mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
pp: case h
G : Type u_1
inst✝⁴ : Group G
inst✝³ : TopologicalSpace G
inst✝² : IsTopologicalGroup G
inst✝¹ : MeasurableSpace G
inst✝ : BorelSpace G
K₀ : PositiveCompacts G
⊢ (haarMeasure K₀) ↑K₀ ≠ 0

Tactic:
simp only [haarMeasure_self]; exact one_ne_zero

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

Goal state before:
pp: G : Type u_1
inst✝⁷ : Group G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : SigmaFinite μ
inst✝ : μ.IsMulLeftInvariant
K : Set G
hK : IsCompact K
h2K : (interior K).Nonempty
hμK : μ K ≠ ∞
⊢ μ.Regular

Tactic:
by
  rw [haarMeasure_unique μ ⟨⟨K, hK⟩, h2K⟩]; exact Regular.smul hμK

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.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
x : E
r : 𝕜
hx : x ≠ 0
hr : r ≠ 0
⊢ ‖x‖ ≠ 0

Tactic:
simp [hx]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.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
x y : E
⊢ ⟪x, y⟫ = ↑‖x‖ * ↑‖y‖ ↔ ↑‖y‖ • x = ↑‖x‖ • y

Tactic:
rcases eq_or_ne x 0 with (rfl | h₀)

Goal state after:
case inl
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
y : E
⊢ ⟪0, y⟫ = ↑‖0‖ * ↑‖y‖ ↔ ↑‖y‖ • 0 = ↑‖0‖ • y

case inr
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
⊢ ⟪x, y⟫ = ↑‖x‖ * ↑‖y‖ ↔ ↑‖y‖ • x = ↑‖x‖ • y
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
⊢ ⟪x, -y⟫_ℝ / (‖x‖ * ‖y‖) = 1 ↔ x ≠ 0 ∧ ∃ r < 0, y = r • x

Tactic:
← norm_neg y,

Goal state after:
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
⊢ ⟪x, -y⟫_ℝ / (‖x‖ * ‖-y‖) = 1 ↔ x ≠ 0 ∧ ∃ r < 0, y = r • x
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
⊢ ‖x + y‖ = ‖x‖ + ‖y‖ ↔ ‖y‖ • x = ‖x‖ • y

Tactic:
by
  rw [← pow_left_inj₀ (norm_nonneg _) (Left.add_nonneg (norm_nonneg _) (norm_nonneg _)) two_ne_zero,
    norm_add_sq (𝕜 := ℝ), add_pow_two, add_left_inj, add_right_inj, re_to_real, mul_assoc,
    mul_right_inj' two_ne_zero, ← inner_eq_norm_mul_iff_real]

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

Goal state before:
pp: 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✝ : Subsingleton ι
v : ι → E
⊢ Orthonormal 𝕜 v ↔ ∀ (i : ι), ‖v i‖ = 1

Tactic:
simp [orthonormal_iff_ite, ← map_pow, pow_eq_one_iff_of_nonneg]

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
ι : Type u_4
v : ι → E
hv : Orthonormal 𝕜 v
i : ι
⊢ ‖v i‖ ≠ ‖0‖

Tactic:
rw [hv.1 i, norm_zero]

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
i : ι
⊢ 1 ≠ 0
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: 𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
F : Type u_8
H : Type u_9
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
a c : F
b d : H
⊢ (∀ (x : H), (((rankOne 𝕜) a) b) x = (((rankOne 𝕜) c) d) x) ↔ ∀ (x : F), (((rankOne 𝕜) b) a) x = (((rankOne 𝕜) d) c) x

Tactic:
ext_iff_inner_left 𝕜 (E := F),

Goal state after:
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
F : Type u_8
H : Type u_9
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
a c : F
b d : H
⊢ (∀ (x : H) (v : F), ⟪v, (((rankOne 𝕜) a) b) x⟫_𝕜 = ⟪v, (((rankOne 𝕜) c) d) x⟫_𝕜) ↔
    ∀ (x : F), (((rankOne 𝕜) b) a) x = (((rankOne 𝕜) d) c) 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
x : E
v : ι → E
s : Finset ι
hv : Orthonormal 𝕜 v
h₂ : ∑ i ∈ s, ∑ j ∈ s, ⟪v i, x⟫ * ⟪x, v j⟫ * ⟪v j, v i⟫ = ∑ k ∈ s, ⟪v k, x⟫ * ⟪x, v k⟫
z : 𝕜
⊢ re (z * (starRingEnd 𝕜) z) = ‖z‖ ^ 2

Tactic:
simp only [mul_conj]

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
x : E
v : ι → E
s : Finset ι
hv : Orthonormal 𝕜 v
h₂ : ∑ i ∈ s, ∑ j ∈ s, ⟪v i, x⟫ * ⟪x, v j⟫ * ⟪v j, v i⟫ = ∑ k ∈ s, ⟪v k, x⟫ * ⟪x, v k⟫
z : 𝕜
⊢ re (↑‖z‖ ^ 2) = ‖z‖ ^ 2
Module: Mathlib.Analysis.InnerProductSpace.Orthogonal

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
x : E
⊢ x ∈ K ∧ x ∈ Kᗮ → x ∈ ⊥

Tactic:
exact fun ⟨hx, ho⟩ => inner_self_eq_zero.1 (ho x hx)

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 : E →ₗ[𝕜] E
hT : T.IsSymmetric
n : ℕ
⊢ (T ^ n).IsSymmetric

Tactic:
by
  refine Nat.le_induction (by simp [Module.End.one_eq_id]) (fun k _ ih ↦ ?_) n n.zero_le
  rw [Module.End.iterate_succ, ← Module.End.mul_eq_comp]
  exact ih.mul_of_commute hT <| .pow_left rfl k

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✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : InnerProductSpace ℝ F
K : Submodule 𝕜 E
inst✝ : CompleteSpace ↥K
v w : E
hwK : w ∈ K
hw : ‖v - w‖ = ⨅ w, ‖v - ↑w‖
⊢ ∃ w ∈ K, v - w ∈ Kᗮ

Tactic:
refine ⟨w, hwK, (K.mem_orthogonal' _).2 ?_⟩

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : InnerProductSpace ℝ F
K : Submodule 𝕜 E
inst✝ : CompleteSpace ↥K
v w : E
hwK : w ∈ K
hw : ‖v - w‖ = ⨅ w, ‖v - ↑w‖
⊢ ∀ u ∈ K, ⟪v - w, u⟫ = 0
Module: Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional

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
ι : Type u_4
inst✝ : DecidableEq ι
V : ι → Submodule 𝕜 E
hV : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ
hc : IsComplete ↑(iSup V)
this : CompleteSpace ↥(iSup V)
⊢ DirectSum.IsInternal V ↔ (iSup V)ᗮ = ⊥

Tactic:
simp only [DirectSum.isInternal_submodule_iff_iSupIndep_and_iSup_eq_top, hV.independent,
    true_and, orthogonal_eq_bot_iff]

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

Goal state before:
pp: case intro
ι : Type u_5
inst✝¹ : Finite ι
α : ι → Type u_6
inst✝ : (i : ι) → PseudoMetricSpace (α i)
f g : PiLp ∞ α
val✝ : Fintype ι
M : { r // 0 ≤ r }
hM : ∀ (i : ι), ⟨dist (f.ofLp i) (g.ofLp i), ⋯⟩ ≤ M
i : ι
⊢ ↑(NNReal.mk (dist (f.ofLp i) (g.ofLp i)) ⋯) ≤ ↑M

Tactic:
exact mod_cast hM i

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

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : SeminormedAddCommGroup α
inst✝ : SeminormedAddCommGroup β
x : WithLp 1 (α × β)
⊢ ‖x‖ = ‖idemFst x‖ + ‖idemSnd x‖

Tactic:
by
  rw [prod_norm_eq_add_idemFst (lt_of_lt_of_eq zero_lt_one toReal_one.symm)]
  simp only [toReal_one, Real.rpow_one, ne_eq, one_ne_zero, not_false_eq_true, div_self]

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

Goal state before:
pp: case mp
ι : Type u_1
E : Type u_3
inst✝² : Fintype ι
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
v : ι → E
x✝ : E
⊢ (∃ t, (∀ (i : ι), t i ∈ Icc (0 i) (1 i)) ∧ x✝ = ∑ i, t i • v i) →
    ∃ g, ∃ (_ : ∀ {i : ι}, g i ∈ (fun a ↦ a • v i) '' Icc 0 1), ∑ i, g i = x✝

Tactic:
rintro ⟨t, ht, rfl⟩

Goal state after:
case mp
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)
⊢ ∃ g, ∃ (_ : ∀ {i : ι}, g i ∈ (fun a ↦ a • v i) '' Icc 0 1), ∑ i, g i = ∑ i, t i • v i
Module: Mathlib.MeasureTheory.Measure.Haar.OfBasis

Goal state before:
pp: 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
b : Basis ι ℝ E
H : (interior (⇑b.equivFunL.symm.toHomeomorph '' Icc 0 1)).Nonempty
⊢ (interior ((fun t ↦ ∑ i, t i • b i) '' Icc 0 1)).Nonempty

Tactic:
convert! H

Goal state after:
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
b : Basis ι ℝ E
H : (interior (⇑b.equivFunL.symm.toHomeomorph '' Icc 0 1)).Nonempty
a✝¹ : ι → ℝ
a✝ : a✝¹ ∈ Icc 0 1
⊢ ∑ i, a✝¹ i • b i = b.equivFunL.symm.toHomeomorph a✝¹
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
⊢ ↑(Pi.basisFun ℝ ι).parallelepiped = ↑(PositiveCompacts.piIcc01 ι)

Tactic:
refine Eq.trans ?_ ((uIcc_of_le ?_).trans (Set.pi_univ_Icc _ _).symm)
    · classical convert! parallelepiped_single (ι := ι) 1
    · exact zero_le_one

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

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f g : α → ℝ
s : Set α
hf : Measurable f
hg : Measurable g
hs : MeasurableSet s
⊢ MeasurableSet ({a | a.1 ∈ s} ∩ {a | a.2 ∈ {a_1 | f a.1 < a_1} ∩ {a_1 | a_1 < g a.1}})

Tactic:
refine
    MeasurableSet.inter ?_
      ((measurableSet_lt (hf.comp measurable_fst) measurable_snd).inter
        (measurableSet_lt measurable_snd (hg.comp measurable_fst)))

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f g : α → ℝ
s : Set α
hf : Measurable f
hg : Measurable g
hs : MeasurableSet s
⊢ MeasurableSet {a | a.1 ∈ s}
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
A : Tendsto (fun r ↦ ENNReal.ofReal (r ^ finrank ℝ E) * μ (closedBall 0 1)) (𝓝[<] 1) (𝓝 (μ (closedBall 0 1)))
r : ℝ
hr : r ∈ Ioo 0 1
⊢ μ (closedBall 0 r) ≤ μ (ball 0 1)

Tactic:
exact measure_mono (closedBall_subset_ball hr.2)

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

Goal state before:
pp: ι : Type u_1
I J : Box ι
tfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J
tfae_2_to_3 : ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper
tfae_3_iff_4 : Icc I.lower I.upper ⊆ Icc J.lower J.upper ↔ J.lower ≤ I.lower ∧ I.upper ≤ J.upper
⊢ [I ≤ J, ↑I ⊆ ↑J, Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper].TFAE

Tactic:
tfae_have 4 → 2
  | h, x, hx, i => Ioc_subset_Ioc (h.1 i) (h.2 i) (hx i)

Goal state after:
ι : Type u_1
I J : Box ι
tfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J
tfae_2_to_3 : ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper
tfae_3_iff_4 : Icc I.lower I.upper ⊆ Icc J.lower J.upper ↔ J.lower ≤ I.lower ∧ I.upper ≤ J.upper
tfae_4_to_2 : J.lower ≤ I.lower ∧ I.upper ≤ J.upper → ↑I ⊆ ↑J
⊢ [I ≤ J, ↑I ⊆ ↑J, Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper].TFAE
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: E : Type u_1
ι : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝³ : Fintype ι
inst✝² : MeasurableSpace E
μ : Measure E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
this✝ : FiniteDimensional ℝ E
x : E
hx : (∀ (i : ι), 0 ≤ (b.repr x) i ∧ (b.repr x) i ≤ 1) ∧ ∃ x_1, 0 ≤ (b.repr x) x_1 → 1 ≤ (b.repr x) x_1
i : ι
hi : 0 ≤ (b.repr x) i → 1 ≤ (b.repr x) i
this : (b.repr x) i = 1
⊢ ∑ x_1 ∈ Finset.univ.erase i, (b.repr x) x_1 • b x_1 + b i ∈
    ⋃ i, ↑(AffineSubspace.mk' (b i) (span ℝ (⇑b '' (Set.univ \ {i}))))

Tactic:
← vadd_eq_add

Goal state after:
E : Type u_1
ι : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝³ : Fintype ι
inst✝² : MeasurableSpace E
μ : Measure E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
this✝ : FiniteDimensional ℝ E
x : E
hx : (∀ (i : ι), 0 ≤ (b.repr x) i ∧ (b.repr x) i ≤ 1) ∧ ∃ x_1, 0 ≤ (b.repr x) x_1 → 1 ≤ (b.repr x) x_1
i : ι
hi : 0 ≤ (b.repr x) i → 1 ≤ (b.repr x) i
this : (b.repr x) i = 1
⊢ ∑ x_1 ∈ Finset.univ.erase i, (b.repr x) x_1 • b x_1 +ᵥ b i ∈
    ⋃ i, ↑(AffineSubspace.mk' (b i) (span ℝ (⇑b '' (Set.univ \ {i}))))
Module: Mathlib.Analysis.BoxIntegral.Box.SubboxInduction

Goal state before:
pp: ι : Type u_1
I✝ J I : Box ι
s : Set ι
i : ι
⊢ s.piecewise (fun i ↦ (I.lower i + I.upper i) / 2) I.lower i <
    s.piecewise I.upper (fun i ↦ (I.lower i + I.upper i) / 2) i

Tactic:
dsimp only [Set.piecewise]
    split_ifs <;> simp only [left_lt_add_div_two, add_div_two_lt_right, I.lower_lt_upper]

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✝² : Module.Finite ℤ ↥L
this✝¹ : 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
this : LinearIndependent ℤ b
h_spanL : span ℤ (Set.range b) = L
⊢ span K (Set.range b) = ⊤

Tactic:
← span_span_of_tower (R := ℤ),

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✝² : Module.Finite ℤ ↥L
this✝¹ : 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
this : LinearIndependent ℤ b
h_spanL : span ℤ (Set.range b) = L
⊢ span K ↑(span ℤ (Set.range b)) = ⊤
Module: Mathlib.Analysis.BoxIntegral.Partition.Basic

Goal state before:
pp: ι : Type u_1
I J : Box ι
π₁ π₂ : Prepartition I
Hle : π₁ ≤ π₂
J₁ : WithBot (Box ι)
hJ₁ : J₁ ∈ Finset.image (fun J' ↦ ↑J ⊓ ↑J') π₁.boxes
hne : J₁ ≠ ⊥
⊢ ∃ J' ∈ Finset.image (fun J' ↦ ↑J ⊓ ↑J') π₂.boxes, J₁ ≤ J'

Tactic:
rw [Finset.mem_image] at hJ₁

Goal state after:
ι : Type u_1
I J : Box ι
π₁ π₂ : Prepartition I
Hle : π₁ ≤ π₂
J₁ : WithBot (Box ι)
hJ₁ : ∃ a ∈ π₁.boxes, ↑J ⊓ ↑a = J₁
hne : J₁ ≠ ⊥
⊢ ∃ J' ∈ Finset.image (fun J' ↦ ↑J ⊓ ↑J') π₂.boxes, J₁ ≤ J'
Module: Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
I : Box ι
x : ι → ℝ
hx : x ∈ I
⊢ ∃ J ∈ splitCenter I, x ∈ J

Tactic:
simp [hx]

Goal state after:
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
i j : ι
⊢ (starRingEnd 𝕜) 1 * (EuclideanSpace.single j 1).ofLp i = if i = j then 1 else 0

Tactic:
PiLp.single_apply,

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
i j : ι
⊢ ((starRingEnd 𝕜) 1 * if i = j then 1 else 0) = 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⟫‖ ^ 2) ^ 2

Tactic:
Real.sq_sqrt

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

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
⊢ 0 ≤ ∑ x_1, ‖⟪b x_1, x⟫‖ ^ 2
Module: Mathlib.Analysis.BoxIntegral.Partition.Split

Goal state before:
pp: case insert
ι : Type u_1
I : Box ι
π : Prepartition I
p : ι × ℝ
s : Finset (ι × ℝ)
a✝ : p ∉ s
ihp : π ⊓ splitMany I s = π.biUnion fun J ↦ splitMany J s
⊢ π ⊓ splitMany I s ⊓ split I p.1 p.2 = π.biUnion fun J ↦ splitMany J s ⊓ split J p.1 p.2

Tactic:
ihp,

Goal state after:
case insert
ι : Type u_1
I : Box ι
π : Prepartition I
p : ι × ℝ
s : Finset (ι × ℝ)
a✝ : p ∉ s
ihp : π ⊓ splitMany I s = π.biUnion fun J ↦ splitMany J s
⊢ (π.biUnion fun J ↦ splitMany J s) ⊓ split I p.1 p.2 = π.biUnion fun J ↦ splitMany J s ⊓ split J p.1 p.2
Module: Mathlib.Analysis.Oscillation

Goal state before:
pp: E : Type u
F : Type v
inst✝¹ : PseudoEMetricSpace F
inst✝ : TopologicalSpace E
f : E → F
D : Set E
x : E
hD : D ∈ 𝓝 x
⊢ ⨅ S ∈ Filter.map f (𝓝[D] x), ediam S = ⨅ S ∈ Filter.map f (𝓝 x), ediam S

Tactic:
nhdsWithin_eq_nhds.2 hD

Goal state after:
E : Type u
F : Type v
inst✝¹ : PseudoEMetricSpace F
inst✝ : TopologicalSpace E
f : E → F
D : Set E
x : E
hD : D ∈ 𝓝 x
⊢ ⨅ S ∈ Filter.map f (𝓝 x), ediam S = ⨅ S ∈ Filter.map f (𝓝 x), ediam S
Module: Mathlib.Analysis.Oscillation

Goal state before:
pp: E : Type u
F : Type v
inst✝¹ : PseudoEMetricSpace F
inst✝ : TopologicalSpace E
f : E → F
D : Set E
x : E
xD : x ∈ D
hf : oscillationWithin f D x = 0
ε : ℝ≥0∞
ε0 : ε > ⨅ S ∈ Filter.map f (𝓝[D] x), ediam S
⊢ ∀ᶠ (x_1 : E) in 𝓝[D] x, edist (f x_1) (f x) < ε

Tactic:
iInf_lt_iff

Goal state after:
E : Type u
F : Type v
inst✝¹ : PseudoEMetricSpace F
inst✝ : TopologicalSpace E
f : E → F
D : Set E
x : E
xD : x ∈ D
hf : oscillationWithin f D x = 0
ε : ℝ≥0∞
ε0 : ∃ i, ∃ (_ : i ∈ Filter.map f (𝓝[D] x)), ediam i < ε
⊢ ∀ᶠ (x_1 : E) in 𝓝[D] x, edist (f x_1) (f x) < ε
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'
inst✝² : Fintype ι
v : Set E
A : ι → Submodule 𝕜 E
inst✝¹ : FiniteDimensional 𝕜 E
n : ℕ
hn : finrank 𝕜 E = n
inst✝ : DecidableEq ι
V : ι → Submodule 𝕜 E
hV : IsInternal V
hV' : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ
i : ι
a : { a // subordinateOrthonormalBasisIndex hn hV a hV' = i }
⊢ finrank 𝕜 ↥(V ((sigmaOrthonormalBasisIndexEquiv hn hV hV').symm ↑a).fst) = finrank 𝕜 ↥(V i)

Tactic:
← subordinateOrthonormalBasisIndex_def,

Goal state after:
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'
inst✝² : Fintype ι
v : Set E
A : ι → Submodule 𝕜 E
inst✝¹ : FiniteDimensional 𝕜 E
n : ℕ
hn : finrank 𝕜 E = n
inst✝ : DecidableEq ι
V : ι → Submodule 𝕜 E
hV : IsInternal V
hV' : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ
i : ι
a : { a // subordinateOrthonormalBasisIndex hn hV a hV' = i }
⊢ finrank 𝕜 ↥(V (subordinateOrthonormalBasisIndex hn hV (↑a) hV')) = finrank 𝕜 ↥(V i)
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
x : ℝ
a : ℤ
h : 0 < ↑n
⊢ ↑a < x ↔ ↑n * a < ⌈↑n * x⌉

Tactic:
Int.lt_ceil,

Goal state after:
n : ℕ
inst✝ : NeZero n
x : ℝ
a : ℤ
h : 0 < ↑n
⊢ ↑a < x ↔ ↑(↑n * a) < ↑n * x
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: case refine_2
ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
inst✝ : Fintype ι
B : Box ι
x : ι → ℝ
hx : x ∈ B
l u : ι → ℤ
hl : ∀ (i : ι), B.lower i = ↑(l i)
hu : ∀ (i : ι), B.upper i = ↑(u i)
i : ι
⊢ (↑⌈↑n * x i⌉ - 1 + 1) / ↑n ≤ ↑(u i)

Tactic:
· exact (mem_admissibleIndex_of_mem_box_aux₂ n (x i) (u i)).mp ((hu i) ▸ (hx i).2)

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

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
ι : Type u_2
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
e f : OrthonormalBasis ι ℝ E
h : e.toBasis.orientation ≠ f.toBasis.orientation
⊢ e.toBasis.det = -f.toBasis.det

Tactic:
by
  rw [e.toBasis.det.eq_smul_basis_det f.toBasis]
  simp [e.det_to_matrix_orthonormalBasis_of_opposite_orientation f h]

Goal state after:
Module: Mathlib.LinearAlgebra.Orientation

Goal state before:
pp: case neg
R : Type u_1
inst✝⁷ : CommRing R
inst✝⁶ : LinearOrder R
inst✝⁵ : IsStrictOrderedRing R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
ι : Type u_3
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : Nonempty ι
e : Basis ι R M
x : Orientation R M ι
h : ¬e.orientation = x
⊢ (e.unitsSMul (Function.update 1 (Classical.arbitrary ι) (-1))).orientation = x

Tactic:
orientation_neg_single,

Goal state after:
case neg
R : Type u_1
inst✝⁷ : CommRing R
inst✝⁶ : LinearOrder R
inst✝⁵ : IsStrictOrderedRing R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
ι : Type u_3
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : Nonempty ι
e : Basis ι R M
x : Orientation R M ι
h : ¬e.orientation = x
⊢ -e.orientation = x