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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁶ : RCLike 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝³ : (i : ι) → NormedAddCommGroup (G i)
inst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝¹ : (i : ι) → (x : G i) → Decidable (x ≠ 0)
inst✝ : DecidableEq ι
l : ⨁ (i : ι), G i
i : ι
v : G i
⊢ (if l i ≠ 0 then ⟪(V i) v, (V i) (l i)⟫ else 0) = ⟪v, l i⟫

Tactic:
split_ifs with h

Goal state after:
case pos
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : RCLike 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝³ : (i : ι) → NormedAddCommGroup (G i)
inst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝¹ : (i : ι) → (x : G i) → Decidable (x ≠ 0)
inst✝ : DecidableEq ι
l : ⨁ (i : ι), G i
i : ι
v : G i
h : l i ≠ 0
⊢ ⟪(V i) v, (V i) (l i)⟫ = ⟪v, l i⟫

case neg
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : RCLike 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝³ : (i : ι) → NormedAddCommGroup (G i)
inst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝¹ : (i : ι) → (x : G i) → Decidable (x ≠ 0)
inst✝ : DecidableEq ι
l : ⨁ (i : ι), G i
i : ι
v : G i
h : ¬l i ≠ 0
⊢ 0 = ⟪v, l i⟫
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : E
⊢ ‖((innerₛₗ 𝕜) x) y‖ ≤ 1 * ‖x‖ * ‖y‖

Tactic:
by
    simp only [norm_inner_le_norm, one_mul, innerₛₗ_apply_apply]

Goal state after:
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
x : E
⊢ x ∈ ⊤ᗮ ↔ x ∈ ⊥

Tactic:
mem_bot,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x : E
⊢ x ∈ ⊤ᗮ ↔ x = 0
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
n : ℕ
a : F
ε : ℝ
εpos : ε > 0
H : ball a ε ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (⇑f '' ball 0 ↑n)
c : 𝕜
hc : 1 < ‖c‖
y : F
hy : y ≠ 0
hc' : 1 < ‖σ c‖
d : 𝕜'
hd : d ≠ 0
ydlt : ‖d • y‖ < ε / 2
dinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖σ c‖ * ‖y‖
δ : ℝ := ‖d‖ * ‖y‖ / 4
δpos : 0 < δ
this✝ : a + d • y ∈ ball a ε
z₁ : F
z₁im : z₁ ∈ ⇑f '' ball 0 ↑n
x₁ : E
h₁ : dist (a + d • y) (f x₁) < δ
hx₁ : ‖x₁‖ < ↑n
xz₁ : f x₁ = z₁
this : a ∈ ball a ε
z₂ : F
z₂im : z₂ ∈ ⇑f '' ball 0 ↑n
x₂ : E
h₂ : dist a (f x₂) < δ
hx₂ : ‖x₂‖ < ↑n
xz₂ : f x₂ = z₂
x : E := x₁ - x₂
I : ‖f x - d • y‖ ≤ 2 * δ
⊢ ‖d⁻¹ • (f x - d • y)‖ = ‖d‖⁻¹ * ‖f x - d • y‖

Tactic:
by rw [norm_smul, norm_inv]

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

Goal state before:
pp: case inl.inl
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
inst✝³ : (i : ι) → PseudoMetricSpace (α i)
inst✝² : (i : ι) → PseudoEMetricSpace (β i)
inst✝¹ : Fintype ι
inst✝ : Fact (1 ≤ ∞)
f g h : PiLp ∞ β
h✝ : IsEmpty ι
⊢ ⨆ i, edist (f.ofLp i) (h.ofLp i) ≤ (⨆ i, edist (f.ofLp i) (g.ofLp i)) + ⨆ i, edist (g.ofLp i) (h.ofLp i)

pp: case inl.inr
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
inst✝³ : (i : ι) → PseudoMetricSpace (α i)
inst✝² : (i : ι) → PseudoEMetricSpace (β i)
inst✝¹ : Fintype ι
inst✝ : Fact (1 ≤ ∞)
f g h : PiLp ∞ β
h✝ : Nonempty ι
⊢ ⨆ i, edist (f.ofLp i) (h.ofLp i) ≤ (⨆ i, edist (f.ofLp i) (g.ofLp i)) + ⨆ i, edist (g.ofLp i) (h.ofLp i)

Tactic:
· simp only [ciSup_of_empty, ENNReal.bot_eq_zero, add_zero, nonpos_iff_eq_zero]

Goal state after:
case inl.inr
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
inst✝³ : (i : ι) → PseudoMetricSpace (α i)
inst✝² : (i : ι) → PseudoEMetricSpace (β i)
inst✝¹ : Fintype ι
inst✝ : Fact (1 ≤ ∞)
f g h : PiLp ∞ β
h✝ : Nonempty ι
⊢ ⨆ i, edist (f.ofLp i) (h.ofLp i) ≤ (⨆ i, edist (f.ofLp i) (g.ofLp i)) + ⨆ i, edist (g.ofLp i) (h.ofLp i)
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
p : ℝ≥0∞
⊢ 1 - idemFst = idemSnd

Tactic:
← idemFst_add_idemSnd,

Goal state after:
α : Type u_2
β : Type u_3
inst✝¹ : SeminormedAddCommGroup α
inst✝ : SeminormedAddCommGroup β
p : ℝ≥0∞
⊢ idemFst + idemSnd - idemFst = idemSnd
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
p : ℝ≥0∞
⊢ 1 - idemSnd = idemFst

Tactic:
← idemFst_add_idemSnd,

Goal state after:
α : Type u_2
β : Type u_3
inst✝¹ : SeminormedAddCommGroup α
inst✝ : SeminormedAddCommGroup β
p : ℝ≥0∞
⊢ idemFst + idemSnd - idemSnd = idemFst
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
inst✝³ : Fact (1 ≤ p)
inst✝² : (i : ι) → PseudoMetricSpace (α i)
inst✝¹ : (i : ι) → PseudoEMetricSpace (β i)
inst✝ : Fintype ι
f g : PiLp p α
⊢ edist f g = ENNReal.ofReal (dist f g)

Tactic:
rcases p.dichotomy with (rfl | h)
    · rw [edist_eq_iSup, dist_eq_iSup]
      cases isEmpty_or_nonempty ι
      · simp
      · refine ENNReal.eq_of_forall_le_nnreal_iff fun r ↦ ?_
        have : BddAbove <| .range fun i ↦ dist (f i) (g i) := Finite.bddAbove_range _
        simp [ciSup_le_iff this]
    · have : 0 < p.toReal := by rw [ENNReal.toReal_pos_iff_ne_top]; rintro rfl; norm_num at h
      simp only [edist_eq_sum, edist_dist, dist_eq_sum, this]
      rw [← ENNReal.ofReal_rpow_of_nonneg (by simp [Finset.sum_nonneg, Real.rpow_nonneg]) (by simp)]
      simp [Real.rpow_nonneg, ENNReal.ofReal_sum_of_nonneg, ← ENNReal.ofReal_rpow_of_nonneg]

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

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
α✝ : Type u_2
β✝ : Type u_3
hp : Fact (1 ≤ p)
inst✝³ : SeminormedAddCommGroup α✝
inst✝² : SeminormedAddCommGroup β✝
α : Type u_4
β : Type u_5
inst✝¹ : NormedAddCommGroup α
inst✝ : NormedAddCommGroup β
x y : α × β
h : dist x y = 0
⊢ x = y

Tactic:
rw [dist_pseudoMetricSpaceToProd] at h

Goal state after:
p : ℝ≥0∞
𝕜 : Type u_1
α✝ : Type u_2
β✝ : Type u_3
hp : Fact (1 ≤ p)
inst✝³ : SeminormedAddCommGroup α✝
inst✝² : SeminormedAddCommGroup β✝
α : Type u_4
β : Type u_5
inst✝¹ : NormedAddCommGroup α
inst✝ : NormedAddCommGroup β
x y : α × β
h : dist (toLp p x) (toLp p y) = 0
⊢ x = y
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
hp : Fact (1 ≤ p)
inst✝⁴ : Fintype ι
inst✝³ : SeminormedRing 𝕜
inst✝² : (i : ι) → SeminormedAddCommGroup (β i)
inst✝¹ : (i : ι) → Module 𝕜 (β i)
inst✝ : ∀ (i : ι), IsBoundedSMul 𝕜 (β i)
c : 𝕜
f : PiLp p β
⊢ ‖c • f‖₊ ≤ ‖c‖₊ * ‖f‖₊

Tactic:
rcases p.dichotomy with (rfl | hp)
    · rw [← nnnorm_ofLp, ← nnnorm_ofLp, ofLp_smul]
      exact nnnorm_smul_le c (ofLp f)
    · have hp0 : 0 < p.toReal := zero_lt_one.trans_le hp
      have hpt : p ≠ ⊤ := p.toReal_pos_iff_ne_top.mp hp0
      rw [nnnorm_eq_sum hpt, nnnorm_eq_sum hpt, one_div, NNReal.rpow_inv_le_iff hp0,
        NNReal.mul_rpow, ← NNReal.rpow_mul, inv_mul_cancel₀ hp0.ne', NNReal.rpow_one,
        Finset.mul_sum]
      simp_rw [← NNReal.mul_rpow, smul_apply]
      gcongr
      apply nnnorm_smul_le

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

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
b : OrthonormalBasis ι ℝ ℝ
⊢ parallelepiped ⇑b = Icc 0 1 ∨ parallelepiped ⇑b = Icc (-1) 0

Tactic:
have e : ι ≃ Fin 1 := by
    apply Fintype.equivFinOfCardEq
    simp only [← finrank_eq_card_basis b.toBasis, finrank_self]

Goal state after:
ι : Type u_1
inst✝ : Fintype ι
b : OrthonormalBasis ι ℝ ℝ
e : ι ≃ Fin 1
⊢ parallelepiped ⇑b = Icc 0 1 ∨ parallelepiped ⇑b = Icc (-1) 0
Module: Mathlib.MeasureTheory.Measure.Haar.OfBasis

Goal state before:
pp: case inl
ι : Type u_1
inst✝ : Fintype ι
b : OrthonormalBasis ι ℝ ℝ
e : ι ≃ Fin 1
B : parallelepiped ⇑(b.reindex e) = parallelepiped ⇑b
F : ℝ → Fin 1 → ℝ := fun t _i ↦ t
A : Icc 0 1 = F '' Icc 0 1
H : ⇑(b.reindex e) = fun x ↦ 1
⊢ (fun t ↦ ∑ i, t i • (b.reindex e) i) '' Icc 0 1 = Icc 0 1

Tactic:
H,

Goal state after:
case inl
ι : Type u_1
inst✝ : Fintype ι
b : OrthonormalBasis ι ℝ ℝ
e : ι ≃ Fin 1
B : parallelepiped ⇑(b.reindex e) = parallelepiped ⇑b
F : ℝ → Fin 1 → ℝ := fun t _i ↦ t
A : Icc 0 1 = F '' Icc 0 1
H : ⇑(b.reindex e) = fun x ↦ 1
⊢ (fun a ↦ ∑ x, a x • 1) '' Icc 0 1 = Icc 0 1
Module: Mathlib.MeasureTheory.Measure.Haar.OfBasis

Goal state before:
pp: case inr
ι : Type u_1
inst✝ : Fintype ι
b : OrthonormalBasis ι ℝ ℝ
e : ι ≃ Fin 1
B : parallelepiped ⇑(b.reindex e) = parallelepiped ⇑b
F : ℝ → Fin 1 → ℝ := ⋯
A : Icc 0 1 = F '' Icc 0 1
H : ⇑(b.reindex e) = fun x ↦ -1
⊢ parallelepiped ⇑(b.reindex e) = Icc (-1) 0

Tactic:
H,

Goal state after:
case inr
ι : Type u_1
inst✝ : Fintype ι
b : OrthonormalBasis ι ℝ ℝ
e : ι ≃ Fin 1
B : parallelepiped ⇑(b.reindex e) = parallelepiped ⇑b
F : ℝ → Fin 1 → ℝ := fun t _i ↦ t
A : Icc 0 1 = F '' Icc 0 1
H : ⇑(b.reindex e) = fun x ↦ -1
⊢ (parallelepiped fun x ↦ -1) = Icc (-1) 0
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: case refine_2
p : ℝ≥0∞
ι : Type u_2
α : ι → Type u_3
inst✝⁵ : Fact (1 ≤ p)
inst✝⁴ : Fintype ι
inst✝³ : (i : ι) → SeminormedAddCommGroup (α i)
R : Type u_5
inst✝² : SeminormedRing R
inst✝¹ : (i : ι) → Module R (α i)
inst✝ : ∀ (i : ι), IsBoundedSMul R (α i)
this : PseudoMetricSpace ((i : ι) → α i) := pseudoMetricSpaceToPi p α
x y : R
z : (i : ι) → α i
⊢ dist (x • z) (y • z) ≤ dist x y * dist z 0

Tactic:
simpa [dist_pseudoMetricSpaceToPi] using dist_pair_smul x y (toLp p z)

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

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f g : α → ℝ
s : Set α
hf : Measurable f
hg : Measurable g
hs : MeasurableSet s
x : α
⊢ volume {a | x ∈ s ∧ a ∈ Ioo (f x) (g x)} = if x ∈ s then ofReal (g x - f x) else 0

Tactic:
split_ifs with h

Goal state after:
case pos
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f g : α → ℝ
s : Set α
hf : Measurable f
hg : Measurable g
hs : MeasurableSet s
x : α
h : x ∈ s
⊢ volume {a | x ∈ s ∧ a ∈ Ioo (f x) (g x)} = ofReal (g x - f x)

case neg
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f g : α → ℝ
s : Set α
hf : Measurable f
hg : Measurable g
hs : MeasurableSet s
x : α
h : x ∉ s
⊢ volume {a | x ∈ s ∧ a ∈ Ioo (f x) (g x)} = 0
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁵ : NormedField K
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace K E
b : Basis ι K E
inst✝² : LinearOrder K
F : Type u_4
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace K F
f : E ≃ₗ[K] F
x : F
⊢ x ∈ ⇑f '' fundamentalDomain b ↔ ∀ (i : ι), (b.repr (f.symm x)) i ∈ Set.Ico 0 1

Tactic:
← mem_fundamentalDomain,

Goal state after:
E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁵ : NormedField K
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace K E
b : Basis ι K E
inst✝² : LinearOrder K
F : Type u_4
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace K F
f : E ≃ₗ[K] F
x : F
⊢ x ∈ ⇑f '' fundamentalDomain b ↔ f.symm x ∈ fundamentalDomain b
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: 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
m n : E
⊢ (∀ (i : ι), (b.repr (fract b n)) i = (b.repr (fract b m)) i) ↔ -m + n ∈ span ℤ (Set.range ⇑b)

Tactic:
simp_rw [repr_fract_apply, Int.fract_eq_fract, eq_comm, Basis.mem_span_iff_repr_mem,
    sub_eq_neg_add, map_add, map_neg, Finsupp.coe_add, Finsupp.coe_neg, Pi.add_apply,
    Pi.neg_apply, ← eq_intCast (algebraMap ℤ K) _, Set.mem_range]

Goal state after:
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, ‖Int.fract ((b.repr m) i) • b i‖ = ∑ i, ‖Int.fract ((b.repr m) i)‖ * ‖b i‖

Tactic:
by simp_rw [norm_smul]

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
hs : NullMeasurableSet s μ
r : ℝ
⊢ NullMeasurableSet (r • s) μ

Tactic:
obtain rfl | hs' := s.eq_empty_or_nonempty

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
hs : NullMeasurableSet ∅ μ
⊢ NullMeasurableSet (r • ∅) μ

case inr
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
hs : NullMeasurableSet s μ
r : ℝ
hs' : s.Nonempty
⊢ NullMeasurableSet (r • s) μ
Module: Mathlib.Analysis.BoxIntegral.Box.Basic

Goal state before:
pp: ι : Type u_1
l u : ι → ℝ
⊢ (if h : ∀ (i : ι), l i < u i then ↑{ lower := l, upper := u, lower_lt_upper := h } else ⊥) = ⊥ ↔ ∃ i, u i ≤ l i

Tactic:
split_ifs with h

Goal state after:
case pos
ι : Type u_1
l u : ι → ℝ
h : ∀ (i : ι), l i < u i
⊢ ↑{ lower := l, upper := u, lower_lt_upper := h } = ⊥ ↔ ∃ i, u i ≤ l i

case neg
ι : Type u_1
l u : ι → ℝ
h : ¬∀ (i : ι), l i < u i
⊢ ⊥ = ⊥ ↔ ∃ i, u i ≤ l i
Module: Mathlib.Analysis.BoxIntegral.Box.Basic

Goal state before:
pp: ι : Type u_1
l u lI uI : ι → ℝ
hI : ∀ (i : ι), lI i < uI i
⊢ (if h : ∀ (i : ι), l i < u i then ↑{ lower := l, upper := u, lower_lt_upper := h } else ⊥) =
      ↑{ lower := lI, upper := uI, lower_lt_upper := hI } ↔
    l = { lower := lI, upper := uI, lower_lt_upper := hI }.lower ∧
      u = { lower := lI, upper := uI, lower_lt_upper := hI }.upper

Tactic:
split_ifs with h

Goal state after:
case pos
ι : Type u_1
l u lI uI : ι → ℝ
hI : ∀ (i : ι), lI i < uI i
h : ∀ (i : ι), l i < u i
⊢ ↑{ lower := l, upper := u, lower_lt_upper := h } = ↑{ lower := lI, upper := uI, lower_lt_upper := hI } ↔
    l = { lower := lI, upper := uI, lower_lt_upper := hI }.lower ∧
      u = { lower := lI, upper := uI, lower_lt_upper := hI }.upper

case neg
ι : Type u_1
l u lI uI : ι → ℝ
hI : ∀ (i : ι), lI i < uI i
h : ¬∀ (i : ι), l i < u i
⊢ ⊥ = ↑{ lower := lI, upper := uI, lower_lt_upper := hI } ↔
    l = { lower := lI, upper := uI, lower_lt_upper := hI }.lower ∧
      u = { lower := lI, upper := uI, lower_lt_upper := hI }.upper
Module: Mathlib.Analysis.BoxIntegral.Box.Basic

Goal state before:
pp: ι : Type u_1
l u : ι → ℝ
⊢ ↑(if h : ∀ (i : ι), l i < u i then ↑{ lower := l, upper := u, lower_lt_upper := h } else ⊥) =
    univ.pi fun i ↦ Ioc (l i) (u i)

Tactic:
split_ifs with h

Goal state after:
case pos
ι : Type u_1
l u : ι → ℝ
h : ∀ (i : ι), l i < u i
⊢ ↑↑{ lower := l, upper := u, lower_lt_upper := h } = univ.pi fun i ↦ Ioc (l i) (u i)

case neg
ι : Type u_1
l u : ι → ℝ
h : ¬∀ (i : ι), l i < u i
⊢ ↑⊥ = univ.pi fun i ↦ Ioc (l i) (u i)
Module: Mathlib.Analysis.BoxIntegral.Partition.Basic

Goal state before:
pp: ι : Type u_1
I J J₁✝ J₂ : Box ι
π π₁ π₂ : Prepartition I
x✝ : ι → ℝ
πi : (J : Box ι) → Prepartition J
J₁' J₁ : Box ι
hJ₁ : J₁ ∈ π.boxes
hJ₁' : J₁' ∈ (πi J₁).boxes
J₂' : Box ι
Hne : J₁' ≠ J₂'
x : ι → ℝ
hx₁ : x ∈ ↑J₁'
hx₂ : x ∈ ↑J₂'
hJ₂ : J₁ ∈ π.boxes
hJ₂' : J₂' ∈ (πi J₁).boxes
⊢ J₁' = J₂'

Tactic:
exact (πi J₁).eq_of_mem_of_mem hJ₁' hJ₂' hx₁ hx₂

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

Goal state before:
pp: ι : Type u_1
I J : Box ι
π : Prepartition I
p : Box ι → Prop
⊢ J ∈ π.filter p ↔ J ∈ π ∧ p J

Tactic:
exact Finset.mem_filter

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

Goal state before:
pp: ι : Type u_1
I J : Box ι
π : TaggedPrepartition I
p : Box ι → Prop
⊢ J ∈ π.filter p ↔ J ∈ π ∧ p J

Tactic:
exact Finset.mem_filter

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

Goal state before:
pp: case refine_3
ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
inst✝ : Fintype ι
s : Set (ι → ℝ)
hs₁ : NullMeasurableSet s volume
hs₂ : volume s ≠ ⊤
⊢ volume (⋃ i, ↑(box n i) ∩ s) ≠ ⊤

pp: case refine_4
ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
inst✝ : Fintype ι
s : Set (ι → ℝ)
hs₁ : NullMeasurableSet s volume
hs₂ : volume s ≠ ⊤
x✝ : ι → ℤ
hν : x✝ ∈ {ν | ↑(box n ν) ⊆ s}
⊢ x✝ ∈ {i | 1 / ↑n ^ card ι ≤ volume (↑(box n i) ∩ s)}

Tactic:
· exact lt_top_iff_ne_top.mp <| measure_lt_top_of_subset
      (by simp only [Set.iUnion_subset_iff, Set.inter_subset_right, implies_true]) hs₂

Goal state after:
case refine_4
ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
inst✝ : Fintype ι
s : Set (ι → ℝ)
hs₁ : NullMeasurableSet s volume
hs₂ : volume s ≠ ⊤
x✝ : ι → ℤ
hν : x✝ ∈ {ν | ↑(box n ν) ⊆ s}
⊢ x✝ ∈ {i | 1 / ↑n ^ card ι ≤ volume (↑(box n i) ∩ s)}
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: ι : Type u_1
s : Set (ι → ℝ)
inst✝ : Finite ι
hs₁ : IsBounded s
hs₄ : ∀ ⦃x y : ℝ⦄, 0 < x → x ≤ y → x • s ⊆ y • s
x y : ℝ
hx : 0 < x
hy : x ≤ y
⊢ Nat.card ↑(s ∩ x⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))) ≤
    Nat.card ↑(s ∩ y⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))))

Tactic:
by
  have := Fintype.ofFinite ι
  rw [Nat.card_congr (tendsto_card_div_pow₁ s hx.ne'),
      Nat.card_congr (tendsto_card_div_pow₁ s (hx.trans_le hy).ne')]
  refine Nat.card_mono ?_ ?_
  · exact ZSpan.setFinite_inter _ (IsBounded.smul₀ hs₁ y)
  · exact Set.inter_subset_inter_left _ <| hs₄ hx hy

Goal state after:
Module: Mathlib.LinearAlgebra.Orientation

Goal state before:
pp: 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
⊢ (if e.orientation = x then e else e.unitsSMul (Function.update 1 (Classical.arbitrary ι) (-1))).orientation = x

Tactic:
split_ifs with h

Goal state after:
case pos
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

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
Module: Mathlib.LinearAlgebra.Orientation

Goal state before:
pp: 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
i : ι
⊢ (if e.orientation = x then e else e.unitsSMul (Function.update 1 (Classical.arbitrary ι) (-1))) i = e i ∨
    (if e.orientation = x then e else e.unitsSMul (Function.update 1 (Classical.arbitrary ι) (-1))) i = -e i

Tactic:
split_ifs with h

Goal state after:
case pos
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
i : ι
h : e.orientation = x
⊢ e i = e i ∨ e i = -e i

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
i : ι
h : ¬e.orientation = x
⊢ (e.unitsSMul (Function.update 1 (Classical.arbitrary ι) (-1))) i = e i ∨
    (e.unitsSMul (Function.update 1 (Classical.arbitrary ι) (-1))) i = -e i
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: case e'_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
φ : E ≃ₗᵢ[ℝ] E
hφ : 0 < LinearMap.det ↑φ.toLinearEquiv
n : ℕ
_i : Fact (finrank ℝ E = n + 1)
o : Orientation ℝ E (Fin (n + 1))
x : Fin (n + 1) → E
this : FiniteDimensional ℝ E
⊢ (map (Fin (n + 1)) φ.toLinearEquiv) o = o

Tactic:
rwa [← o.map_eq_iff_det_pos φ.toLinearEquiv] at hφ

Goal state after:
case e'_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
φ : E ≃ₗᵢ[ℝ] E
hφ : 0 < LinearMap.det ↑φ.toLinearEquiv
n : ℕ
_i : Fact (finrank ℝ E = n + 1)
o : Orientation ℝ E (Fin (n + 1))
x : Fin (n + 1) → E
this : FiniteDimensional ℝ E
⊢ Fintype.card (Fin (n + 1)) = finrank ℝ E
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: ι : Type u_2
inst✝⁴ : Fintype ι
L₁ L₂ : Submodule ℤ (ι → ℝ)
inst✝³ : DiscreteTopology ↥L₁
inst✝² : IsZLattice ℝ L₁
inst✝¹ : DiscreteTopology ↥L₂
inst✝ : IsZLattice ℝ L₂
h : L₁ ≤ L₂
b₁ : Basis ι ℤ ↥L₁ := IsZLattice.basis L₁
b₂ : Basis ι ℤ ↥L₂ := IsZLattice.basis L₂
i j : ι
⊢ (Basis.ofZLatticeBasis ℝ L₂ b₂).toMatrix (⇑(Basis.ofZLatticeBasis ℝ L₁ b₁)) i j =
    ↑(b₂.toMatrix (fun i ↦ ⟨↑(b₁ i), ⋯⟩) i j)

Tactic:
Basis.toMatrix_apply,

Goal state after:
ι : Type u_2
inst✝⁴ : Fintype ι
L₁ L₂ : Submodule ℤ (ι → ℝ)
inst✝³ : DiscreteTopology ↥L₁
inst✝² : IsZLattice ℝ L₁
inst✝¹ : DiscreteTopology ↥L₂
inst✝ : IsZLattice ℝ L₂
h : L₁ ≤ L₂
b₁ : Basis ι ℤ ↥L₁ := IsZLattice.basis L₁
b₂ : Basis ι ℤ ↥L₂ := IsZLattice.basis L₂
i j : ι
⊢ ((Basis.ofZLatticeBasis ℝ L₂ b₂).repr ((Basis.ofZLatticeBasis ℝ L₁ b₁) j)) i =
    ↑(b₂.toMatrix (fun i ↦ ⟨↑(b₁ i), ⋯⟩) i j)
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: ι : Type u_2
inst✝⁴ : Fintype ι
L₁ L₂ : Submodule ℤ (ι → ℝ)
inst✝³ : DiscreteTopology ↥L₁
inst✝² : IsZLattice ℝ L₁
inst✝¹ : DiscreteTopology ↥L₂
inst✝ : IsZLattice ℝ L₂
h : L₁ ≤ L₂
b₁ : Basis ι ℤ ↥L₁ := IsZLattice.basis L₁
b₂ : Basis ι ℤ ↥L₂ := IsZLattice.basis L₂
i j : ι
⊢ ((Basis.ofZLatticeBasis ℝ L₂ b₂).repr ((Basis.ofZLatticeBasis ℝ L₁ b₁) j)) i =
    ↑(b₂.toMatrix (fun i ↦ ⟨↑(b₁ i), ⋯⟩) i j)

Tactic:
Basis.toMatrix_apply,

Goal state after:
ι : Type u_2
inst✝⁴ : Fintype ι
L₁ L₂ : Submodule ℤ (ι → ℝ)
inst✝³ : DiscreteTopology ↥L₁
inst✝² : IsZLattice ℝ L₁
inst✝¹ : DiscreteTopology ↥L₂
inst✝ : IsZLattice ℝ L₂
h : L₁ ≤ L₂
b₁ : Basis ι ℤ ↥L₁ := IsZLattice.basis L₁
b₂ : Basis ι ℤ ↥L₂ := IsZLattice.basis L₂
i j : ι
⊢ ((Basis.ofZLatticeBasis ℝ L₂ b₂).repr ((Basis.ofZLatticeBasis ℝ L₁ b₁) j)) i = ↑((b₂.repr ⟨↑(b₁ j), ⋯⟩) i)