mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.LocallyConvex.SeparatingDual

Goal state before:
pp: R : Type u_4
V : Type u_5
W : Type u_6
inst✝¹³ : NormedField R
inst✝¹² : AddCommGroup V
inst✝¹¹ : AddCommGroup W
inst✝¹⁰ : TopologicalSpace R
inst✝⁹ : TopologicalSpace V
inst✝⁸ : TopologicalSpace W
inst✝⁷ : IsTopologicalRing R
inst✝⁶ : Module R V
inst✝⁵ : Module R W
inst✝⁴ : SeparatingDual R V
inst✝³ : IsTopologicalAddGroup V
inst✝² : IsTopologicalAddGroup W
inst✝¹ : ContinuousSMul R V
inst✝ : ContinuousSMul R W
f g : V ≃L[R] W
⊢ g.conjContinuousAlgEquiv = f.conjContinuousAlgEquiv ↔ ∃ α, f = α • g

Tactic:
ContinuousAlgEquiv.ext_iff,

Goal state after:
R : Type u_4
V : Type u_5
W : Type u_6
inst✝¹³ : NormedField R
inst✝¹² : AddCommGroup V
inst✝¹¹ : AddCommGroup W
inst✝¹⁰ : TopologicalSpace R
inst✝⁹ : TopologicalSpace V
inst✝⁸ : TopologicalSpace W
inst✝⁷ : IsTopologicalRing R
inst✝⁶ : Module R V
inst✝⁵ : Module R W
inst✝⁴ : SeparatingDual R V
inst✝³ : IsTopologicalAddGroup V
inst✝² : IsTopologicalAddGroup W
inst✝¹ : ContinuousSMul R V
inst✝ : ContinuousSMul R W
f g : V ≃L[R] W
⊢ ⇑g.conjContinuousAlgEquiv = ⇑f.conjContinuousAlgEquiv ↔ ∃ α, f = α • g
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ
hf : Integrable f μ
ε : ℝ
εpos : 0 < ε
δ : ℝ≥0 := NNReal.mk (ε / 2) ⋯
δpos : 0 < δ
fp : α → ℝ≥0 := fun x ↦ (f x).toNNReal
int_fp : Integrable (fun x ↦ ↑(fp x)) μ
gp : α → ℝ≥0∞
fp_lt_gp : ∀ (x : α), ↑(fp x) < gp x
gpcont : LowerSemicontinuous gp
gp_lt_top : ∀ᵐ (x : α) ∂μ, gp x < ∞
gp_integrable : Integrable (fun x ↦ (gp x).toReal) μ
gpint : ∫ (x : α), (gp x).toReal ∂μ < ∫ (x : α), ↑(fp x) ∂μ + ↑δ
fm : α → ℝ≥0 := fun x ↦ (-f x).toNNReal
int_fm : Integrable (fun x ↦ ↑(fm x)) μ
gm : α → ℝ≥0
gm_le_fm : ∀ (x : α), gm x ≤ fm x
gmcont : UpperSemicontinuous gm
gm_integrable : Integrable (fun x ↦ ↑(gm x)) μ
gmint : ∫ (x : α), ↑(fm x) ∂μ - ↑δ ≤ ∫ (x : α), ↑(gm x) ∂μ
g : α → EReal := fun x ↦ ↑(gp x) - ↑↑(gm x)
ae_g : ∀ᵐ (x : α) ∂μ, (g x).toReal = (↑(gp x)).toReal - (↑↑(gm x)).toReal
x✝ : α
hx : gp x✝ < ∞
⊢ g x✝ < ⊤

Tactic:
simp only [g, sub_eq_add_neg, Ne, (EReal.add_lt_top _ _).ne, lt_top_iff_ne_top,
      lt_top_iff_ne_top.1 hx, EReal.coe_ennreal_eq_top_iff, not_false_iff, EReal.neg_eq_top_iff,
      EReal.coe_ennreal_ne_bot]

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

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
ι : Type u_4
inst✝⁴ : Finite ι
M : ι → Type u_5
inst✝³ : (i : ι) → NormedAddCommGroup (M i)
inst✝² : (i : ι) → NormedSpace 𝕜 (M i)
inst✝¹ : ∀ (i : ι), SeparatingDual 𝕜 (M i)
inst✝ : CompleteSpace (ContinuousMultilinearMap 𝕜 M F)
m : (i : ι) → M i
hm : ∀ (i : ι), m i ≠ 0
⊢ CompleteSpace F

Tactic:
refine Metric.complete_of_cauchySeq_tendsto fun f hf => ?_
  have : ∀ i, ∃ φ : StrongDual 𝕜 (M i), φ (m i) = 1 := fun i ↦ exists_eq_one (hm i)
  choose φ hφ using this
  cases nonempty_fintype ι
  let g : ℕ → (ContinuousMultilinearMap 𝕜 M F) := fun n ↦
    compContinuousLinearMapL φ
    (ContinuousMultilinearMap.smulRightL 𝕜 _ F ((ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι 𝕜)) (f n))
  have : CauchySeq g := by
    refine (ContinuousLinearMap.lipschitz _).cauchySeq_comp ?_
    exact (ContinuousLinearMap.lipschitz _).cauchySeq_comp hf
  obtain ⟨a, ha⟩ : ∃ a, Tendsto g atTop (𝓝 a) := cauchy_iff_exists_le_nhds.mp this
  refine ⟨a m, ?_⟩
  have : Tendsto (fun n ↦ g n m) atTop (𝓝 (a m)) := ((continuous_eval_const _).tendsto _).comp ha
  simpa [g, hφ]

Goal state after:
Module: Mathlib.Topology.ContinuousMap.ContinuousMapZero

Goal state before:
pp: X : Type u_1
R : Type u_2
inst✝³ : Zero R
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace R
inst✝ : Zero X
f : X → R
g : C(X, R)₀
hf : Continuous[inst✝², inst✝¹] f
hf₀ : f 0 = 0
⊢ mkD f g = { toFun := f, continuous_toFun := hf, map_zero' := hf₀ }

Tactic:
simp only [mkD, And.intro hf hf₀, true_and, ↓reduceDIte]

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

Goal state before:
pp: X : Type u_1
E : Type u_3
inst✝⁷ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_6
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedSpace ℝ E
ι : Type u_8
inst✝² : Fintype ι
M : ι → Type u_9
inst✝¹ : (i : ι) → NormedAddCommGroup (M i)
inst✝ : (i : ι) → NormedSpace 𝕜 (M i)
φ : X → ContinuousMultilinearMap 𝕜 M E
φ_int : Integrable φ μ
m : (i : ι) → M i
hE : ¬CompleteSpace E
hm : ∀ (i : ι), m i ≠ 0
⊢ ¬CompleteSpace (ContinuousMultilinearMap 𝕜 M E)

Tactic:
rwa [SeparatingDual.completeSpace_continuousMultilinearMap_iff _ _ hm]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝ : NormedAddCommGroup E
a b : ℝ
f : ℝ → E
c : ℝ
h✝ : ‖f (min a b)‖ₑ ≠ ∞
h : IntervalIntegrable (fun x ↦ f (c - x)) volume (c - a) (c - b)
⊢ IntervalIntegrable f volume a b

Tactic:
by simpa using h.comp_sub_left c

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
μ : Measure ℝ
inst✝ : CompleteSpace E
c : E
⊢ ∫ (x : ℝ) in a..b, c ∂μ = (μ.real (Ioc a b) - μ.real (Ioc b a)) • c

Tactic:
simp only [measureReal_def, intervalIntegral, setIntegral_const, sub_smul]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Goal state before:
pp: g' g φ : ℝ → ℝ
a b : ℝ
hab : a ≤ b
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt g (g' x) (Ioi x) x
φint : IntegrableOn φ (Icc a b) volume
hφg : ∀ x ∈ Ico a b, g' x ≤ φ x
ε : ℝ
εpos : 0 < ε
G' : ℝ → EReal
f_lt_G' : ∀ (x : ℝ), ↑(φ x) < G' x
G'cont : LowerSemicontinuous G'
G'int : Integrable (fun x ↦ (G' x).toReal) (volume.restrict (Icc a b))
G'lt_top : ∀ᵐ (x : ℝ) ∂volume.restrict (Icc a b), G' x < ⊤
hG' : ∫ (x : ℝ) in Icc a b, (G' x).toReal < (∫ (x : ℝ) in Icc a b, φ x) + ε
s : Set ℝ := {t | g t - g a ≤ ∫ (u : ℝ) in a..t, (G' u).toReal} ∩ Icc a b
s_closed : IsClosed s
t : ℝ
ht : t ∈ {t | g t - g a ≤ ∫ (u : ℝ) in a..t, (G' u).toReal} ∩ Ico a b
v : ℝ
t_lt_v : v ∈ Ioi t
y : ℝ
g'_lt_y' : ↑(g' t) < ↑y
y_lt_G' : ↑y < G' t
I1 : ∀ᶠ (u : ℝ) in 𝓝[>] t, (u - t) * y ≤ ∫ (w : ℝ) in t..u, (G' w).toReal
I2 : ∀ᶠ (u : ℝ) in 𝓝[>] t, g u - g t ≤ (u - t) * y
I3 : ∀ᶠ (u : ℝ) in 𝓝[>] t, g u - g t ≤ ∫ (w : ℝ) in t..u, (G' w).toReal
I4 : ∀ᶠ (u : ℝ) in 𝓝[>] t, u ∈ Ioc t (min v b)
x : ℝ
hx : g x - g t ≤ ∫ (w : ℝ) in t..x, (G' w).toReal
h'x : x ∈ Ioc t (min v b)
⊢ x ∈ {t | g t - g a ≤ ∫ (u : ℝ) in a..t, (G' u).toReal}

Tactic:
calc
      g x - g a = g t - g a + (g x - g t) := by abel
      _ ≤ (∫ w in a..t, (G' w).toReal) + ∫ w in t..x, (G' w).toReal := add_le_add ht.1 hx
      _ = ∫ w in a..x, (G' w).toReal := by
        apply integral_add_adjacent_intervals
        · rw [intervalIntegrable_iff_integrableOn_Ioc_of_le ht.2.1]
          exact IntegrableOn.mono_set G'int
            (Ioc_subset_Icc_self.trans (Icc_subset_Icc le_rfl ht.2.2.le))
        · rw [intervalIntegrable_iff_integrableOn_Ioc_of_le h'x.1.le]
          apply IntegrableOn.mono_set G'int
          exact Ioc_subset_Icc_self.trans (Icc_subset_Icc ht.2.1 (h'x.2.trans (min_le_right _ _)))

Goal state after:
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : TopologicalSpace X
μ : Measure ℝ
inst✝¹ : NullSingletonClass μ
inst✝ : IsLocallyFiniteMeasure μ
f : X → ℝ → E
a₀ : ℝ
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
q : X
b₀ ε : ℝ
εpos : ε > 0
a : ℝ
a_lt : a < a₀ ∧ a < b₀
b : ℝ
lt_b : a₀ < b ∧ b₀ < b
this : IsCompact ({q} ×ˢ Icc a b)
M : ℝ
hM : M ∈ upperBounds ((fun x ↦ ‖Function.uncurry f x‖) '' {q} ×ˢ Icc a b)
δ : ℝ
δpos : 0 < δ
hδ : δ < 1
h'δ : Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
h''δ : (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
⊢ ∀ᶠ (x : X × ℝ) in 𝓝 (q, b₀),
    dist (∫ (t : ℝ) in a₀..x.2, f x.1 t ∂μ) (∫ (t : ℝ) in a₀..(q, b₀).2, f (q, b₀).1 t ∂μ) < ε

Tactic:
obtain ⟨v, v_mem, hv⟩ : ∃ v ∈ 𝓝[univ] q, ∀ p ∈ v, ∀ x ∈ Icc a b, dist (f p x) (f q x) < δ :=
    IsCompact.mem_uniformity_of_prod isCompact_Icc hf.continuousOn (mem_univ _)
      (dist_mem_uniformity δpos)

Goal state after:
E : Type u_1
X : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : TopologicalSpace X
μ : Measure ℝ
inst✝¹ : NullSingletonClass μ
inst✝ : IsLocallyFiniteMeasure μ
f : X → ℝ → E
a₀ : ℝ
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
q : X
b₀ ε : ℝ
εpos : ε > 0
a : ℝ
a_lt : a < a₀ ∧ a < b₀
b : ℝ
lt_b : a₀ < b ∧ b₀ < b
this : IsCompact ({q} ×ˢ Icc a b)
M : ℝ
hM : M ∈ upperBounds ((fun x ↦ ‖Function.uncurry f x‖) '' {q} ×ˢ Icc a b)
δ : ℝ
δpos : 0 < δ
hδ : δ < 1
h'δ : Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
h''δ : (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
v : Set X
v_mem : v ∈ 𝓝[univ] q
hv : ∀ p ∈ v, ∀ x ∈ Icc a b, dist (f p x) (f q x) < δ
⊢ ∀ᶠ (x : X × ℝ) in 𝓝 (q, b₀),
    dist (∫ (t : ℝ) in a₀..x.2, f x.1 t ∂μ) (∫ (t : ℝ) in a₀..(q, b₀).2, f (q, b₀).1 t ∂μ) < ε
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
a b : ℝ
μ : Measure ℝ
inst✝¹ : CompleteSpace E
inst✝ : IsFiniteMeasure μ
c : E
⊢ ∫ (x : ℝ) in a..b, c ∂μ = (μ.real (Iic b) - μ.real (Iic a)) • c

Tactic:
simp only [sub_smul, ← setIntegral_const]
  refine (integral_Iic_sub_Iic ?_ ?_).symm <;> simp

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: f : ℝ → ℝ
a b : ℝ
hfi : IntervalIntegrable f volume a b
hpos : ∀ x ∈ Ioo a b, 0 < f x
hab : a < b
hsupp : Ioo a b ⊆ support f ∩ Ioc a b
⊢ 0 ≤ᵐ[volume.restrict (Ι a b)] f

Tactic:
EventuallyLE,

Goal state after:
f : ℝ → ℝ
a b : ℝ
hfi : IntervalIntegrable f volume a b
hpos : ∀ x ∈ Ioo a b, 0 < f x
hab : a < b
hsupp : Ioo a b ⊆ support f ∩ Ioc a b
⊢ ∀ᵐ (x : ℝ) ∂volume.restrict (Ι a b), 0 x ≤ f x
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: f : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
c d : ℝ
hca : c ≤ a
hab : a ≤ b
hbd : b ≤ d
hf : 0 ≤ᵐ[μ.restrict (Ioc c d)] f
hfi : IntervalIntegrable f μ c d
⊢ ∫ (x : ℝ) in a..b, f x ∂μ ≤ ∫ (x : ℝ) in c..d, f x ∂μ

Tactic:
rw [integral_of_le hab, integral_of_le (hca.trans (hab.trans hbd))]
  exact setIntegral_mono_set hfi.1 hf (Ioc_subset_Ioc hca hbd).eventuallyLE

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

Goal state before:
pp: G : Type u_1
inst✝¹⁰ : Group G
inst✝⁹ : MeasurableSpace G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : BorelSpace G
inst✝⁵ : PolishSpace G
Γ : Subgroup G
inst✝⁴ : Γ.Normal
inst✝³ : T2Space (G ⧸ Γ)
inst✝² : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝¹ : ν.IsMulLeftInvariant
hasFun : HasFundamentalDomain (↥Γ.op) G ν
inst✝ : QuotientMeasureEqMeasurePreimage ν μ
x : G ⧸ Γ
⊢ map (fun x_1 ↦ x * x_1) μ = μ

Tactic:
ext A hA

Goal state after:
G : Type u_1
inst✝¹⁰ : Group G
inst✝⁹ : MeasurableSpace G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : BorelSpace G
inst✝⁵ : PolishSpace G
Γ : Subgroup G
inst✝⁴ : Γ.Normal
inst✝³ : T2Space (G ⧸ Γ)
inst✝² : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝¹ : ν.IsMulLeftInvariant
hasFun : HasFundamentalDomain (↥Γ.op) G ν
inst✝ : QuotientMeasureEqMeasurePreimage ν μ
x : G ⧸ Γ
A : Set (G ⧸ Γ)
hA : MeasurableSet A
⊢ (map (fun x_1 ↦ x * x_1) μ) A = μ A
Module: Mathlib.MeasureTheory.Measure.Haar.Quotient

Goal state before:
pp: G : Type u_1
inst✝¹⁰ : Group G
inst✝⁹ : MeasurableSpace G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : BorelSpace G
inst✝⁵ : PolishSpace G
Γ : Subgroup G
inst✝⁴ : Γ.Normal
inst✝³ : T2Space (G ⧸ Γ)
inst✝² : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝¹ : ν.IsMulLeftInvariant
hasFun : HasFundamentalDomain (↥Γ.op) G ν
inst✝ : QuotientMeasureEqMeasurePreimage ν μ
x : G ⧸ Γ
A : Set (G ⧸ Γ)
hA : MeasurableSet A
x₁ : G
h : ⟦x₁⟧ = x
⊢ SMulInvariantMeasure G (G ⧸ Γ) μ

Tactic:
exact smulInvariantMeasure_quotient ν

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

Goal state before:
pp: case convert_3
G : Type u_1
inst✝¹⁴ : Group G
inst✝¹³ : MeasurableSpace G
inst✝¹² : TopologicalSpace G
inst✝¹¹ : IsTopologicalGroup G
inst✝¹⁰ : BorelSpace G
inst✝⁹ : PolishSpace G
Γ : Subgroup G
inst✝⁸ : Γ.Normal
inst✝⁷ : T2Space (G ⧸ Γ)
inst✝⁶ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝⁵ : ν.IsMulLeftInvariant
inst✝⁴ : Countable ↥Γ
inst✝³ : ν.IsMulRightInvariant
inst✝² : SigmaFinite ν
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : SigmaFinite μ
s : Set G
fund_dom_s : IsFundamentalDomain (↥Γ.op) s ν
V : Set (G ⧸ Γ)
meas_V : MeasurableSet V
neZeroV : μ V ≠ 0
hV : μ V = ν (QuotientGroup.mk ⁻¹' V ∩ s)
neTopV : μ V ≠ ∞
U : Set (G ⧸ Γ)
a✝ : MeasurableSet U
meas_π : Measurable QuotientGroup.mk
μ' : Measure (G ⧸ Γ) := ⋯
has_fund : HasFundamentalDomain (↥Γ.op) G ν
i : QuotientMeasureEqMeasurePreimage ν μ'
this✝ : μ'.IsMulLeftInvariant
this : SigmaFinite μ'
⊢ ν (QuotientGroup.mk ⁻¹' V ∩ s) ≠ ∞

Tactic:
· exact trans hV.symm neTopV

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic

Goal state before:
pp: T : ℝ
g : ℝ → ℝ
hg : Periodic g T
h_int : IntervalIntegrable g volume 0 T
hT : 0 < T
t : ℝ
h'_int : ∀ (a₁ a₂ : ℝ), IntervalIntegrable g volume a₁ a₂ := intervalIntegrable₀ hg (LT.lt.ne' hT) h_int
ε : ℝ := Int.fract (t / T) * T
⊢ sInf ((fun t ↦ ∫ (x : ℝ) in 0..t, g x) '' Icc 0 T) + ⌊t / T⌋ • ∫ (x : ℝ) in 0..T, g x ≤
    (∫ (x : ℝ) in 0..ε, g x) + ⌊t / T⌋ • ∫ (x : ℝ) in 0..T, g x

Tactic:
add_le_add_iff_right

Goal state after:
T : ℝ
g : ℝ → ℝ
hg : Periodic g T
h_int : IntervalIntegrable g volume 0 T
hT : 0 < T
t : ℝ
h'_int : ∀ (a₁ a₂ : ℝ), IntervalIntegrable g volume a₁ a₂ := intervalIntegrable₀ hg (LT.lt.ne' hT) h_int
ε : ℝ := Int.fract (t / T) * T
⊢ sInf ((fun t ↦ ∫ (x : ℝ) in 0..t, g x) '' Icc 0 T) ≤ ∫ (x : ℝ) in 0..ε, g x
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic

Goal state before:
pp: T : ℝ
g : ℝ → ℝ
hg : Periodic g T
h_int : IntervalIntegrable g volume 0 T
hT : 0 < T
t : ℝ
h'_int : ∀ (a₁ a₂ : ℝ), IntervalIntegrable g volume a₁ a₂ := intervalIntegrable₀ hg (LT.lt.ne' hT) h_int
ε : ℝ := Int.fract (t / T) * T
⊢ (∫ (x : ℝ) in 0..ε, g x) + ⌊t / T⌋ • ∫ (x : ℝ) in 0..T, g x ≤
    sSup ((fun t ↦ ∫ (x : ℝ) in 0..t, g x) '' Icc 0 T) + ⌊t / T⌋ • ∫ (x : ℝ) in 0..T, g x

Tactic:
add_le_add_iff_right

Goal state after:
T : ℝ
g : ℝ → ℝ
hg : Periodic g T
h_int : IntervalIntegrable g volume 0 T
hT : 0 < T
t : ℝ
h'_int : ∀ (a₁ a₂ : ℝ), IntervalIntegrable g volume a₁ a₂ := intervalIntegrable₀ hg (LT.lt.ne' hT) h_int
ε : ℝ := Int.fract (t / T) * T
⊢ ∫ (x : ℝ) in 0..ε, g x ≤ sSup ((fun t ↦ ∫ (x : ℝ) in 0..t, g x) '' Icc 0 T)
Module: Mathlib.MeasureTheory.Integral.DivergenceTheorem

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
I : Box (Fin (n + 1))
f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hc : ContinuousOn f (Box.Icc I)
Hd : ∀ x ∈ Box.Ioo I \ s, HasFDerivAt f (f' x) x
J : ℕ →o Box (Fin (n + 1))
hJ_sub : ∀ (n_1 : ℕ), Box.Icc (J n_1) ⊆ Box.Ioo I
hJl : Tendsto (Box.lower ∘ ⇑J) atTop (𝓝 I.lower)
hJu : Tendsto (Box.upper ∘ ⇑J) atTop (𝓝 I.upper)
hJ_sub' : ∀ (k : ℕ), Box.Icc (J k) ⊆ Box.Icc I
hJ_le : ∀ (k : ℕ), J k ≤ I
HcJ : ∀ (k : ℕ), ContinuousOn f (Box.Icc (J k))
HdJ : ∀ (k : ℕ), ∀ x ∈ Box.Icc (J k) \ s, HasFDerivWithinAt f (f' x) (Box.Icc (J k)) x
HiJ : ∀ (k : ℕ), IntegrableOn (fun x ↦ ∑ i, (f' x) (e i) i) (Box.Icc (J k)) volume
HJ_eq :
  ∀ (k : ℕ),
    ∫ (x : Fin (n + 1) → ℝ) in Box.Icc (J k), ∑ i, (f' x) (e i) i =
      ∑ i,
        ((∫ (x : Fin n → ℝ) in Box.Icc ((J k).face i), f (i.insertNth ((J k).upper i) x) i) -
          ∫ (x : Fin n → ℝ) in Box.Icc ((J k).face i), f (i.insertNth ((J k).lower i) x) i)
Hi : Integrable (fun x ↦ ∑ i, (f' x) (e i) i) (volume.restrict (Box.Ioo I))
⊢ Tendsto (fun k ↦ ∫ (x : Fin (n + 1) → ℝ) in Box.Ioo (J k), ∑ i, (f' x) (e i) i) atTop
    (𝓝 (∫ (x : Fin (n + 1) → ℝ) in Box.Ioo I, ∑ i, (f' x) (e i) i))

Tactic:
rw [← Box.iUnion_Ioo_of_tendsto J.monotone hJl hJu] at Hi ⊢

Goal state after:
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
I : Box (Fin (n + 1))
f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hc : ContinuousOn f (Box.Icc I)
Hd : ∀ x ∈ Box.Ioo I \ s, HasFDerivAt f (f' x) x
J : ℕ →o Box (Fin (n + 1))
hJ_sub : ∀ (n_1 : ℕ), Box.Icc (J n_1) ⊆ Box.Ioo I
hJl : Tendsto (Box.lower ∘ ⇑J) atTop (𝓝 I.lower)
hJu : Tendsto (Box.upper ∘ ⇑J) atTop (𝓝 I.upper)
hJ_sub' : ∀ (k : ℕ), Box.Icc (J k) ⊆ Box.Icc I
hJ_le : ∀ (k : ℕ), J k ≤ I
HcJ : ∀ (k : ℕ), ContinuousOn f (Box.Icc (J k))
HdJ : ∀ (k : ℕ), ∀ x ∈ Box.Icc (J k) \ s, HasFDerivWithinAt f (f' x) (Box.Icc (J k)) x
HiJ : ∀ (k : ℕ), IntegrableOn (fun x ↦ ∑ i, (f' x) (e i) i) (Box.Icc (J k)) volume
HJ_eq :
  ∀ (k : ℕ),
    ∫ (x : Fin (n + 1) → ℝ) in Box.Icc (J k), ∑ i, (f' x) (e i) i =
      ∑ i,
        ((∫ (x : Fin n → ℝ) in Box.Icc ((J k).face i), f (i.insertNth ((J k).upper i) x) i) -
          ∫ (x : Fin n → ℝ) in Box.Icc ((J k).face i), f (i.insertNth ((J k).lower i) x) i)
Hi : Integrable (fun x ↦ ∑ i, (f' x) (e i) i) (volume.restrict (⋃ n_1, Box.Ioo (J n_1)))
⊢ Tendsto (fun k ↦ ∫ (x : Fin (n + 1) → ℝ) in Box.Ioo (J k), ∑ i, (f' x) (e i) i) atTop
    (𝓝 (∫ (x : Fin (n + 1) → ℝ) in ⋃ n_1, Box.Ioo (J n_1), ∑ i, (f' x) (e i) i))
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
n : ℕ
⊢ (2 * π)⁻¹ * ‖∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤
    (2 * π)⁻¹ *
      ∫ (θ : ℝ) in 0..2 * π,
        ‖deriv (circleMap c R) θ • (circleMap c R θ - c)⁻¹ ^ n • (circleMap c R θ - c)⁻¹ • f (circleMap c R θ)‖

Tactic:
by
      gcongr
      exact intervalIntegral.norm_integral_le_integral_norm (by positivity)

Goal state after:
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ≥0
n : ℕ
hR : ↑R ^ n ≠ 0
⊢ ↑R ^ n ≠ 0

Tactic:
norm_cast at hR ⊢

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Prod

Goal state before:
pp: E : Type u_3
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
X : Type u_5
Y : Type u_6
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ∞)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ∞)
x : X
y : Y
hy : f x y ≠ 0
⊢ y ∈ V

Tactic:
have : (x, y) ∈ Function.support f.uncurry := hy

Goal state after:
E : Type u_3
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
X : Type u_5
Y : Type u_6
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝¹ : Fact (μ U < ∞)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this✝ : Fact (ν V < ∞)
x : X
y : Y
hy : f x y ≠ 0
this : (x, y) ∈ support (uncurry f)
⊢ y ∈ V
Module: Mathlib.Analysis.Complex.CauchyIntegral

Goal state before:
pp: E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → E
y : E
s : Set ℂ
hs : s.Countable
hc : ContinuousOn f (closedBall c R \ {c})
hd : ∀ z ∈ (ball c R \ {c}) \ s, DifferentiableAt ℂ f z
hy : Tendsto f (𝓝[≠] c) (𝓝 y)
⊢ ‖(∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) - (2 * ↑π * I) • y‖ ≤ 0

Tactic:
refine le_of_forall_gt_imp_ge_of_dense fun ε ε0 => ?_

Goal state after:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → E
y : E
s : Set ℂ
hs : s.Countable
hc : ContinuousOn f (closedBall c R \ {c})
hd : ∀ z ∈ (ball c R \ {c}) \ s, DifferentiableAt ℂ f z
hy : Tendsto f (𝓝[≠] c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
⊢ ‖(∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) - (2 * ↑π * I) • y‖ ≤ ε
Module: Mathlib.MeasureTheory.Integral.Prod

Goal state before:
pp: E : Type u_3
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
X : Type u_5
Y : Type u_6
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ∞)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ∞)
y : Y
x : X
hy : f x y ≠ 0
⊢ y ∈ V

Tactic:
have : (x, y) ∈ Function.support f.uncurry := hy

Goal state after:
E : Type u_3
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
X : Type u_5
Y : Type u_6
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝¹ : Fact (μ U < ∞)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this✝ : Fact (ν V < ∞)
y : Y
x : X
hy : f x y ≠ 0
this : (x, y) ∈ support (uncurry f)
⊢ y ∈ V
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: F : Type u_1
inst✝³ : Field F
p : F[X]
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Fact (map (algebraMap F E) p).Splits
ϕ : Gal(E/F)
x : ↑(p.rootSet E)
⊢ ↑((restrict p E) ϕ • x) = ϕ ↑x

Tactic:
by
  let ψ := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F p.SplittingField E)
  change ↑(ψ (ψ.symm _)) = ϕ x
  rw [AlgEquiv.apply_symm_apply ψ]
  change ϕ (rootsEquivRoots p E ((rootsEquivRoots p E).symm x)) = ϕ x
  rw [Equiv.apply_symm_apply (rootsEquivRoots p E)]

Goal state after:
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.Defs

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
σ : Type u_2
f : σ → R
s : Finset σ
n : ℕ
⊢ (map f s.val).esymm n = ∑ t ∈ Finset.powersetCard n s, t.prod f

Tactic:
by
  simp only [esymm, powersetCard_map, ← Finset.map_val_val_powersetCard, map_map]
  simp only [Function.comp_apply, Finset.prod_map_val, Finset.sum_map_val]

Goal state after: