 (α✝ i)
inst✝⁵ : (i : ι✝) → Module 𝕜 (β i)
c : 𝕜
ι : Type u_5
κ : Type u_6
p : ℝ≥0∞
α : ι ⊕ κ → Type u_7
inst✝⁴ : Fintype ι
inst✝³ : Fintype κ
inst✝² : Fact (1 ≤ p)
inst✝¹ : (i : ι ⊕ κ) → SeminormedAddCommGroup (α i)
inst✝ : (i : ι ⊕ κ) → Module 𝕜 (α i)
x : (i : ι ⊕ κ) → α i
hp : 1 ≤ p.toReal
this : 0 < p.toReal
⊢ ‖(WithLp.linearEquiv p 𝕜 ((i : ι ⊕ κ) → α i) ≪≫ₗ LinearEquiv.sumPiEquivProdPi 𝕜 ι κ α ≪≫ₗ
            (WithLp.linearEquiv p 𝕜 ((s : ι) → α (Sum.inl s))).symm.prodCongr
              (WithLp.linearEquiv p 𝕜 ((t : κ) → α (Sum.inr t))).symm ≪≫ₗ
          (WithLp.linearEquiv p 𝕜 (WithLp p ((s : ι) → α (Sum.inl s)) × WithLp p ((t : κ) → α (Sum.inr t)))).symm)
        ((WithLp.linearEquiv p 𝕜 ((i : ι ⊕ κ) → α i)).symm x)‖ =
    ‖(WithLp.linearEquiv p 𝕜 ((i : ι ⊕ κ) → α i)).symm x‖

Tactic:
have hpt : p ≠ ⊤ := (toReal_pos_iff_ne_top p).mp this

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
hp✝ : Fact (1 ≤ p✝)
inst✝¹⁰ : Fintype ι✝
inst✝⁹ : Semiring 𝕜
inst✝⁸ : (i : ι✝) → SeminormedAddCommGroup (α✝ i)
inst✝⁷ : (i : ι✝) → SeminormedAddCommGroup (β i)
inst✝⁶ : (i : ι✝) → Module 𝕜 (α✝ i)
inst✝⁵ : (i : ι✝) → Module 𝕜 (β i)
c : 𝕜
ι : Type u_5
κ : Type u_6
p : ℝ≥0∞
α : ι ⊕ κ → Type u_7
inst✝⁴ : Fintype ι
inst✝³ : Fintype κ
inst✝² : Fact (1 ≤ p)
inst✝¹ : (i : ι ⊕ κ) → SeminormedAddCommGroup (α i)
inst✝ : (i : ι ⊕ κ) → Module 𝕜 (α i)
x : (i : ι ⊕ κ) → α i
hp : 1 ≤ p.toReal
this : 0 < p.toReal
hpt : p ≠ ∞
⊢ ‖(WithLp.linearEquiv p 𝕜 ((i : ι ⊕ κ) → α i) ≪≫ₗ LinearEquiv.sumPiEquivProdPi 𝕜 ι κ α ≪≫ₗ
            (WithLp.linearEquiv p 𝕜 ((s : ι) → α (Sum.inl s))).symm.prodCongr
              (WithLp.linearEquiv p 𝕜 ((t : κ) → α (Sum.inr t))).symm ≪≫ₗ
          (WithLp.linearEquiv p 𝕜 (WithLp p ((s : ι) → α (Sum.inl s)) × WithLp p ((t : κ) → α (Sum.inr t)))).symm)
        ((WithLp.linearEquiv p 𝕜 ((i : ι ⊕ κ) → α i)).symm x)‖ =
    ‖(WithLp.linearEquiv p 𝕜 ((i : ι ⊕ κ) → α i)).symm x‖
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Basic

Goal state before:
pp: ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
t : TransvectionStruct ι ℝ
⊢ MeasurePreserving (⇑(toLin' t.toMatrix)) volume volume

Tactic:
by
  /- We use `lmarginal` to conveniently use Fubini's theorem.
    Along the coordinate where there is a shearing, it acts like a
    translation, and therefore preserves Lebesgue. -/
  have ht : Measurable (toLin' t.toMatrix) :=
    (toLin' t.toMatrix).continuous_of_finiteDimensional.measurable
  refine ⟨ht, ?_⟩
  refine (pi_eq fun s hs ↦ ?_).symm
  have h2s : MeasurableSet (univ.pi s) := .pi countable_univ fun i _ ↦ hs i
  simp_rw [← pi_pi, ← lintegral_indicator_one h2s]
  rw [lintegral_map (measurable_one.indicator h2s) ht, volume_pi]
  refine lintegral_eq_of_lmarginal_eq {t.i} ((measurable_one.indicator h2s).comp ht)
    (measurable_one.indicator h2s) ?_
  simp_rw [lmarginal_singleton]
  ext x
  cases t with | mk t_i t_j t_hij t_c =>
  simp [transvection, single_mulVec, t_hij.symm, ← Function.update_add,
    lintegral_add_right_eq_self fun xᵢ ↦ indicator (univ.pi s) 1 (Function.update x t_i xᵢ)]

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

Goal state before:
pp: ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
a : ι → ℝ
⊢ (parallelepiped fun i ↦ Pi.single i (a i)) = uIcc 0 a

Tactic:
by
  ext x
  simp_rw [Set.uIcc, mem_parallelepiped_iff, Set.mem_Icc, Pi.le_def, ← forall_and, Pi.inf_apply,
    Pi.sup_apply, ← Pi.single_smul', Pi.one_apply, Pi.zero_apply, ← Pi.smul_apply',
    Finset.univ_sum_single (_ : ι → ℝ)]
  constructor
  · rintro ⟨t, ht, rfl⟩ i
    specialize ht i
    simp_rw [smul_eq_mul, Pi.mul_apply]
    rcases le_total (a i) 0 with hai | hai
    · rw [sup_eq_left.mpr hai, inf_eq_right.mpr hai]
      exact ⟨le_mul_of_le_one_left hai ht.2, mul_nonpos_of_nonneg_of_nonpos ht.1 hai⟩
    · rw [sup_eq_right.mpr hai, inf_eq_left.mpr hai]
      exact ⟨mul_nonneg ht.1 hai, mul_le_of_le_one_left hai ht.2⟩
  · intro h
    refine ⟨fun i => x i / a i, fun i => ?_, funext fun i => ?_⟩
    · specialize h i
      rcases le_total (a i) 0 with hai | hai
      · rw [sup_eq_left.mpr hai, inf_eq_right.mpr hai] at h
        exact ⟨div_nonneg_of_nonpos h.2 hai, div_le_one_of_ge h.1 hai⟩
      · rw [sup_eq_right.mpr hai, inf_eq_left.mpr hai] at h
        exact ⟨div_nonneg h.1 hai, div_le_one_of_le₀ h.2 hai⟩
    · specialize h i
      simp only [smul_eq_mul, Pi.mul_apply]
      rcases eq_or_ne (a i) 0 with hai | hai
      · rw [hai, inf_idem, sup_idem, ← le_antisymm_iff] at h
        rw [hai, ← h, zero_div, zero_mul]
      · rw [div_mul_cancel₀ _ hai]

Goal state after:
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
r : ℝ
s : Set E
h : r ≠ 0
⊢ μ (r • s) = ENNReal.ofReal |r ^ finrank ℝ E| * μ s

Tactic:
← preimage_smul_inv₀ h,

Goal state after:
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
r : ℝ
s : Set E
h : r ≠ 0
⊢ μ ((fun x ↦ r⁻¹ • x) ⁻¹' s) = ENNReal.ofReal |r ^ finrank ℝ E| * μ s
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
hxy : ⟦a✝⟧.liftOn (fractRestrict b) ⋯ = ⟦b✝⟧.liftOn (fractRestrict b) ⋯
⊢ ⟦a✝⟧ = ⟦b✝⟧

Tactic:
Quotient.liftOn_mk (s := quotientRel (span ℤ (Set.range b))),

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 : fractRestrict b a✝ = ⟦b✝⟧.liftOn (fractRestrict b) ⋯
⊢ ⟦a✝⟧ = ⟦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
hxy : ⟨fract b a✝, ⋯⟩ = ⟦b✝⟧.liftOn (fractRestrict b) ⋯
⊢ ⟦a✝⟧ = ⟦b✝⟧

Tactic:
Quotient.liftOn_mk (s := quotientRel (span ℤ (Set.range b))),

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 : ⟨fract b a✝, ⋯⟩ = fractRestrict b b✝
⊢ ⟦a✝⟧ = ⟦b✝⟧
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: 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
⊢ s ⊆ Set.range Subtype.val

Tactic:
simp only [Subtype.range_coe_subtype, Set.ofPred_mem_eq, subset_rfl]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: ι✝ : Type u_1
ι' : Type u_2
𝕜 : Type u_3
inst✝⁹ : RCLike 𝕜
E : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : InnerProductSpace 𝕜 E
F : Type u_5
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace ℝ F
F' : Type u_6
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : InnerProductSpace ℝ F'
ι : Type u_7
inst✝² : Fintype ι
f : ι → Type u_8
inst✝¹ : (i : ι) → NormedAddCommGroup (f i)
inst✝ : (i : ι) → InnerProductSpace 𝕜 (f i)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
intro x y
    unfold inner
    rw [map_sum]
    apply Finset.sum_congr rfl
    rintro z -
    apply inner_conj_symm

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

Goal state before:
pp: ι : Type u_1
I : Box ι
π : Prepartition I
p : Box ι → Prop
⊢ ↑π.boxes ∪ ↑(π.filter p).boxes = ↑π.boxes

Tactic:
Set.union_eq_left,

Goal state after:
ι : Type u_1
I : Box ι
π : Prepartition I
p : Box ι → Prop
⊢ ↑(π.filter p).boxes ⊆ ↑π.boxes
Module: Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction

Goal state before:
pp: case refine_2
ι : Type u_1
inst✝ : Fintype ι
I : Box ι
r : (ι → ℝ) → ↑(Ioi 0)
z : ι → ℝ
x✝ : z ∈ Box.Icc I
⊢ ∃ U ∈ 𝓝[Box.Icc I] z,
    ∀ J ≤ I,
      ∀ (m : ℕ),
        z ∈ Box.Icc J →
          Box.Icc J ⊆ U →
            (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) →
              ∃ π,
                π.IsPartition ∧
                  π.IsHenstock ∧
                    π.IsSubordinate r ∧
                      (∀ J_1 ∈ π, ∃ m, ∀ (i : ι), J_1.upper i - J_1.lower i = (J.upper i - J.lower i) / 2 ^ m) ∧
                        π.distortion = J.distortion

Tactic:
refine ⟨Box.Icc I ∩ closedBall z (r z),
      inter_mem_nhdsWithin _ (closedBall_mem_nhds _ (r z).coe_prop), ?_⟩
    intro J _ n Hmem HIcc Hsub
    rw [Set.subset_inter_iff] at HIcc
    refine ⟨single _ _ le_rfl _ Hmem, isPartition_single _, isHenstock_single _,
      (isSubordinate_single _ _).2 HIcc.2, ?_, distortion_single _ _⟩
    simp only [TaggedPrepartition.mem_single, forall_eq]
    refine ⟨0, fun i => ?_⟩
    simp

Goal state after:
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
this✝³ : Module.Finite ℤ ↥L
this✝² : Module ℚ E
this✝¹ : IsAddTorsionFree E
b₀ : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
b : Free.ChooseBasisIndex ℤ ↥L → E := Subtype.val ∘ ⇑b₀
this✝ : LinearIndependent ℤ b
h_spanL : span ℤ (Set.range b) = L
h_spanE : span K (Set.range b) = ⊤
h_card : Fintype.card (Free.ChooseBasisIndex ℤ ↥L) = (Set.range b).toFinset.card
t : Set E
ht_inc : t ⊆ Set.range b
ht_span : span K t = span K (Set.range b)
ht_lin : LinearIndependent K Subtype.val
e : Basis (↑t) K E := Basis.mk ht_lin ⋯
this : Fintype ↑t
h : finrank K E < Fintype.card (Free.ChooseBasisIndex ℤ ↥L)
v : E
hv : v ∈ Set.range b \ Set.range ⇑e
a✝ : LinearIndepOn ℚ id (Set.range ⇑e)
h_mapsto : Set.MapsTo (fun n ↦ fract e (n • v)) Set.univ (Metric.closedBall 0 (∑ i, ‖e i‖) ∩ ↑L)
h_finite : (Metric.closedBall 0 (∑ i, ‖e i‖) ∩ ↑L).Finite
n m : ℤ
h_ne : n ≠ m
h_eq : fract e (n • v) = fract e (m • v)
h_nz : -↑n + ↑m ≠ 0
⊢ -↑n • v + ↑m • v ∈ ↑(span ℤ (Set.range ⇑e))

Tactic:
neg_smul,

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
this✝³ : Module.Finite ℤ ↥L
this✝² : Module ℚ E
this✝¹ : IsAddTorsionFree E
b₀ : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
b : Free.ChooseBasisIndex ℤ ↥L → E := Subtype.val ∘ ⇑b₀
this✝ : LinearIndependent ℤ b
h_spanL : span ℤ (Set.range b) = L
h_spanE : span K (Set.range b) = ⊤
h_card : Fintype.card (Free.ChooseBasisIndex ℤ ↥L) = (Set.range b).toFinset.card
t : Set E
ht_inc : t ⊆ Set.range b
ht_span : span K t = span K (Set.range b)
ht_lin : LinearIndependent K Subtype.val
e : Basis (↑t) K E := Basis.mk ht_lin ⋯
this : Fintype ↑t
h : finrank K E < Fintype.card (Free.ChooseBasisIndex ℤ ↥L)
v : E
hv : v ∈ Set.range b \ Set.range ⇑e
a✝ : LinearIndepOn ℚ id (Set.range ⇑e)
h_mapsto : Set.MapsTo (fun n ↦ fract e (n • v)) Set.univ (Metric.closedBall 0 (∑ i, ‖e i‖) ∩ ↑L)
h_finite : (Metric.closedBall 0 (∑ i, ‖e i‖) ∩ ↑L).Finite
n m : ℤ
h_ne : n ≠ m
h_eq : fract e (n • v) = fract e (m • v)
h_nz : -↑n + ↑m ≠ 0
⊢ -(↑n • v) + ↑m • v ∈ ↑(span ℤ (Set.range ⇑e))
Module: Mathlib.Analysis.BoxIntegral.Partition.Split

Goal state before:
pp: ι : Type u_1
I : Box ι
i : ι
x : ℝ
⊢ Disjoint (I.splitLower i x) (I.splitUpper i x)

Tactic:
by
  rw [← disjoint_withBotCoe, coe_splitLower, coe_splitUpper]
  refine (Disjoint.inf_left' _ ?_).inf_right' _
  rw [Set.disjoint_left]
  exact fun y (hle : y i ≤ x) hlt => not_lt_of_ge hle hlt

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: η : Type u_7
inst✝⁴ : Fintype η
ι : η → Type u_8
inst✝³ : (i : η) → Fintype (ι i)
𝕜 : Type u_9
inst✝² : RCLike 𝕜
E : η → Type u_10
inst✝¹ : (i : η) → NormedAddCommGroup (E i)
inst✝ : (i : η) → InnerProductSpace 𝕜 (E i)
B : (i : η) → OrthonormalBasis (ι i) 𝕜 (E i)
⊢ (Pi.orthonormalBasis B).toBasis = (Pi.basis fun i ↦ (B i).toBasis).map (WithLp.linearEquiv 2 𝕜 ((j : η) → E j)).symm

Tactic:
by ext; rfl

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
E : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
a b : OrthonormalBasis ι 𝕜 E
⊢ a.toBasis.toMatrix ⇑b ∈ Matrix.unitaryGroup ι 𝕜

Tactic:
rw [Matrix.mem_unitaryGroup_iff']
  exact a.toMatrix_orthonormalBasis_conjTranspose_mul_self b

Goal state after: