mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case refine_3
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
μ : Measure X
f g : X → E
s : Set X
inst✝ : ClosedIciTopology E
hf : IntegrableOn f s μ
hg : IntegrableOn g s μ
hs : NullMeasurableSet s μ
h : ∀ᵐ (x : X) ∂μ, x ∈ s → f x ≤ g x
⊢ MeasurableSet (toMeasurable μ s)

Tactic:
exact measurableSet_toMeasurable μ s

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case refine_4
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
μ : Measure X
f g : X → E
s : Set X
inst✝ : ClosedIciTopology E
hf : IntegrableOn f s μ
hg : IntegrableOn g s μ
hs : NullMeasurableSet s μ
h : ∀ᵐ (x : X) ∂μ, x ∈ s → f x ≤ g x
⊢ ∀ᵐ (x : X) ∂μ, x ∈ toMeasurable μ s → f x ≤ g x

Tactic:
filter_upwards [hs.toMeasurable_ae_eq.mem_iff, h] with x hx h
    rwa [hx]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁸ : LinearOrder R
inst✝⁷ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁶ : OrderTopology R
inst✝⁵ : CompactIccSpace R
inst✝⁴ : MeasurableSpace R
inst✝³ : BorelSpace R
inst✝² : SecondCountableTopology R
inst✝¹ : DenselyOrdered R
inst✝ : Nonempty R
hf : Tendsto (↑f) atTop atTop
inhabited_h : Inhabited R
⊢ f.measure (Ioi default) ≤ f.measure univ

Tactic:
exact measure_mono (subset_univ _)

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁸ : LinearOrder R
inst✝⁷ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁶ : OrderTopology R
inst✝⁵ : CompactIccSpace R
inst✝⁴ : MeasurableSpace R
inst✝³ : BorelSpace R
inst✝² : SecondCountableTopology R
inst✝¹ : DenselyOrdered R
inst✝ : Nonempty R
hf : Tendsto (↑f) atBot atBot
inhabited_h : Inhabited R
⊢ f.measure (Iio default) ≤ f.measure univ

Tactic:
exact measure_mono (subset_univ _)

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
F : Type u_4
mX : MeasurableSpace X
𝕜 : Type u_5
inst✝³ : NormedRing 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : Module 𝕜 F
inst✝ : IsBoundedSMul 𝕜 F
p : ℝ≥0∞
μ : Measure X
c : 𝕜
f : ↥(Lp F p μ)
s : Set X
⊢ MemLp.toLp ↑↑(c • f) ⋯ = c • MemLp.toLp ↑↑f ⋯

Tactic:
by
  ext1
  refine (ae_restrict_of_ae (Lp.coeFn_smul c f)).mp ?_
  refine (MemLp.coeFn_toLp ((Lp.memLp f).restrict s)).mp ?_
  refine (MemLp.coeFn_toLp ((Lp.memLp (c • f)).restrict s)).mp ?_
  refine
    (Lp.coeFn_smul c (MemLp.toLp f ((Lp.memLp f).restrict s))).mono fun x hx1 hx2 hx3 hx4 => ?_
  simp only [hx2, hx1, hx3, hx4, Pi.smul_apply]

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 : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x : E
⊢ re ⟪x, x⟫ = ‖⟪x, x⟫‖

Tactic:
by
  conv_rhs => rw [← inner_self_ofReal_re]
  symm
  exact norm_of_nonneg inner_self_nonneg

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: Y : Type u_2
E : Type u_3
F : Type u_4
X : Type u_5
G : Type u_6
𝕜 : Type u_7
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : TopologicalSpace Y
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : OpensMeasurableSpace Y
μ : Measure Y
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : NormedSpace 𝕜 E
L : F →L[𝕜] G →L[𝕜] E
f : X → Y → G
s : Set X
k : Set Y
g : Y → F
hk : IsCompact k
hf : ContinuousOn (uncurry f) (s ×ˢ univ)
hfs : ∀ (p : X) (x : Y), p ∈ s → x ∉ k → f p x = 0
hg : IntegrableOn g k μ
A : ∀ p ∈ s, Continuous[inst✝¹⁰, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f p)
q : X
hq : q ∈ s
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : 0 < δ
hδ : ∫ (x : Y) in k, ‖L‖ * ‖g x‖ * δ ∂μ < ε
v : Set X
v_mem : v ∈ 𝓝[s] q
hv : ∀ p ∈ v, ∀ x ∈ k, ‖f p x - f q x‖ < δ
p : X
hp : p ∈ s
C : ℝ
hC : ∀ (y : Y), ‖f p y‖ ≤ C
this : IntegrableOn (fun y ↦ ‖L‖ * ‖g y‖ * C) k μ
this✝¹ : MeasurableSpace G := ⋯
this✝ : BorelSpace G
⊢ support (f p) ⊆ k

Tactic:
apply support_subset_iff'.2 (fun y hy ↦ hfs p y hp hy)

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 ↑{ carrier := K, isCompact' := hK, interior_nonempty' := h2K } •
      haarMeasure { carrier := K, isCompact' := hK, interior_nonempty' := h2K }).Regular

Tactic:
exact Regular.smul hμK

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
ι : Type u_4
E' : Type u_7
inst✝¹ : SeminormedAddCommGroup E'
inst✝ : InnerProductSpace 𝕜 E'
v : ι → E
f : E →ₗᵢ[𝕜] E'
⊢ Orthonormal 𝕜 (⇑f ∘ v) ↔ Orthonormal 𝕜 v

Tactic:
orthonormal_iff_ite,

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
ι : Type u_4
E' : Type u_7
inst✝¹ : SeminormedAddCommGroup E'
inst✝ : InnerProductSpace 𝕜 E'
v : ι → E
f : E →ₗᵢ[𝕜] E'
⊢ Orthonormal 𝕜 (⇑f ∘ v) ↔ Orthonormal 𝕜 v

Tactic:
simp_rw [orthonormal_iff_ite, Function.comp_apply, LinearIsometry.inner_map_map]

Goal state after:
Module: Mathlib.Topology.Baire.CompleteMetrizable

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : IsCompletelyPseudoMetrizableSpace X
x✝ : UpgradedIsCompletelyPseudoMetrizableSpace X := upgradeIsCompletelyPseudoMetrizable X
f : ℕ → Set X
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n ↦ 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → X → ℝ≥0∞ → X
radius : ℕ → X → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : X) (δ : ℝ≥0∞), δ ≠ 0 → 0 < radius n x δ
HB : ∀ (n : ℕ) (x : X) (δ : ℝ≥0∞), δ ≠ 0 → radius n x δ ≤ B (n + 1)
Hball : ∀ (n : ℕ) (x : X) (δ : ℝ≥0∞), δ ≠ 0 → closedEBall (center n x δ) (radius n x δ) ⊆ closedEBall x δ ∩ f n
x : X
ε : ℝ≥0∞
εpos : 0 < ε
F : ℕ → X × ℝ≥0∞ := fun n ↦ Nat.recOn n (x, min ε (B 0)) fun n p ↦ (center n p.1 p.2, radius n p.1 p.2)
c : ℕ → X := fun n ↦ (F n).1
r : ℕ → ℝ≥0∞ := fun n ↦ (F n).2
rpos : ∀ (n : ℕ), 0 < r n
r0 : ∀ (n : ℕ), r n ≠ 0
n : ℕ
⊢ r n ≤ B n

Tactic:
cases n with
    | zero => exact min_le_right _ _
    | succ n => exact HB n (c n) (r n) (r0 n)

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

Goal state before:
pp: case neg
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
T : E →L[𝕜] F
C : ℝ
hC : 0 ≤ C
x : E
h : ‖T x‖ * ‖T x‖ / ‖T x‖ ≤ C
hx : ‖x‖ = 1
hTx : ¬‖T x‖ = 0
⊢ ‖T x‖ ≤ C

Tactic:
mul_self_div_self

Goal state after:
case neg
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
T : E →L[𝕜] F
C : ℝ
hC : 0 ≤ C
x : E
h : ‖T x‖ ≤ C
hx : ‖x‖ = 1
hTx : ¬‖T x‖ = 0
⊢ ‖T x‖ ≤ C
Module: Mathlib.Topology.Algebra.Module.ClosedSubmodule

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

Tactic:
simp only [carrier_eq_coe, ClosedSubmodule.coe_toSubmodule, coe_closure, SetLike.mem_coe]

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

Goal state before:
pp: R : Type u_2
N : Type u_4
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid N
inst✝³ : TopologicalSpace N
inst✝² : Module R N
inst✝¹ : ContinuousAdd N
inst✝ : ContinuousConstSMul R N
S : Set (ClosedSubmodule R N)
⊢ ↑(sSup S) = closure (⨆ s ∈ S, ↑s).carrier

Tactic:
simp only [← coe_toSubmodule, toSubmodule_sSup]

Goal state after:
R : Type u_2
N : Type u_4
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid N
inst✝³ : TopologicalSpace N
inst✝² : Module R N
inst✝¹ : ContinuousAdd N
inst✝ : ContinuousConstSMul R N
S : Set (ClosedSubmodule R N)
⊢ ↑↑(⨆ s ∈ S, ↑s).closure = closure (⨆ s ∈ S, ↑s).carrier
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜' F
f : E →SL[σ] F
σ' : 𝕜' →+* 𝕜
inst✝³ : RingHomInvPair σ σ'
inst✝² : RingHomIsometric σ
inst✝¹ : RingHomIsometric σ'
inst✝ : CompleteSpace F
surj : Surjective ⇑f
A : ⋃ n, closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (⇑f '' ball 0 ↑n) = Set.univ
this : ∃ n x, x ∈ interior (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (⇑f '' ball 0 ↑n))
⊢ ∃ C ≥ 0, ∀ (y : F), ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖

Tactic:
simp only [mem_interior_iff_mem_nhds, Metric.mem_nhds_iff] at this

Goal state after:
𝕜 : Type u_1
𝕜' : Type u_2
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜' F
f : E →SL[σ] F
σ' : 𝕜' →+* 𝕜
inst✝³ : RingHomInvPair σ σ'
inst✝² : RingHomIsometric σ
inst✝¹ : RingHomIsometric σ'
inst✝ : CompleteSpace F
surj : Surjective ⇑f
A : ⋃ n, closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (⇑f '' ball 0 ↑n) = Set.univ
this : ∃ n x, ∃ ε > 0, ball x ε ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (⇑f '' ball 0 ↑n)
⊢ ∃ C ≥ 0, ∀ (y : F), ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
ι : Type u_2
β : ι → Type u_4
inst✝² : Fact (1 ≤ p)
inst✝¹ : (i : ι) → PseudoEMetricSpace (β i)
inst✝ : Fintype ι
x y : PiLp p β
i : ι
h : 1 ≤ p.toReal
cancel : p.toReal * (1 / p.toReal) = 1
⊢ edist (x.ofLp i) (y.ofLp i) = (edist (x.ofLp i) (y.ofLp i) ^ p.toReal) ^ (1 / p.toReal)

Tactic:
by
        simp [← ENNReal.rpow_mul, cancel, -one_div]

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹⁴ : NontriviallyNormedField 𝕜
inst✝¹³ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
F✝ : Type u_4
inst✝¹⁰ : NormedAddCommGroup F✝
inst✝⁹ : NormedSpace 𝕜' F✝
f✝ : E →SL[σ] F✝
σ' : 𝕜' →+* 𝕜
inst✝⁸ : RingHomInvPair σ σ'
inst✝⁷ : RingHomIsometric σ
inst✝⁶ : RingHomIsometric σ'
inst✝⁵ : CompleteSpace F✝
inst✝⁴ : CompleteSpace E
F : Type u_5
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace F
f : E →L[𝕜] F
G : Submodule 𝕜 F
h : IsCompl (↑f).range G
inst✝ : CompleteSpace ↥G
hker : (↑f).ker = ⊥
⊢ (↑(f.coprod G.subtypeL)).ker = ⊥

Tactic:
by
      rw [ker_coprod_of_disjoint_range]
      · simp [hker]
      · simp [h.disjoint]

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

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
⊢ Orthonormal 𝕜 fun i ↦ single i 1

Tactic:
orthonormal_iff_ite,

Goal state after:
ι : Type u_1
𝕜 : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
⊢ ∀ (i j : ι), ⟪single i 1, single j 1⟫ = if i = j then 1 else 0
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: case a
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 : ι
⊢ ‖⟪x, b i⟫‖ ^ 2 = ‖⟪b i, x⟫‖ ^ 2

Tactic:
rw [← inner_conj_symm, RCLike.norm_conj]

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
ι : Type u_3
hs : IsZLattice K L
b : Basis ι ℤ ↥L
x : ↥L
i : ι
⊢ ((ofZLatticeBasis K L b).repr ↑x) i = ↑((b.repr x) i)

Tactic:
suffices ((b.ofZLatticeBasis K L).repr.toLinearMap.restrictScalars ℤ) ∘ₗ L.subtype
      = Finsupp.mapRange.linearMap (Algebra.linearMap ℤ K) ∘ₗ b.repr.toLinearMap by
    exact DFunLike.congr_fun (LinearMap.congr_fun this x) i
  refine Basis.ext b fun i ↦ ?_
  simp_rw [LinearMap.coe_comp, Function.comp_apply, LinearMap.coe_restrictScalars,
    LinearEquiv.coe_coe, coe_subtype, ← b.ofZLatticeBasis_apply K, repr_self,
    Finsupp.mapRange.linearMap_apply, Finsupp.mapRange_single, Algebra.linearMap_apply, map_one]

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✝⁴ : RCLike 𝕜
E : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : Fintype ι
inst✝ : Fintype ι'
b : OrthonormalBasis ι 𝕜 E
e : ι ≃ ι'
i' : ι'
⊢ b.repr.symm ((LinearIsometryEquiv.piLpCongrLeft 2 𝕜 𝕜 e).symm (EuclideanSpace.single i' 1)) =
    b.repr.symm (EuclideanSpace.single (e.symm i') 1)

Tactic:
LinearIsometryEquiv.piLpCongrLeft_symm,

Goal state after:
ι : Type u_1
ι' : Type u_2
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✝ : Fintype ι'
b : OrthonormalBasis ι 𝕜 E
e : ι ≃ ι'
i' : ι'
⊢ b.repr.symm ((LinearIsometryEquiv.piLpCongrLeft 2 𝕜 𝕜 e.symm) (EuclideanSpace.single i' 1)) =
    b.repr.symm (EuclideanSpace.single (e.symm i') 1)
Module: Mathlib.Analysis.BoxIntegral.Partition.Filter

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
l : IntegrationParams
I : Box ι
⊢ (l.toFilter I).HasBasis (fun r ↦ ∀ (c : ℝ≥0), l.RCond (r c)) fun r ↦ {π | ∃ c, l.MemBaseSet I c (r c) π}

Tactic:
by
  simpa only [ofPred_exists] using! hasBasis_iSup (l.hasBasis_toFilterDistortion I)

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✝¹⁰ : RCLike 𝕜
E✝ : Type u_4
inst✝⁹ : NormedAddCommGroup E✝
inst✝⁸ : InnerProductSpace 𝕜 E✝
F : Type u_5
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : InnerProductSpace ℝ F
F' : Type u_6
inst✝⁵ : NormedAddCommGroup F'
inst✝⁴ : InnerProductSpace ℝ F'
inst✝³ : Fintype ι
V : Type u_7
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace 𝕜 V
inst✝ : FiniteDimensional 𝕜 V
S : Submodule 𝕜 V
L✝ L : ↥S →ₗᵢ[𝕜] V
d : ℕ := finrank 𝕜 ↥Sᗮ
LS : Submodule 𝕜 V := L.range
E : ↥Sᗮ ≃ₗᵢ[𝕜] ↥LSᗮ
L3 : ↥Sᗮ →ₗᵢ[𝕜] V := LSᗮ.subtypeₗᵢ.comp E.toLinearIsometry
this✝ : CompleteSpace ↥S
this : CompleteSpace V
p1 : V →ₗ[𝕜] ↥S := ↑S.orthogonalProjectionOnto
p2 : V →ₗ[𝕜] ↥Sᗮ := ↑Sᗮ.orthogonalProjectionOnto
M : V →ₗ[𝕜] V := L.toLinearMap ∘ₗ p1 + L3.toLinearMap ∘ₗ p2
x : V
Mx_decomp : M x = L (p1 x) + L3 (p2 x)
Lp1x : L (p1 x) ∈ L.range
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
have Lp2x : L3 (p2 x) ∈ (LinearMap.range L.toLinearMap)ᗮ := by
        simp only [LS,
          ← Submodule.range_subtype LSᗮ]
        apply LinearMap.mem_range_self

Goal state after:
ι : Type u_1
ι' : Type u_2
𝕜 : Type u_3
inst✝¹⁰ : RCLike 𝕜
E✝ : Type u_4
inst✝⁹ : NormedAddCommGroup E✝
inst✝⁸ : InnerProductSpace 𝕜 E✝
F : Type u_5
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : InnerProductSpace ℝ F
F' : Type u_6
inst✝⁵ : NormedAddCommGroup F'
inst✝⁴ : InnerProductSpace ℝ F'
inst✝³ : Fintype ι
V : Type u_7
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace 𝕜 V
inst✝ : FiniteDimensional 𝕜 V
S : Submodule 𝕜 V
L✝ L : ↥S →ₗᵢ[𝕜] V
d : ℕ := finrank 𝕜 ↥Sᗮ
LS : Submodule 𝕜 V := L.range
E : ↥Sᗮ ≃ₗᵢ[𝕜] ↥LSᗮ
L3 : ↥Sᗮ →ₗᵢ[𝕜] V := LSᗮ.subtypeₗᵢ.comp E.toLinearIsometry
this✝ : CompleteSpace ↥S
this : CompleteSpace V
p1 : V →ₗ[𝕜] ↥S := ↑S.orthogonalProjectionOnto
p2 : V →ₗ[𝕜] ↥Sᗮ := ↑Sᗮ.orthogonalProjectionOnto
M : V →ₗ[𝕜] V := L.toLinearMap ∘ₗ p1 + L3.toLinearMap ∘ₗ p2
x : V
Mx_decomp : M x = L (p1 x) + L3 (p2 x)
Lp1x : L (p1 x) ∈ L.range
Lp2x : L3 (p2 x) ∈ L.rangeᗮ
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: ι : Type u_1
inst✝ : Finite ι
s : Set (ι → ℝ)
h : IsBounded s
this : Fintype ι
R : ℝ
hR₁ : 0 < R
hR₂ : s ⊆ Metric.ball 0 R
C : ℕ := ⌈R⌉₊
hC : 0 < ⌈R⌉₊
x✝ : ι
⊢ -↑C < ↑C

Tactic:
by simp [C, neg_lt_self_iff, Nat.cast_pos, hC]

Goal state after:
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: ι : Type u_1
inst✝ : Finite ι
s : Set (ι → ℝ)
h : IsBounded s
this : Fintype ι
R : ℝ
hR₁ : 0 < R
hR₂ : s ⊆ Metric.ball 0 R
C : ℕ := ⌈R⌉₊
hC : 0 < ⌈R⌉₊
I : Box ι := { lower := fun x ↦ -↑C, upper := fun x ↦ ↑C, lower_lt_upper := ⋯ }
x : ι → ℝ
hx : x ∈ Metric.ball 0 ↑C
⊢ x ∈ ↑I

Tactic:
C,

Goal state after:
ι : Type u_1
inst✝ : Finite ι
s : Set (ι → ℝ)
h : IsBounded s
this : Fintype ι
R : ℝ
hR₁ : 0 < R
hR₂ : s ⊆ Metric.ball 0 R
C : ℕ := ⌈R⌉₊
hC : 0 < ⌈R⌉₊
I : Box ι := { lower := fun x ↦ -↑C, upper := fun x ↦ ↑C, lower_lt_upper := ⋯ }
x : ι → ℝ
hx : x ∈ Metric.ball 0 ↑⌈R⌉₊
⊢ x ∈ ↑I
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: ι : Type u_1
n : ℕ
inst✝ : NeZero n
⊢ Function.Injective fun ν ↦ tag n ν

Tactic:
refine fun _ _ h ↦ funext_iff.mpr fun i ↦ ?_

Goal state after:
ι : Type u_1
n : ℕ
inst✝ : NeZero n
x✝¹ x✝ : ι → ℤ
h : (fun ν ↦ tag n ν) x✝¹ = (fun ν ↦ tag n ν) x✝
i : ι
⊢ x✝¹ i = x✝ i
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
inst✝ : Fintype ι
B : Box ι
⊢ Finset (ι → ℤ)

Tactic:
by
  refine (setFinite_index n B.measurableSet_coe.nullMeasurableSet ?_).toFinset
  exact lt_top_iff_ne_top.mp (IsBounded.measure_lt_top B.isBounded)

Goal state after:
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
inst✝ : Fintype ι
ν : ι → ℤ
B : Box ι
hν : ν ∈ admissibleIndex n B
⊢ (prepartition n B).tag (box n ν) = tag n ν

Tactic:
by
  dsimp only [prepartition]
  have h : ∃ ν' ∈ admissibleIndex n B, box n ν = box n ν' := ⟨ν, hν, rfl⟩
  rw [dif_pos h, (tag_injective n).eq_iff, ← (box_injective n).eq_iff]
  exact h.choose_spec.2.symm

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

Goal state before:
pp: ι : Type u
E : Type v
F : Type w
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
I : Box ι
inst✝ : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
⊢ integral I l (-f) vol = -integral I l f vol

Tactic:
exact if h : Integrable I l f vol then h.hasIntegral.neg.integral_eq
  else by rw [integral, integral, dif_neg h, dif_neg (mt Integrable.of_neg h), neg_zero]

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

Goal state before:
pp: case succ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
n✝ : ℕ
_i : Fact (finrank ℝ E = n✝ + 1)
o : Orientation ℝ E (Fin (n✝ + 1))
b : OrthonormalBasis (Fin (n✝ + 1)) ℝ E
hb : b.toBasis.orientation = o
⊢ o.volumeForm = b.toBasis.det

Tactic:
volumeForm

Goal state after:
case succ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
n✝ : ℕ
_i : Fact (finrank ℝ E = n✝ + 1)
o : Orientation ℝ E (Fin (n✝ + 1))
b : OrthonormalBasis (Fin (n✝ + 1)) ℝ E
hb : b.toBasis.orientation = o
⊢ Nat.casesAuxOn (motive := fun a ↦ n✝ + 1 = a → E [⋀^Fin (n✝ + 1)]→ₗ[ℝ] ℝ) (n✝ + 1)
      (fun h ↦
        Eq.rec (motive := fun x x_1 ↦ [_i : Fact (finrank ℝ E = x)] → Orientation ℝ E (Fin x) → E [⋀^Fin x]→ₗ[ℝ] ℝ)
          (fun [Fact (finrank ℝ E = 0)] o ↦
            ⋯.by_cases (fun x ↦ AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1) fun x ↦
              -AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
          ⋯ o)
      (fun n h ↦
        Eq.rec (motive := fun x x_1 ↦ [_i : Fact (finrank ℝ E = x)] → Orientation ℝ E (Fin x) → E [⋀^Fin x]→ₗ[ℝ] ℝ)
          (fun [Fact (finrank ℝ E = n + 1)] o ↦ (Orientation.finOrthonormalBasis ⋯ ⋯ o).toBasis.det) ⋯ o)
      ⋯ =
    b.toBasis.det
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: case succ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
n : ℕ
_i : Fact (finrank ℝ E = n + 1)
o : Orientation ℝ E (Fin (n + 1))
b : OrthonormalBasis (Fin (n + 1)) ℝ E
hb : b.toBasis.orientation ≠ o
e : OrthonormalBasis (Fin n.succ) ℝ E := Orientation.finOrthonormalBasis ⋯ ⋯ o
⊢ o.volumeForm = -b.toBasis.det

Tactic:
volumeForm

Goal state after:
case succ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
n : ℕ
_i : Fact (finrank ℝ E = n + 1)
o : Orientation ℝ E (Fin (n + 1))
b : OrthonormalBasis (Fin (n + 1)) ℝ E
hb : b.toBasis.orientation ≠ o
e : OrthonormalBasis (Fin n.succ) ℝ E := Orientation.finOrthonormalBasis ⋯ ⋯ o
⊢ Nat.casesAuxOn (motive := fun a ↦ n + 1 = a → E [⋀^Fin (n + 1)]→ₗ[ℝ] ℝ) (n + 1)
      (fun h ↦
        Eq.rec (motive := fun x x_1 ↦ [_i : Fact (finrank ℝ E = x)] → Orientation ℝ E (Fin x) → E [⋀^Fin x]→ₗ[ℝ] ℝ)
          (fun [Fact (finrank ℝ E = 0)] o ↦
            ⋯.by_cases (fun x ↦ AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1) fun x ↦
              -AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
          ⋯ o)
      (fun n_1 h ↦
        Eq.rec (motive := fun x x_1 ↦ [_i : Fact (finrank ℝ E = x)] → Orientation ℝ E (Fin x) → E [⋀^Fin x]→ₗ[ℝ] ℝ)
          (fun [Fact (finrank ℝ E = n_1 + 1)] o ↦ (Orientation.finOrthonormalBasis ⋯ ⋯ o).toBasis.det) ⋯ o)
      ⋯ =
    -b.toBasis.det
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
n : ℕ
_i : Fact (finrank ℝ E = n)
o : Orientation ℝ E (Fin n)
v : Fin n → E
hv : Pairwise fun i j ↦ ⟪v i, v j⟫ = 0
⊢ |o.volumeForm v| = ∏ i, ‖v i‖

Tactic:
by
  rcases n with - | n
  · refine o.eq_or_eq_neg_of_isEmpty.elim ?_ ?_ <;> rintro rfl <;> simp
  have : FiniteDimensional ℝ E := .of_fact_finrank_eq_succ n
  have hdim : finrank ℝ E = Fintype.card (Fin n.succ) := by simpa using _i.out
  let b : OrthonormalBasis (Fin n.succ) ℝ E := gramSchmidtOrthonormalBasis hdim v
  have hb : b.toBasis.det v = ∏ i, ⟪b i, v i⟫ := gramSchmidtOrthonormalBasis_det hdim v
  rw [o.volumeForm_robust' b, hb, Finset.abs_prod]
  by_cases! h : ∃ i, v i = 0
  · obtain ⟨i, hi⟩ := h
    rw [Finset.prod_eq_zero (Finset.mem_univ i), Finset.prod_eq_zero (Finset.mem_univ i)] <;>
      simp [hi]
  congr
  ext i
  have hb : b i = ‖v i‖⁻¹ • v i := gramSchmidtOrthonormalBasis_apply_of_orthogonal hdim hv (h i)
  simp only [hb, inner_smul_left, real_inner_self_eq_norm_mul_norm, RCLike.conj_to_real]
  rw [abs_of_nonneg]
  · field
  · positivity

Goal state after: