mathlib-initiative/mathlib-tactics huggingface dataset:
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
ht : Measurable ⇑(toLin' t.toMatrix)
s : ι → Set ℝ
hs : ∀ (i : ι), MeasurableSet (s i)
h2s : MeasurableSet (univ.pi s)
⊢ (Measure.map (⇑(toLin' t.toMatrix)) volume) (univ.pi s) = ∏ i, volume (s i)

Tactic:
simp_rw [← pi_pi, ← lintegral_indicator_one h2s]

Goal state after:
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
t : TransvectionStruct ι ℝ
ht : Measurable ⇑(toLin' t.toMatrix)
s : ι → Set ℝ
hs : ∀ (i : ι), MeasurableSet (s i)
h2s : MeasurableSet (univ.pi s)
⊢ ∫⁻ (a : ι → ℝ), (univ.pi s).indicator 1 a ∂Measure.map (⇑(toLin' t.toMatrix)) volume =
    ∫⁻ (a : ι → ℝ), (univ.pi s).indicator 1 a ∂Measure.pi fun i ↦ volume
Module: Mathlib.MeasureTheory.Measure.Haar.OfBasis

Goal state before:
pp: case mpr.refine_2.inl
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
a x : ι → ℝ
i : ι
h : min 0 (a i) ≤ x i ∧ x i ≤ max 0 (a i)
hai : a i = 0
⊢ x i = x i / a i * a i

Tactic:
hai,

Goal state after:
case mpr.refine_2.inl
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
a x : ι → ℝ
i : ι
h : min 0 0 ≤ x i ∧ x i ≤ max 0 0
hai : a i = 0
⊢ x i = x i / a i * a i
Module: Mathlib.MeasureTheory.Measure.Haar.OfBasis

Goal state before:
pp: case mpr.refine_2.inl
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
a x : ι → ℝ
i : ι
h : 0 = x i
hai : a i = 0
⊢ x i = x i / a i * a i

Tactic:
hai,

Goal state after:
case mpr.refine_2.inl
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
a x : ι → ℝ
i : ι
h : 0 = x i
hai : a i = 0
⊢ x i = x i / 0 * 0
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁷ : NormedField K
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace K E
b : Basis ι K E
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : FloorRing K
inst✝¹ : Fintype ι
inst✝ : HasSolidNorm K
m : E
i : ι
x✝ : i ∈ Finset.univ
⊢ ‖Int.fract ((b.repr m) i)‖ ≤ 1

Tactic:
(norm_one.symm : 1 = ‖(1 : K)‖)

Goal state after:
E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁷ : NormedField K
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace K E
b : Basis ι K E
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : FloorRing K
inst✝¹ : Fintype ι
inst✝ : HasSolidNorm K
m : E
i : ι
x✝ : i ∈ Finset.univ
⊢ ‖Int.fract ((b.repr m) i)‖ ≤ ‖1‖
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: case inr
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
s : ℝ
h : 0 < r
⊢ μ (ball x (r * s)) = ENNReal.ofReal (r ^ finrank ℝ E) * μ (ball 0 s)

Tactic:
· exact addHaar_ball_mul_of_pos μ x h s

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

Goal state before:
pp: ι : Type u_1
I J : WithBot (Box ι)
⊢ Disjoint ↑I ↑J ↔ Disjoint I J

Tactic:
simp only [disjoint_iff_inf_le, ← withBotCoe_subset_iff, coe_inf]
  rfl

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

Goal state before:
pp: case inl
ι : Type u_1
I : Box ι
π : Prepartition I
x : ι → ℝ
J₁ : Box ι
h₁ : J₁ ∈ π
hx₁ : x ∈ Box.Icc J₁
J₂ : Box ι
h₂ : J₂ ∈ π
hx₂ : x ∈ Box.Icc J₂
i : ι
H : ∀ (x_1 : ι), J₁.lower x_1 = x x_1 ↔ J₂.lower x_1 = x x_1
hi₁ : J₁.lower i = x i
⊢ (Set.Ioc (J₁.lower i) (J₁.upper i) ∩ Set.Ioc (J₂.lower i) (J₂.upper i)).Nonempty

Tactic:
have hi₂ : J₂.lower i = x i := (H _).1 hi₁
    have H₁ : x i < J₁.upper i := by simpa only [hi₁] using J₁.lower_lt_upper i
    have H₂ : x i < J₂.upper i := by simpa only [hi₂] using J₂.lower_lt_upper i
    rw [Set.Ioc_inter_Ioc, hi₁, hi₂, sup_idem, Set.nonempty_Ioc]
    exact lt_min H₁ H₂

Goal state after:
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
this✝¹ : Module.Finite ℤ ↥L
this✝ : Module ℚ E
this : IsAddTorsionFree E
b₀ : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
⊢ finrank ℤ ↥L = finrank K E

Tactic:
let b := Subtype.val ∘ b₀

Goal state after:
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✝¹ : Module.Finite ℤ ↥L
this✝ : Module ℚ E
this : IsAddTorsionFree E
b₀ : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
b : Free.ChooseBasisIndex ℤ ↥L → E := Subtype.val ∘ ⇑b₀
⊢ finrank ℤ ↥L = finrank K E
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: case h2
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 : Set E
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t u : Set E
h'u : μ u ≠ 0
R : ℝ
Rpos : 0 < R
t_bound : t ⊆ closedBall 0 R
t' : Set E := ⋯
ht' : t' = R⁻¹ • t
u' : Set E := ⋯
hu' : u' = R⁻¹ • u
A : Tendsto (fun r ↦ μ (s ∩ ({x} + r • t')) / μ ({x} + r • u')) (𝓝[>] 0) (𝓝 0)
⊢ ∀ᶠ (x : ℝ) in 𝓝[>] 0, R * x ∈ Ioi (R * 0)

Tactic:
· filter_upwards [self_mem_nhdsWithin]
      intro r rpos
      rw [mul_zero]
      exact mul_pos Rpos rpos

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

Goal state before:
pp: ι : Type u_1
I J : Box ι
π : Prepartition I
h : I ≤ J
J' : Box ι
hJ' : J' ∈ π.boxes
⊢ ↑J' ≤ ↑J

Tactic:
exact WithBot.coe_le_coe.2 ((π.le_of_mem hJ').trans h)

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

Goal state before:
pp: case mp
ι : Type u_1
I : Box ι
π : Prepartition I
πi : (J : Box ι) → Prepartition J
π' : Prepartition I
H : π.biUnion πi ≤ π'
J : Box ι
hJ : J ∈ π
⊢ πi J ≤ π'.restrict J

Tactic:
rw [← π.restrict_biUnion πi hJ]

Goal state after:
case mp
ι : Type u_1
I : Box ι
π : Prepartition I
πi : (J : Box ι) → Prepartition J
π' : Prepartition I
H : π.biUnion πi ≤ π'
J : Box ι
hJ : J ∈ π
⊢ (π.biUnion πi).restrict J ≤ π'.restrict J
Module: Mathlib.Analysis.BoxIntegral.Partition.Basic

Goal state before:
pp: case refine_1
ι : Type u_1
I : Box ι
π : Prepartition I
πi : (J : Box ι) → Prepartition J
π' : Prepartition I
H : π' ≤ π.biUnion πi
J : Box ι
hJ : J ∈ π
⊢ π'.restrict J ≤ πi J

Tactic:
rw [← π.restrict_biUnion πi hJ]

Goal state after:
case refine_1
ι : Type u_1
I : Box ι
π : Prepartition I
πi : (J : Box ι) → Prepartition J
π' : Prepartition I
H : π' ≤ π.biUnion πi
J : Box ι
hJ : J ∈ π
⊢ π'.restrict J ≤ (π.biUnion πi).restrict J
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
b : OrthonormalBasis ι 𝕜 E
i j : ι
⊢ ⟪b i, b j⟫ = if i = j then 1 else 0

Tactic:
by_cases h : i = j

Goal state after:
case pos
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
b : OrthonormalBasis ι 𝕜 E
i j : ι
h : i = j
⊢ ⟪b i, b j⟫ = if i = j then 1 else 0

case neg
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
b : OrthonormalBasis ι 𝕜 E
i j : ι
h : ¬i = j
⊢ ⟪b i, b j⟫ = if i = j then 1 else 0
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_3
inst✝³ : RCLike 𝕜
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : Fintype ι
b : OrthonormalBasis ι 𝕜 E
x : E
⊢ ∑ i, ‖⟪b i, x⟫‖ ^ 2 = √(∑ x_1, ‖⟪b x_1, x⟫‖ * ‖⟪b x_1, x⟫‖) ^ 2

Tactic:
← pow_two

Goal state after:
ι : Type u_1
𝕜 : Type u_3
inst✝³ : RCLike 𝕜
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : Fintype ι
b : OrthonormalBasis ι 𝕜 E
x : E
⊢ ∑ i, ‖⟪b i, x⟫‖ ^ 2 = √(∑ x_1, ‖⟪b x_1, x⟫‖ ^ 2) ^ 2
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
this : (a.toBasis.toMatrix ⇑b).det * (starRingEnd 𝕜) (a.toBasis.toMatrix ⇑b).det = 1
⊢ ‖a.toBasis.det ⇑b‖ = 1

Tactic:
RCLike.mul_conj

Goal state after:
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
this : ↑‖(a.toBasis.toMatrix ⇑b).det‖ ^ 2 = 1
⊢ ‖a.toBasis.det ⇑b‖ = 1
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
𝕜 : Type u_3
inst✝⁸ : RCLike 𝕜
E : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
F : Type u_5
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace ℝ F
F' : Type u_6
inst✝³ : NormedAddCommGroup F'
inst✝² : InnerProductSpace ℝ F'
inst✝¹ : Fintype ι
v : Set E
A : ι → Submodule 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
b : OrthonormalBasis (↥(Classical.choose ⋯)) 𝕜 E := Classical.choose ⋯
⊢ OrthonormalBasis (Fin (finrank 𝕜 E)) 𝕜 E

Tactic:
finrank_eq_card_basis b.toBasis

Goal state after:
ι : Type u_1
ι' : Type u_2
𝕜 : Type u_3
inst✝⁸ : RCLike 𝕜
E : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
F : Type u_5
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace ℝ F
F' : Type u_6
inst✝³ : NormedAddCommGroup F'
inst✝² : InnerProductSpace ℝ F'
inst✝¹ : Fintype ι
v : Set E
A : ι → Submodule 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
b : OrthonormalBasis (↥(Classical.choose ⋯)) 𝕜 E := Classical.choose ⋯
⊢ OrthonormalBasis (Fin (Fintype.card ↥(Classical.choose ⋯))) 𝕜 E
Module: Mathlib.Analysis.Oscillation

Goal state before:
pp: E : Type u
F : Type v
inst✝¹ : PseudoEMetricSpace F
inst✝ : PseudoEMetricSpace E
K : Set E
f : E → F
D : Set E
ε : ℝ≥0∞
comp : IsCompact K
hK : ∀ x ∈ K, oscillationWithin f D x < ε
S : ℝ → Set E := fun r ↦ {x | ∃ a > r, ediam (f '' (eball x (ENNReal.ofReal a) ∩ D)) ≤ ε}
S_open : ∀ r > 0, IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (S r)
S_cover : K ⊆ ⋃ r, ⋃ (_ : r > 0), S r
S_antitone : ∀ (r₁ r₂ : ℝ), r₁ ≤ r₂ → S r₂ ⊆ S r₁
T : Set ℝ
Tb : T ⊆ Real.lt✝ 0
Tfin : T.Finite
hT : K ⊆ ⋃ i ∈ T, S i
⊢ ∃ r > 0, K ⊆ S r

Tactic:
by_cases T_nonempty : T.Nonempty

Goal state after:
case pos
E : Type u
F : Type v
inst✝¹ : PseudoEMetricSpace F
inst✝ : PseudoEMetricSpace E
K : Set E
f : E → F
D : Set E
ε : ℝ≥0∞
comp : IsCompact K
hK : ∀ x ∈ K, oscillationWithin f D x < ε
S : ℝ → Set E := fun r ↦ {x | ∃ a > r, ediam (f '' (eball x (ENNReal.ofReal a) ∩ D)) ≤ ε}
S_open : ∀ r > 0, IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (S r)
S_cover : K ⊆ ⋃ r, ⋃ (_ : r > 0), S r
S_antitone : ∀ (r₁ r₂ : ℝ), r₁ ≤ r₂ → S r₂ ⊆ S r₁
T : Set ℝ
Tb : T ⊆ Real.lt✝ 0
Tfin : T.Finite
hT : K ⊆ ⋃ i ∈ T, S i
T_nonempty : T.Nonempty
⊢ ∃ r > 0, K ⊆ S r

case neg
E : Type u
F : Type v
inst✝¹ : PseudoEMetricSpace F
inst✝ : PseudoEMetricSpace E
K : Set E
f : E → F
D : Set E
ε : ℝ≥0∞
comp : IsCompact K
hK : ∀ x ∈ K, oscillationWithin f D x < ε
S : ℝ → Set E := fun r ↦ {x | ∃ a > r, ediam (f '' (eball x (ENNReal.ofReal a) ∩ D)) ≤ ε}
S_open : ∀ r > 0, IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (S r)
S_cover : K ⊆ ⋃ r, ⋃ (_ : r > 0), S r
S_antitone : ∀ (r₁ r₂ : ℝ), r₁ ≤ r₂ → S r₂ ⊆ S r₁
T : Set ℝ
Tb : T ⊆ Real.lt✝ 0
Tfin : T.Finite
hT : K ⊆ ⋃ i ∈ T, S i
T_nonempty : ¬T.Nonempty
⊢ ∃ r > 0, K ⊆ S r
Module: Mathlib.Analysis.BoxIntegral.Partition.Additive

Goal state before:
pp: ι : Type u_1
M : Type u_2
n : ℕ
N : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : AddCommMonoid N
I₀✝ : WithTop (Box ι)
I : Box ι
i✝ : ι
G : Type u
inst✝ : AddCommGroup G
I₀ : Box (Fin (n + 1))
i : Fin (n + 1)
f : ℝ → Box (Fin n) → G
fb : ↑(Set.Icc (I₀.lower i) (I₀.upper i)) → Fin n →ᵇᵃ[↑(I₀.face i)] G
hf : ∀ (x : ℝ) (hx : x ∈ Set.Icc (I₀.lower i) (I₀.upper i)) (J : Box (Fin n)), f x J = (fb ⟨x, hx⟩) J
⊢ ∀ (I : Box (Fin (n + 1))),
    ↑I ≤ ↑I₀ →
      ∀ {i_1 : Fin (n + 1)} {x : ℝ},
        x ∈ Set.Ioo (I.lower i_1) (I.upper i_1) →
          Option.elim' 0 (fun J ↦ f (J.upper i) (J.face i) - f (J.lower i) (J.face i)) (I.splitLower i_1 x) +
              Option.elim' 0 (fun J ↦ f (J.upper i) (J.face i) - f (J.lower i) (J.face i)) (I.splitUpper i_1 x) =
            f (I.upper i) (I.face i) - f (I.lower i) (I.face i)

Tactic:
intro J hJ j x
      rw [WithTop.coe_le_coe] at hJ
      refine i.succAboveCases (fun hx => ?_) (fun j hx => ?_) j
      · simp only [Box.splitLower_def hx, Box.splitUpper_def hx, update_self, ← WithBot.some_eq_coe,
          Option.elim', Box.face, Function.comp_def, update_of_ne (Fin.succAbove_ne _ _)]
        abel
      · have : (J.face i : WithTop (Box (Fin n))) ≤ I₀.face i :=
          WithTop.coe_le_coe.2 (face_mono hJ i)
        rw [le_iff_Icc, @Box.Icc_eq_pi _ I₀] at hJ
        rw [hf _ (hJ J.upper_mem_Icc _ trivial), hf _ (hJ J.lower_mem_Icc _ trivial),
          ← (fb _).map_split_add this j x, ← (fb _).map_split_add this j x]
        have hx' : x ∈ Ioo ((J.face i).lower j) ((J.face i).upper j) := hx
        simp only [Box.splitLower_def hx, Box.splitUpper_def hx, Box.splitLower_def hx',
          Box.splitUpper_def hx', ← WithBot.some_eq_coe, Option.elim', Box.face_mk,
          update_of_ne (Fin.succAbove_ne _ _).symm, sub_add_sub_comm,
          update_comp_eq_of_injective _ (Fin.strictMono_succAbove i).injective j x, ← hf]
        simp only [Box.face]

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

Goal state before:
pp: 𝕜 : Type u_3
inst✝⁵ : RCLike 𝕜
E : Type u_4
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
m : Type u_7
n : Type u_8
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : Fintype m
b : OrthonormalBasis n 𝕜 E
b₂ : OrthonormalBasis m 𝕜 𝕜
x : E
i✝ : m
j✝ : n
⊢ (toMatrix b.toBasis b₂.toBasis) ((innerₛₗ 𝕜) x) i✝ j✝ = vecMulVec (star ⇑b₂) (star (b.repr x).ofLp) i✝ j✝

Tactic:
simp [LinearMap.toMatrix_apply, vecMulVec_apply, OrthonormalBasis.repr_apply_apply, mul_comm]

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

Goal state before:
pp: 𝕜 : Type u_7
E : Type u_8
F : Type u_9
ι : Type u_10
ι' : Type u_11
inst✝⁷ : RCLike 𝕜
inst✝⁶ : SeminormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 F
inst✝² : Finite ι
inst✝¹ : Fintype ι'
inst✝ : DecidableEq ι'
x : E
y : F
b : Basis ι 𝕜 E
b' : OrthonormalBasis ι' 𝕜 F
this : Fintype ι
⊢ Matrix.replicateCol Unit ⇑(b.repr x) * Matrix.vecMulVec (star 1) (star (b'.repr y).ofLp) =
    Matrix.vecMulVec (⇑(b.repr x)) (star (b'.repr y).ofLp)

Tactic:
star_one,

Goal state after:
𝕜 : Type u_7
E : Type u_8
F : Type u_9
ι : Type u_10
ι' : Type u_11
inst✝⁷ : RCLike 𝕜
inst✝⁶ : SeminormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 F
inst✝² : Finite ι
inst✝¹ : Fintype ι'
inst✝ : DecidableEq ι'
x : E
y : F
b : Basis ι 𝕜 E
b' : OrthonormalBasis ι' 𝕜 F
this : Fintype ι
⊢ Matrix.replicateCol Unit ⇑(b.repr x) * Matrix.vecMulVec 1 (star (b'.repr y).ofLp) =
    Matrix.vecMulVec (⇑(b.repr x)) (star (b'.repr y).ofLp)
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: case pos
ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
inst✝ : Fintype ι
B I : Box ι
hI : ∃ ν ∈ admissibleIndex n B, I = box n ν
⊢ tag n hI.choose ∈ Box.Icc B

Tactic:
exact Box.coe_subset_Icc <| (mem_admissibleIndex_iff.mp hI.choose_spec.1) (tag_mem n _)

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
hf : Pairwise fun x1 x2 ↦ ⟪f x1, f x2⟫_𝕜 = 0
i : ι
⊢ f i - ∑ i_1 < i, (𝕜 ∙ gramSchmidt 𝕜 f i_1).starProjection (f i) = f i

Tactic:
trans f i - 0

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
hf : Pairwise fun x1 x2 ↦ ⟪f x1, f x2⟫_𝕜 = 0
i : ι
⊢ f i - ∑ i_1 < i, (𝕜 ∙ gramSchmidt 𝕜 f i_1).starProjection (f i) = f i - 0

𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
hf : Pairwise fun x1 x2 ↦ ⟪f x1, f x2⟫_𝕜 = 0
i : ι
⊢ f i - 0 = f i
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: case refine_1
ι : Type u_1
s : Set (ι → ℝ)
inst✝ : Fintype ι
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
hs₄ : ∀ ⦃x y : ℝ⦄, 0 < x → x ≤ y → x • s ⊆ y • s
⊢ Tendsto (fun b ↦ ↑(Nat.card ↑(s ∩ (↑⌊b⌋₊)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))))) / b ^ card ι) atTop
    (𝓝 (volume.real s * 1 ^ card ι))

Tactic:
refine Tendsto.congr' (tendsto_card_div_pow₅ s) (Tendsto.mul ?_ (Tendsto.pow ?_ _))

Goal state after:
case refine_1.refine_1
ι : Type u_1
s : Set (ι → ℝ)
inst✝ : Fintype ι
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
hs₄ : ∀ ⦃x y : ℝ⦄, 0 < x → x ≤ y → x • s ⊆ y • s
⊢ Tendsto (fun x ↦ ↑(Nat.card ↑(s ∩ (↑⌊x⌋₊)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))))) / ↑⌊x⌋₊ ^ card ι) atTop
    (𝓝 (volume.real s))

case refine_1.refine_2
ι : Type u_1
s : Set (ι → ℝ)
inst✝ : Fintype ι
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
hs₄ : ∀ ⦃x y : ℝ⦄, 0 < x → x ≤ y → x • s ⊆ y • s
⊢ Tendsto (fun x ↦ ↑⌊x⌋₊ / x) atTop (𝓝 1)
Module: Mathlib.LinearAlgebra.Orientation

Goal state before:
pp: R : Type u_1
inst✝⁶ : Field R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : FiniteDimensional R M
x₁ x₂ : Orientation R M ι
h : Fintype.card ι = finrank R M
e : Basis ι R M
this : DecidableEq ι := Classical.decEq ι
⊢ x₁ = x₂ ∨ x₁ = -x₂

Tactic:
rcases e.orientation_eq_or_eq_neg x₁ with (h₁ | h₁) <;>
    rcases e.orientation_eq_or_eq_neg x₂ with (h₂ | h₂)

Goal state after:
case inl.inl
R : Type u_1
inst✝⁶ : Field R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : FiniteDimensional R M
x₁ x₂ : Orientation R M ι
h : Fintype.card ι = finrank R M
e : Basis ι R M
this : DecidableEq ι := Classical.decEq ι
h₁ : x₁ = e.orientation
h₂ : x₂ = e.orientation
⊢ x₁ = x₂ ∨ x₁ = -x₂

case inl.inr
R : Type u_1
inst✝⁶ : Field R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : FiniteDimensional R M
x₁ x₂ : Orientation R M ι
h : Fintype.card ι = finrank R M
e : Basis ι R M
this : DecidableEq ι := Classical.decEq ι
h₁ : x₁ = e.orientation
h₂ : x₂ = -e.orientation
⊢ x₁ = x₂ ∨ x₁ = -x₂

case inr.inl
R : Type u_1
inst✝⁶ : Field R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : FiniteDimensional R M
x₁ x₂ : Orientation R M ι
h : Fintype.card ι = finrank R M
e : Basis ι R M
this : DecidableEq ι := Classical.decEq ι
h₁ : x₁ = -e.orientation
h₂ : x₂ = e.orientation
⊢ x₁ = x₂ ∨ x₁ = -x₂

case inr.inr
R : Type u_1
inst✝⁶ : Field R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : FiniteDimensional R M
x₁ x₂ : Orientation R M ι
h : Fintype.card ι = finrank R M
e : Basis ι R M
this : DecidableEq ι := Classical.decEq ι
h₁ : x₁ = -e.orientation
h₂ : x₂ = -e.orientation
⊢ x₁ = x₂ ∨ x₁ = -x₂
Module: Mathlib.Analysis.InnerProductSpace.ProdL2

Goal state before:
pp: 𝕜 : Type u_1
ι₁ : Type u_2
ι₂ : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
K : Submodule 𝕜 E
inst✝ : K.HasOrthogonalProjection
x✝ : E
x y z : E ⧸ K
⊢ ⟪K.quotientEquivOrthogonal x + K.quotientEquivOrthogonal y, K.quotientEquivOrthogonal z⟫_𝕜 =
    ⟪K.quotientEquivOrthogonal x, K.quotientEquivOrthogonal z⟫_𝕜 +
      ⟪K.quotientEquivOrthogonal y, K.quotientEquivOrthogonal z⟫_𝕜

Tactic:
inner_add_left

Goal state after:
𝕜 : Type u_1
ι₁ : Type u_2
ι₂ : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
K : Submodule 𝕜 E
inst✝ : K.HasOrthogonalProjection
x✝ : E
x y z : E ⧸ K
⊢ ⟪K.quotientEquivOrthogonal x, K.quotientEquivOrthogonal z⟫_𝕜 +
      ⟪K.quotientEquivOrthogonal y, K.quotientEquivOrthogonal z⟫_𝕜 =
    ⟪K.quotientEquivOrthogonal x, K.quotientEquivOrthogonal z⟫_𝕜 +
      ⟪K.quotientEquivOrthogonal y, K.quotientEquivOrthogonal z⟫_𝕜
Module: Mathlib.Analysis.Normed.Lp.MeasurableSpace

Goal state before:
pp: p : ℝ≥0∞
X✝ : Type u_1
inst✝⁵ : MeasurableSpace X✝
ι : Type u_2
X : ι → Type u_3
inst✝⁴ : Countable ι
inst✝³ : (i : ι) → MeasurableSpace (X i)
inst✝² : (i : ι) → TopologicalSpace (X i)
inst✝¹ : ∀ (i : ι), BorelSpace (X i)
inst✝ : ∀ (i : ι), SecondCountableTopology (X i)
⊢ WithLp.measurableSpace p ((i : ι) → X i) = MeasurableSpace.comap WithLp.ofLp (borel ((i : ι) → X i))

Tactic:
WithLp.measurableSpace,

Goal state after:
p : ℝ≥0∞
X✝ : Type u_1
inst✝⁵ : MeasurableSpace X✝
ι : Type u_2
X : ι → Type u_3
inst✝⁴ : Countable ι
inst✝³ : (i : ι) → MeasurableSpace (X i)
inst✝² : (i : ι) → TopologicalSpace (X i)
inst✝¹ : ∀ (i : ι), BorelSpace (X i)
inst✝ : ∀ (i : ι), SecondCountableTopology (X i)
⊢ MeasurableSpace.comap WithLp.ofLp inferInstance = MeasurableSpace.comap WithLp.ofLp (borel ((i : ι) → X i))
Module: Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace

Goal state before:
pp: ι : Type u_1
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace ℝ F
inst✝³ : MeasurableSpace F
inst✝² : BorelSpace F
inst✝¹ : Fintype ι
inst✝ : FiniteDimensional ℝ F
n : ℕ
_i : Fact (finrank ℝ F = n)
o : Orientation ℝ F (Fin n)
b : OrthonormalBasis ι ℝ F
⊢ Fintype.card ι = finrank ℝ F

Tactic:
finrank_eq_card_basis b.toBasis

Goal state after:
ι : Type u_1
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace ℝ F
inst✝³ : MeasurableSpace F
inst✝² : BorelSpace F
inst✝¹ : Fintype ι
inst✝ : FiniteDimensional ℝ F
n : ℕ
_i : Fact (finrank ℝ F = n)
o : Orientation ℝ F (Fin n)
b : OrthonormalBasis ι ℝ F
⊢ Fintype.card ι = Fintype.card ι
Module: Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace

Goal state before:
pp: ι : Type u_1
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace ℝ F
inst✝³ : MeasurableSpace F
inst✝² : BorelSpace F
inst✝¹ : Fintype ι
inst✝ : FiniteDimensional ℝ F
b : OrthonormalBasis ι ℝ F
this : Fact (finrank ℝ F = finrank ℝ F)
o : Orientation ℝ F (Fin (finrank ℝ F)) := (stdOrthonormalBasis ℝ F).toBasis.orientation
⊢ volume (parallelepiped ⇑b) = 1

Tactic:
← o.measure_eq_volume

Goal state after:
ι : Type u_1
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace ℝ F
inst✝³ : MeasurableSpace F
inst✝² : BorelSpace F
inst✝¹ : Fintype ι
inst✝ : FiniteDimensional ℝ F
b : OrthonormalBasis ι ℝ F
this : Fact (finrank ℝ F = finrank ℝ F)
o : Orientation ℝ F (Fin (finrank ℝ F)) := (stdOrthonormalBasis ℝ F).toBasis.orientation
⊢ o.volumeForm.measure (parallelepiped ⇑b) = 1