mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: case mp.refine_2
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
hx : x ∈ s
h : ContDiffWithinAt 𝕜 0 f s x
u : Set E
H : u ∈ 𝓝[insert x s] x
p : E → FormalMultilinearSeries 𝕜 E F
hp : HasFTaylorSeriesUpToOn (↑0) f p u
⊢ ContinuousOn f (s ∩ u)

Tactic:
simp only [Nat.cast_zero, hasFTaylorSeriesUpToOn_zero_iff] at hp
      exact hp.1.mono inter_subset_right

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: 𝕜 : Type u
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type uF
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
x : E
n : ℕ∞ω
inst✝ : CompleteSpace F
h : AnalyticAt 𝕜 f x
⊢ ContDiffAt 𝕜 n f x

Tactic:
by
  rw [← contDiffWithinAt_univ]
  rw [← analyticWithinAt_univ] at h
  exact h.contDiffWithinAt

Goal state after:
Module: Mathlib.Analysis.Complex.RealDeriv

Goal state before:
pp: z : ℝ
f : ℝ → ℝ
s : Set ℝ
u : ℝ
hf : HasDerivWithinAt f u s z
⊢ HasDerivWithinAt (fun y ↦ ↑(f y)) (↑u) s z

Tactic:
by
  simpa only [Function.comp_apply, ofRealCLM_apply] using!
    ofRealCLM.hasFDerivAt.comp_hasDerivWithinAt z hf

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: case cons
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
ι : Type u_3
f : ι → E → F
i : ℕ
x : E
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
a : ι
u : Finset ι
ha : a ∉ u
IH :
  (∀ j ∈ u, ContDiffWithinAt 𝕜 (↑i) (f j) s x) →
    iteratedFDerivWithin 𝕜 i (fun x ↦ ∑ j ∈ u, f j x) s x = ∑ j ∈ u, iteratedFDerivWithin 𝕜 i (f j) s x
h : ContDiffWithinAt 𝕜 (↑i) (f a) s x ∧ ∀ a ∈ u, ContDiffWithinAt 𝕜 (↑i) (f a) s x
⊢ iteratedFDerivWithin 𝕜 i (fun x ↦ ∑ j ∈ Finset.cons a u ha, f j x) s x =
    ∑ j ∈ Finset.cons a u ha, iteratedFDerivWithin 𝕜 i (f j) s x

Tactic:
simp only [Finset.sum_cons]

Goal state after:
case cons
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
ι : Type u_3
f : ι → E → F
i : ℕ
x : E
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
a : ι
u : Finset ι
ha : a ∉ u
IH :
  (∀ j ∈ u, ContDiffWithinAt 𝕜 (↑i) (f j) s x) →
    iteratedFDerivWithin 𝕜 i (fun x ↦ ∑ j ∈ u, f j x) s x = ∑ j ∈ u, iteratedFDerivWithin 𝕜 i (f j) s x
h : ContDiffWithinAt 𝕜 (↑i) (f a) s x ∧ ∀ a ∈ u, ContDiffWithinAt 𝕜 (↑i) (f a) s x
⊢ iteratedFDerivWithin 𝕜 i (fun x ↦ f a x + ∑ j ∈ u, f j x) s x =
    iteratedFDerivWithin 𝕜 i (f a) s x + ∑ j ∈ u, iteratedFDerivWithin 𝕜 i (f j) s x
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: case top
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type uF
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
n : ℕ∞ω
inst✝ : CompleteSpace E
f : OpenPartialHomeomorph E F
f₀' : E ≃L[𝕜] F
a : F
ha : a ∈ f.target
hf₀' : HasFDerivAt (↑f) (↑f₀') (↑f.symm a)
Itop : ∀ (n : ℕ), ContDiffAt 𝕜 (↑↑n) (↑f) (↑f.symm a) → ContDiffAt 𝕜 (↑↑n) (↑f.symm) a
hf : ContDiffAt 𝕜 ∞ (↑f) (↑f.symm a)
⊢ ContDiffAt 𝕜 ∞ (↑f.symm) a

Tactic:
exact contDiffAt_infty.mpr fun n ↦ Itop n (contDiffAt_infty.mp hf n)

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

Goal state before:
pp: X : Type u_1
E : Type u_2
ι : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure X
l : Filter X
inst✝ : l.IsMeasurablyGenerated
f : X → E
b : E
h : Tendsto f (l ⊓ ae μ) (𝓝 b)
hfm✝ : StronglyMeasurableAtFilter f l μ
hμ : μ.FiniteAtFilter l
s✝ : ι → Set X
li : Filter ι
hs : Tendsto s✝ li l.smallSets
m : ι → ℝ
hsμ : (fun i ↦ μ.real (s✝ i)) =ᶠ[li] m
ε : ℝ
ε₀ : 0 < ε
this : ∀ᶠ (s : Set X) in l.smallSets, ∀ᵐ (x : X) ∂μ, x ∈ s → f x ∈ closedBall b ε
s : Set X
hμs : μ s < ⊤
h_integrable : IntegrableOn f s μ
hfm : AEStronglyMeasurable f (μ.restrict s)
⊢ (∀ᵐ (x : X) ∂μ, x ∈ s → ‖f x - b‖ ≤ ε) → ‖∫ (x : X) in s, f x ∂μ - μ.real s • b‖ ≤ ε * ‖μ.real s‖

Tactic:
h_norm

Goal state after:
X : Type u_1
E : Type u_2
ι : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure X
l : Filter X
inst✝ : l.IsMeasurablyGenerated
f : X → E
b : E
h : Tendsto f (l ⊓ ae μ) (𝓝 b)
hfm✝ : StronglyMeasurableAtFilter f l μ
hμ : μ.FiniteAtFilter l
s✝ : ι → Set X
li : Filter ι
hs : Tendsto s✝ li l.smallSets
m : ι → ℝ
hsμ : (fun i ↦ μ.real (s✝ i)) =ᶠ[li] m
ε : ℝ
ε₀ : 0 < ε
this : ∀ᶠ (s : Set X) in l.smallSets, ∀ᵐ (x : X) ∂μ, x ∈ s → f x ∈ closedBall b ε
s : Set X
hμs : μ s < ⊤
h_integrable : IntegrableOn f s μ
hfm : AEStronglyMeasurable f (μ.restrict s)
h_norm : ∀ᵐ (x : X) ∂μ, x ∈ s → ‖f x - b‖ ≤ ε
⊢ ‖∫ (x : X) in s, f x ∂μ - μ.real s • b‖ ≤ ε * ‖μ.real s‖
Module: Mathlib.Analysis.Calculus.Deriv.MeanValue

Goal state before:
pp: D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : DifferentiableOn ℝ f (interior D)
C : ℝ
lt_hf' : ∀ x ∈ interior D, deriv f x < C
x : ℝ
hx : x ∈ D
y : ℝ
hy : y ∈ D
hxy : x < y
hf'_gt : ∀ x ∈ interior D, -C < deriv (fun y ↦ -f y) x
⊢ f y - f x < C * (y - x)

Tactic:
linarith [hD.mul_sub_lt_image_sub_of_lt_deriv hf.fun_neg hf'.neg hf'_gt x hx y hy hxy]

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: case h₂
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
c : 𝕜
hc : 1 < ‖c‖
r ε : ℝ
hε : 0 < ε
hr : 0 < r
x : E
L₁ L₂ : E →L[𝕜] F
h₁ : x ∈ A f L₁ r ε
h₂ : x ∈ A f L₂ r ε
y : E
ley : r ≤ 2 * ‖c‖ * ‖y‖
ylt : ‖y‖ < r / 2
⊢ ‖f (x + y) - f x - L₁ (x + y - x)‖ ≤ ε * r

Tactic:
apply le_of_mem_A h₁
        · simp only [le_of_lt (half_pos hr), mem_closedBall, dist_self]
        · simp only [dist_eq_norm, add_sub_cancel_left, mem_closedBall, ylt.le]

Goal state after:
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≥0
fint : Integrable (fun x ↦ ↑(f x)) μ
fmeas : AEMeasurable f μ
ε : ℝ≥0
εpos : 0 < ↑ε
δ : ℝ≥0
δpos : 0 < δ
hδε : δ < ε
int_f_ne_top : ∫⁻ (a : α), ↑(f a) ∂μ ≠ ∞
g : α → ℝ≥0∞
f_lt_g : ∀ (x : α), ↑(f x) < g x
gcont : LowerSemicontinuous g
gint : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ↑δ
gint_ne : ∫⁻ (x : α), g x ∂μ ≠ ∞
g_lt_top : ∀ᵐ (x : α) ∂μ, g x < ∞
Ig : ∫⁻ (a : α), ENNReal.ofReal (g a).toReal ∂μ = ∫⁻ (a : α), g a ∂μ
⊢ (∫⁻ (a : α), g a ∂μ).toReal ≤ (∫⁻ (a : α), ↑(f a) ∂μ + ↑δ).toReal

Tactic:
by
          apply ENNReal.toReal_mono _ gint
          simpa using int_f_ne_top

Goal state after:
Module: Mathlib.Analysis.Convex.Cone.Extension

Goal state before:
pp: case refine_2.inl
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : PointedCone ℝ E
f : E →ₗ.[ℝ] ℝ
nonneg : ∀ (x : ↥f.domain), ↑x ∈ s → 0 ≤ ↑f x
dense : ∀ (y : E), ∃ x, ↑x + y ∈ s
hdom : f.domain ≠ ⊤
y : E
hy : y ∉ f.domain
c : ℝ
le_c : ∀ (x : ↥f.domain), -↑x - y ∈ s → ↑f x ≤ c
c_le : ∀ (x : ↥f.domain), ↑x + y ∈ s → c ≤ ↑f x
x : E
hx : x ∈ f.domain
r : ℝ
hy' : r • y ∈ (LinearPMap.mkSpanSingleton y (-c) ⋯).domain
hz : x + r • y ∈ (f.supSpanSingleton y (-c) hy).domain
hzs : x + r • y ∈ s
hr : r < 0
this : ↑f (r⁻¹ • ⟨x, hx⟩) ≤ c
⊢ (RingHom.id ℝ) r • c ≤ ↑f ⟨x, hx⟩

Tactic:
rwa [← mul_le_mul_iff_right₀ (neg_pos.2 hr), neg_mul, neg_mul, neg_le_neg_iff, f.map_smul,
        smul_eq_mul, ← mul_assoc, mul_inv_cancel₀ hr.ne, one_mul] at this

Goal state after:
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: case inr.mpr
E : Type u_2
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module ℝ E
α : Type u_3
inst✝⁶ : Field α
inst✝⁵ : LinearOrder α
inst✝⁴ : IsStrictOrderedRing α
inst✝³ : MulActionWithZero α ℝ
inst✝² : IsStrictOrderedModule α ℝ
inst✝¹ : MulActionWithZero α E
inst✝ : IsScalarTower α ℝ (Set E)
s : Set E
a : α
ha : 0 ≤ a
x : E
ha' : 0 < a
r : ℝ
⊢ (∃ y, (y ∈ Ioi 0 ∧ y⁻¹ • x ∈ s) ∧ a • y = r) → r ∈ Ioi 0 ∧ r⁻¹ • a • x ∈ s

Tactic:
rintro ⟨r, ⟨hr, hx⟩, rfl⟩

Goal state after:
case inr.mpr
E : Type u_2
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module ℝ E
α : Type u_3
inst✝⁶ : Field α
inst✝⁵ : LinearOrder α
inst✝⁴ : IsStrictOrderedRing α
inst✝³ : MulActionWithZero α ℝ
inst✝² : IsStrictOrderedModule α ℝ
inst✝¹ : MulActionWithZero α E
inst✝ : IsScalarTower α ℝ (Set E)
s : Set E
a : α
ha : 0 ≤ a
x : E
ha' : 0 < a
r : ℝ
hr : r ∈ Ioi 0
hx : r⁻¹ • x ∈ s
⊢ a • r ∈ Ioi 0 ∧ (a • r)⁻¹ • a • x ∈ s
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: case inr.mpr
E : Type u_2
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module ℝ E
α : Type u_3
inst✝⁸ : Field α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : MulActionWithZero α ℝ
inst✝⁴ : IsStrictOrderedModule α ℝ
inst✝³ : MulActionWithZero α E
inst✝² : SMulCommClass α ℝ ℝ
inst✝¹ : IsScalarTower α ℝ ℝ
inst✝ : IsScalarTower α ℝ E
s : Set E
a : α
ha : 0 ≤ a
ha' : 0 < a
x : E
r : ℝ
⊢ (∃ y, (y ∈ Ioi 0 ∧ y⁻¹ • x ∈ s) ∧ a⁻¹ • y = r) → r ∈ Ioi 0 ∧ ∃ y ∈ s, a • y = r⁻¹ • x

Tactic:
rintro ⟨r, ⟨hr, hx⟩, rfl⟩

Goal state after:
case inr.mpr
E : Type u_2
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module ℝ E
α : Type u_3
inst✝⁸ : Field α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : MulActionWithZero α ℝ
inst✝⁴ : IsStrictOrderedModule α ℝ
inst✝³ : MulActionWithZero α E
inst✝² : SMulCommClass α ℝ ℝ
inst✝¹ : IsScalarTower α ℝ ℝ
inst✝ : IsScalarTower α ℝ E
s : Set E
a : α
ha : 0 ≤ a
ha' : 0 < a
x : E
r : ℝ
hr : r ∈ Ioi 0
hx : r⁻¹ • x ∈ s
⊢ a⁻¹ • r ∈ Ioi 0 ∧ ∃ y ∈ s, a • y = (a⁻¹ • r)⁻¹ • x
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: E : Type u_2
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module ℝ E
α : Type u_3
inst✝⁸ : Field α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : MulActionWithZero α ℝ
inst✝⁴ : IsStrictOrderedModule α ℝ
inst✝³ : Module α E
inst✝² : SMulCommClass α ℝ ℝ
inst✝¹ : IsScalarTower α ℝ ℝ
inst✝ : IsScalarTower α ℝ E
s : Set E
symmetric : ∀ x ∈ s, -x ∈ s
a : α
⊢ gauge (a • s) = |a|⁻¹ • gauge s

Tactic:
← gauge_smul_left_of_nonneg (abs_nonneg a)

Goal state after:
E : Type u_2
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module ℝ E
α : Type u_3
inst✝⁸ : Field α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : MulActionWithZero α ℝ
inst✝⁴ : IsStrictOrderedModule α ℝ
inst✝³ : Module α E
inst✝² : SMulCommClass α ℝ ℝ
inst✝¹ : IsScalarTower α ℝ ℝ
inst✝ : IsScalarTower α ℝ E
s : Set E
symmetric : ∀ x ∈ s, -x ∈ s
a : α
⊢ gauge (a • s) = gauge (|a| • s)
Module: Mathlib.Analysis.Normed.Operator.CompleteCodomain

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : SeparatingDual 𝕜 E
inst✝¹ : Nontrivial E
inst✝ : CompleteSpace (E →L[𝕜] F)
⊢ CompleteSpace F

Tactic:
by
  refine Metric.complete_of_cauchySeq_tendsto fun f hf => ?_
  obtain ⟨v, hv⟩ : ∃ (v : E), v ≠ 0 := exists_ne 0
  obtain ⟨φ, hφ⟩ : ∃ φ : StrongDual 𝕜 E, φ v = 1 := exists_eq_one hv
  let g : ℕ → (E →L[𝕜] F) := fun n ↦ ContinuousLinearMap.smulRightL 𝕜 E F φ (f n)
  have : CauchySeq g := (ContinuousLinearMap.smulRightL 𝕜 E F φ).lipschitz.cauchySeq_comp hf
  obtain ⟨a, ha⟩ : ∃ a, Tendsto g atTop (𝓝 a) := cauchy_iff_exists_le_nhds.mp this
  refine ⟨a v, ?_⟩
  have : Tendsto (fun n ↦ g n v) atTop (𝓝 (a v)) := by
    have : Continuous (fun (i : E →L[𝕜] F) ↦ i v) := by fun_prop
    exact (this.tendsto _).comp ha
  simpa [g, ContinuousLinearMap.smulRightL, hφ]

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

Goal state before:
pp: case hcont
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 := ⋯
δpos : 0 < δ
fp : α → ℝ≥0 := ⋯
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 := ⋯
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 := ⋯
ae_g : ∀ᵐ (x : α) ∂μ, (g x).toReal = (↑(gp x)).toReal - (↑↑(gm x)).toReal
⊢ ∀ (x : α), ContinuousAt (fun p ↦ p.1 + p.2) (↑(gp x), -↑↑(gm x))

Tactic:
· intro x
      exact EReal.continuousAt_add (by simp) (by simp)

Goal state after:
Module: Mathlib.MeasureTheory.Topology

Goal state before:
pp: α : Type u_1
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : NullSingletonClass μ
U : Set α
hU : MeasurableSet U
s : Set α
hs : s ∈ codiscreteWithin U
⊢ s ∈ ae (μ.restrict U)

Tactic:
have : DiscreteTopology ↑(sᶜ ∩ U) := isDiscrete_iff_discreteTopology.mp
    <| isDiscrete_of_codiscreteWithin ((compl_compl s).symm ▸ hs)

Goal state after:
α : Type u_1
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : NullSingletonClass μ
U : Set α
hU : MeasurableSet U
s : Set α
hs : s ∈ codiscreteWithin U
this : DiscreteTopology ↑(sᶜ ∩ U)
⊢ s ∈ ae (μ.restrict U)
Module: Mathlib.Topology.ContinuousMap.ContinuousMapZero

Goal state before:
pp: X : Type u_1
Y : Type u_2
R : Type u_3
inst✝⁵ : Zero X
inst✝⁴ : Zero Y
inst✝³ : Zero R
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace R
f : C(X, Y)₀
⊢ Continuous fun g ↦ g.comp f

Tactic:
rw [continuous_induced_rng]
  change Continuous fun g : C(Y, R)₀ ↦ (g : C(Y, R)).comp (f : C(X, Y))
  fun_prop

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

Goal state before:
pp: case h_closed
X : Type u_1
E : Type u_3
F : Type u_4
inst✝¹¹ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_6
𝕜' : Type u_7
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : RCLike 𝕜'
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
inst✝⁴ : NormedSpace ℝ F
σ : 𝕜 →+* 𝕜'
inst✝³ : RingHomIsometric σ
inst✝² : CompleteSpace F
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hσ : ∀ (r : ℝ) (x : 𝕜), σ (r • x) = r • σ x
L : E →SL[σ] F
φ : X → E
φ_int : Integrable φ μ
⊢ IsClosed {f | ∫ (x : X), L (↑↑f x) ∂μ = L (∫ (x : X), ↑↑f x ∂μ)}

pp: case h_ae
X : Type u_1
E : Type u_3
F : Type u_4
inst✝¹¹ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_6
𝕜' : Type u_7
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : RCLike 𝕜'
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
inst✝⁴ : NormedSpace ℝ F
σ : 𝕜 →+* 𝕜'
inst✝³ : RingHomIsometric σ
inst✝² : CompleteSpace F
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hσ : ∀ (r : ℝ) (x : 𝕜), σ (r • x) = r • σ x
L : E →SL[σ] F
φ : X → E
φ_int : Integrable φ μ
⊢ ∀ ⦃f g : X → E⦄,
    f =ᵐ[μ] g →
      Integrable f μ → ∫ (x : X), L (f x) ∂μ = L (∫ (x : X), f x ∂μ) → ∫ (x : X), L (g x) ∂μ = L (∫ (x : X), g x ∂μ)

Tactic:
· exact isClosed_eq L.continuous_integral_comp_L1 (L.continuous.comp continuous_integral)

Goal state after:
case h_ae
X : Type u_1
E : Type u_3
F : Type u_4
inst✝¹¹ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_6
𝕜' : Type u_7
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : RCLike 𝕜'
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
inst✝⁴ : NormedSpace ℝ F
σ : 𝕜 →+* 𝕜'
inst✝³ : RingHomIsometric σ
inst✝² : CompleteSpace F
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hσ : ∀ (r : ℝ) (x : 𝕜), σ (r • x) = r • σ x
L : E →SL[σ] F
φ : X → E
φ_int : Integrable φ μ
⊢ ∀ ⦃f g : X → E⦄,
    f =ᵐ[μ] g →
      Integrable f μ → ∫ (x : X), L (f x) ∂μ = L (∫ (x : X), f x ∂μ) → ∫ (x : X), L (g x) ∂μ = L (∫ (x : X), g x ∂μ)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: ε : Type u_3
inst✝³ : TopologicalSpace ε
inst✝² : ENormedAddMonoid ε
inst✝¹ : PseudoMetrizableSpace ε
f : ℝ → ε
a b : ℝ
μ : Measure ℝ
inst✝ : NullSingletonClass μ
hab : a ≤ b
ha : ‖f a‖ₑ ≠ ∞
hb : ‖f b‖ₑ ≠ ∞
⊢ IntervalIntegrable f μ a b ↔ IntegrableOn f (Ioo a b) μ

Tactic:
by
  rw [intervalIntegrable_iff_integrableOn_Icc_of_le hab ha,
    integrableOn_Icc_iff_integrableOn_Ioo ha hb]

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
inst✝ : CompleteSpace E
c : E
⊢ ∫ (x : ℝ) in a..b, c = (b - a) • c

Tactic:
by
  simp only [integral_const', Real.volume_Ioc, ENNReal.toReal_ofReal', ← neg_sub b,
    max_zero_sub_eq_self, measureReal_def]

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

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : TopologicalSpace X
μ : Measure ℝ
inst✝ : FirstCountableTopology X
F : X → ℝ → E
bound : ℝ → ℝ
a b a₀ b₀ : ℝ
x₀ : X
hF_meas : ∀ (x : X), AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
h_bound : ∀ᶠ (x : X) in 𝓝 x₀, ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x t‖ ≤ bound t
bound_integrable : IntervalIntegrable bound μ a b
h_cont : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ContinuousAt (fun x ↦ F x t) x₀
ha₀ : a₀ ∈ Ioo a b
hb₀ : b₀ ∈ Ioo a b
hμb₀ : μ {b₀} = 0
hsub : ∀ {a₀ b₀ : ℝ}, a₀ ∈ Ioo a b → b₀ ∈ Ioo a b → Ι a₀ b₀ ⊆ Ι a b
Ioo_nhds : Ioo a b ∈ 𝓝 b₀
Icc_nhds : Icc a b ∈ 𝓝 b₀
hx₀ : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x₀ t‖ ≤ bound t
x : X
t : ℝ
hx : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x t‖ ≤ bound t
ht : t ∈ Ioo a b
⊢ ∀ (x_1 : ℝ), ‖F x x_1‖ ≤ bound x_1 ∧ ‖F x₀ x_1‖ ≤ bound x_1 → ‖F x x_1 - F x₀ x_1‖ ≤ 2 * bound x_1

Tactic:
rintro s ⟨hs₁, hs₂⟩

Goal state after:
E : Type u_1
X : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : TopologicalSpace X
μ : Measure ℝ
inst✝ : FirstCountableTopology X
F : X → ℝ → E
bound : ℝ → ℝ
a b a₀ b₀ : ℝ
x₀ : X
hF_meas : ∀ (x : X), AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
h_bound : ∀ᶠ (x : X) in 𝓝 x₀, ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x t‖ ≤ bound t
bound_integrable : IntervalIntegrable bound μ a b
h_cont : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ContinuousAt (fun x ↦ F x t) x₀
ha₀ : a₀ ∈ Ioo a b
hb₀ : b₀ ∈ Ioo a b
hμb₀ : μ {b₀} = 0
hsub : ∀ {a₀ b₀ : ℝ}, a₀ ∈ Ioo a b → b₀ ∈ Ioo a b → Ι a₀ b₀ ⊆ Ι a b
Ioo_nhds : Ioo a b ∈ 𝓝 b₀
Icc_nhds : Icc a b ∈ 𝓝 b₀
hx₀ : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x₀ t‖ ≤ bound t
x : X
t : ℝ
hx : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x t‖ ≤ bound t
ht : t ∈ Ioo a b
s : ℝ
hs₁ : ‖F x s‖ ≤ bound s
hs₂ : ‖F x₀ s‖ ≤ bound s
⊢ ‖F x s - F x₀ s‖ ≤ 2 * bound s
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c : ℝ
f : ℝ → E
μ : Measure ℝ
hab : IntervalIntegrable f μ a b
hbc : IntervalIntegrable f μ b c
hac : IntervalIntegrable f μ a c
this : Ioc a b ∪ Ioc b c ∪ Ioc c a = Ioc b a ∪ Ioc c b ∪ Ioc a c
⊢ ∫ (x : ℝ) in Ioc a b ∪ Ioc b c ∪ Ioc c a, f x ∂μ = ∫ (x : ℝ) in Ioc b a ∪ Ioc c b ∪ Ioc a c, f x ∂μ

Tactic:
by rw [this]

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

Goal state before:
pp: E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
inst✝ : CompleteSpace E
f f' : ℝ → E
hab : a < b
fa fb : E
hderiv : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hint : IntervalIntegrable f' volume a b
ha : Tendsto f (𝓝[>] a) (𝓝 fa)
hb : Tendsto f (𝓝[<] b) (𝓝 fb)
F : ℝ → E := update (update f a fa) b fb
Fderiv : ∀ x ∈ Ioo a b, HasDerivAt F (f' x) x
⊢ (ContinuousOn f ((Icc a b \ {b}) \ {a}) ∧ (a ∈ Icc a b \ {b} → Tendsto f (𝓝[(Icc a b \ {b}) \ {a}] a) (𝓝 fa))) ∧
    (b ∈ Icc a b → Tendsto (update f a fa) (𝓝[Icc a b \ {b}] b) (𝓝 fb))

Tactic:
Icc_sdiff_right,

Goal state after:
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
inst✝ : CompleteSpace E
f f' : ℝ → E
hab : a < b
fa fb : E
hderiv : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hint : IntervalIntegrable f' volume a b
ha : Tendsto f (𝓝[>] a) (𝓝 fa)
hb : Tendsto f (𝓝[<] b) (𝓝 fb)
F : ℝ → E := update (update f a fa) b fb
Fderiv : ∀ x ∈ Ioo a b, HasDerivAt F (f' x) x
⊢ (ContinuousOn f (Ico a b \ {a}) ∧ (a ∈ Ico a b → Tendsto f (𝓝[Ico a b \ {a}] a) (𝓝 fa))) ∧
    (b ∈ Icc a b → Tendsto (update f a fa) (𝓝[Ico a b] b) (𝓝 fb))
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Goal state before:
pp: g' g : ℝ → ℝ
a b : ℝ
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x
g'pos : ∀ x ∈ Ioo a b, 0 ≤ g' x
hab : a < b
meas_g' : AEMeasurable g' (volume.restrict (Ioo a b))
H : ENNReal.ofReal (g b - g a) < ∫⁻ (x : ℝ) in Ioo a b, ↑‖g' x‖₊
f : SimpleFunc ℝ ℝ≥0
fle : ∀ (x : ℝ), f x ≤ ‖g' x‖₊
fint : ∫⁻ (x : ℝ) in Ioo a b, ↑(f x) < ∞
F : ℝ → ℝ := NNReal.toReal ∘ ⇑f
hf : ENNReal.ofReal (g b - g a) < ENNReal.ofReal (∫ (x : ℝ) in Ioo a b, F x)
intF : IntegrableOn F (Ioo a b) volume
A : ∫⁻ (x : ℝ) in Ioo a b, ↑(f x) = ENNReal.ofReal (∫ (x : ℝ) in Ioo a b, F x)
⊢ ∫ (x : ℝ) in a..b, F x ≤ g b - g a

Tactic:
refine integral_le_sub_of_hasDeriv_right_of_le hab.le hcont hderiv ?_ fun x hx => ?_

Goal state after:
case refine_1
g' g : ℝ → ℝ
a b : ℝ
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x
g'pos : ∀ x ∈ Ioo a b, 0 ≤ g' x
hab : a < b
meas_g' : AEMeasurable g' (volume.restrict (Ioo a b))
H : ENNReal.ofReal (g b - g a) < ∫⁻ (x : ℝ) in Ioo a b, ↑‖g' x‖₊
f : SimpleFunc ℝ ℝ≥0
fle : ∀ (x : ℝ), f x ≤ ‖g' x‖₊
fint : ∫⁻ (x : ℝ) in Ioo a b, ↑(f x) < ∞
F : ℝ → ℝ := NNReal.toReal ∘ ⇑f
hf : ENNReal.ofReal (g b - g a) < ENNReal.ofReal (∫ (x : ℝ) in Ioo a b, F x)
intF : IntegrableOn F (Ioo a b) volume
A : ∫⁻ (x : ℝ) in Ioo a b, ↑(f x) = ENNReal.ofReal (∫ (x : ℝ) in Ioo a b, F x)
⊢ IntegrableOn F (Icc a b) volume

case refine_2
g' g : ℝ → ℝ
a b : ℝ
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x
g'pos : ∀ x ∈ Ioo a b, 0 ≤ g' x
hab : a < b
meas_g' : AEMeasurable g' (volume.restrict (Ioo a b))
H : ENNReal.ofReal (g b - g a) < ∫⁻ (x : ℝ) in Ioo a b, ↑‖g' x‖₊
f : SimpleFunc ℝ ℝ≥0
fle : ∀ (x : ℝ), f x ≤ ‖g' x‖₊
fint : ∫⁻ (x : ℝ) in Ioo a b, ↑(f x) < ∞
F : ℝ → ℝ := NNReal.toReal ∘ ⇑f
hf : ENNReal.ofReal (g b - g a) < ENNReal.ofReal (∫ (x : ℝ) in Ioo a b, F x)
intF : IntegrableOn F (Ioo a b) volume
A : ∫⁻ (x : ℝ) in Ioo a b, ↑(f x) = ENNReal.ofReal (∫ (x : ℝ) in Ioo a b, F x)
x : ℝ
hx : x ∈ Ioo a b
⊢ F x ≤ g' x
Module: Mathlib.Topology.Algebra.Order.Floor

Goal state before:
pp: α : Type u_1
inst✝³ : Ring α
inst✝² : LinearOrder α
inst✝¹ : FloorRing α
inst✝ : IsStrictOrderedRing α
b : ℤ
⊢ b ≤ ⌊↑(b + 1)⌋

Tactic:
by rw [floor_intCast]; exact (lt_add_one _).le

Goal state after:
Module: Mathlib.Topology.Algebra.Order.Floor

Goal state before:
pp: case inl.left.refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁷ : Ring α
inst✝⁶ : LinearOrder α
inst✝⁵ : FloorRing α
inst✝⁴ : TopologicalSpace α
inst✝³ : IsStrictOrderedRing α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : β → α → γ
h : ContinuousOn (uncurry f) (univ ×ˢ Icc 0 1)
hf : ∀ (s : β), f s 0 = f s 1
s : β
n : ℤ
⊢ 𝓝 (uncurry f (s, 1)) = 𝓝 ((uncurry f ∘ Prod.map id fract) (s, ↑n))

Tactic:
· simp [hf]

Goal state after: