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

Goal state before:
pp: E : Type u_2
inst✝⁵ : TopologicalSpace E
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
s : Set E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : LocallyConvexSpace ℝ E
hs₁ : Convex ℝ s
hs₂ : IsCompact s
hs : s.Nonempty
ht₁ : Convex ℝ ∅
ht₂ : IsClosed ∅
disj : Disjoint s ∅
⊢ (∀ a ∈ s, 0 a < 1) ∧ 1 < 2 ∧ ∀ b ∈ ∅, 2 < 0 b

Tactic:
simp

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.Separation

Goal state before:
pp: E : Type u_2
inst✝⁵ : TopologicalSpace E
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
s✝ t✝ : Set E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : LocallyConvexSpace ℝ E
hs₁ : Convex ℝ s✝
hs₂ : IsClosed s✝
ht₁ : Convex ℝ t✝
ht₂ : IsCompact t✝
disj : Disjoint s✝ t✝
f : StrongDual ℝ E
s t : ℝ
hs : ∀ a ∈ t✝, f a < s
st : s < t
ht : ∀ b ∈ s✝, t < f b
⊢ ∀ a ∈ s✝, (-f) a < -t

Tactic:
simpa using ht

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
f : E → F
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
inst✝ : CompleteSpace F
this : IsComplete univ
x✝ : E
⊢ DifferentiableAt 𝕜 f x✝ ↔ DifferentiableAt 𝕜 f x✝ ∧ fderiv 𝕜 f x✝ ∈ univ

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
f : E → F
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
inst✝ : CompleteSpace F
⊢ MeasurableSet {x | DifferentiableAt 𝕜 f x}

Tactic:
have : IsComplete (univ : Set (E →L[𝕜] F)) := isComplete_univ
  convert! measurableSet_of_differentiableAt_of_isComplete 𝕜 f this
  simp

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : CompleteSpace F
inst✝³ : MeasurableSpace 𝕜
inst✝² : OpensMeasurableSpace 𝕜
inst✝¹ : MeasurableSpace F
inst✝ : BorelSpace F
f : 𝕜 → F
⊢ Measurable (deriv f)

Tactic:
by
  simpa only [fderiv_apply_one_eq_deriv] using measurable_fderiv_apply_const 𝕜 f 1

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

Goal state before:
pp: case refine_3
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
⊢ ↑↑(gm x) ≠ ⊥

Tactic:
simp

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

Goal state before:
pp: case refine_4
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
⊢ ↑↑(-f x).toNNReal ≠ ⊤

Tactic:
simp

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.Separation

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁷ : TopologicalSpace E
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
s t : Set E
inst✝⁴ : RCLike 𝕜
inst✝³ : Module 𝕜 E
inst✝² : IsScalarTower ℝ 𝕜 E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul 𝕜 E
hs : Convex ℝ s
ht : Convex ℝ t
hst : Disjoint (interior s) t
hsint : (interior s).Nonempty
htne : ¬t.Nonempty
⊢ (∀ a ∈ s, re (0 a) ≤ 0) ∧ ∀ b ∈ t, 0 ≤ re (0 b)

Tactic:
simp

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

Goal state before:
pp: case hf
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
⊢ LowerSemicontinuous fun z ↦ ↑(gp z)

Tactic:
exact continuous_coe_ennreal_ereal.comp_lowerSemicontinuous gpcont fun x y hxy =>
          EReal.coe_ennreal_le_coe_ennreal_iff.2 hxy

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
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
⊢ (↑(gp x), -↑↑(gm x)).1 ≠ ⊤ ∨ (↑(gp x), -↑↑(gm x)).2 ≠ ⊥

Tactic:
simp

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.Separation

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁸ : TopologicalSpace E
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
s✝ t✝ : Set E
inst✝⁵ : RCLike 𝕜
inst✝⁴ : Module 𝕜 E
inst✝³ : IsScalarTower ℝ 𝕜 E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
hs₁ : Convex ℝ s✝
hs₂ : IsClosed s✝
ht₁ : Convex ℝ t✝
ht₂ : IsCompact t✝
disj : Disjoint s✝ t✝
f : StrongDual 𝕜 E
s t : ℝ
hs : ∀ a ∈ t✝, re (f a) < s
st : s < t
ht : ∀ b ∈ s✝, t < re (f b)
⊢ ∀ a ∈ s✝, re ((-f) a) < -t

Tactic:
simpa using ht

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
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
⊢ (↑(gp x), -↑↑(gm x)).1 ≠ ⊥ ∨ (↑(gp x), -↑↑(gm x)).2 ≠ ⊤

Tactic:
simp

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
hf : Integrable f μ
ε : ℝ
εpos : 0 < ε
g : α → EReal
g_lt_f : ∀ (x : α), ↑(-f x) < g x
gcont : LowerSemicontinuous g
g_integrable : Integrable (fun x ↦ (g x).toReal) μ
g_lt_top : ∀ᵐ (x : α) ∂μ, g x < ⊤
gint : ∫ (x : α), (g x).toReal ∂μ < ∫ (x : α), -f x ∂μ + ε
x : α
⊢ -↑(f x) < g x

Tactic:
simpa only [EReal.coe_neg] using g_lt_f x

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
hf : Integrable f μ
ε : ℝ
εpos : 0 < ε
g : α → EReal
g_lt_f : ∀ (x : α), ↑(-f x) < g x
gcont : LowerSemicontinuous g
g_integrable : Integrable (fun x ↦ (g x).toReal) μ
g_lt_top : ∀ᵐ (x : α) ∂μ, g x < ⊤
gint : ∫ (x : α), (g x).toReal ∂μ < ∫ (x : α), -f x ∂μ + ε
x✝ : α
⊢ ((fun x ↦ -g x) x✝).toReal = (-fun x ↦ (g x).toReal) x✝

Tactic:
simp

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.Separation

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁹ : TopologicalSpace E
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module ℝ E
s : Set E
inst✝⁶ : RCLike 𝕜
inst✝⁵ : Module 𝕜 E
inst✝⁴ : IsScalarTower ℝ 𝕜 E
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : LocallyConvexSpace ℝ E
inst✝ : HereditarilyLindelofSpace E
hs₁ : Convex ℝ s
hs₂ : IsClosed s
ι : Type (max u_1 u_2) := (l : StrongDual 𝕜 E) × { c // ∀ y ∈ s, re (l y) ≤ c }
l : ι → StrongDual 𝕜 E := fun lc ↦ lc.fst
c : ι → ℝ := fun lc ↦ ↑lc.snd
hc : ∀ (i : ι), ∀ y ∈ s, re ((l i) y) ≤ c i := fun lc ↦ Subtype.prop lc.snd
x✝¹ : E
x✝ : x✝¹ ∈ s
⊢ re (0 x✝¹) ≤ 0

Tactic:
simp

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

Goal state before:
pp: X : Type u_1
R : Type u_2
inst✝⁴ : Zero X
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace R
inst✝¹ : AddZeroClass R
inst✝ : ContinuousAdd R
f g : C(X, R)₀
⊢ (↑f + ↑g) 0 = 0

Tactic:
simp

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

Goal state before:
pp: X : Type u_1
R : Type u_2
inst✝⁴ : Zero X
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace R
inst✝¹ : NegZeroClass R
inst✝ : ContinuousNeg R
f : C(X, R)₀
⊢ (-↑f) 0 = 0

Tactic:
simp

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

Goal state before:
pp: X : Type u_1
R : Type u_2
inst✝⁴ : Zero X
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace R
inst✝¹ : SubNegZeroMonoid R
inst✝ : ContinuousSub R
f g : C(X, R)₀
⊢ (↑f - ↑g) 0 = 0

Tactic:
simp

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

Goal state before:
pp: X : Type u_1
R : Type u_2
inst✝⁴ : Zero X
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace R
inst✝¹ : MulZeroClass R
inst✝ : ContinuousMul R
f g : C(X, R)₀
⊢ (↑f * ↑g) 0 = 0

Tactic:
simp

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

Goal state before:
pp: X : Type u_1
R : Type u_2
inst✝⁵ : Zero X
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace R
M : Type u_3
inst✝² : Zero R
inst✝¹ : SMulZeroClass M R
inst✝ : ContinuousConstSMul M R
m : M
f : C(X, R)₀
⊢ (m • ↑f) 0 = 0

Tactic:
simp

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

Goal state before:
pp: X : Type u_1
R : Type u_2
inst✝⁶ : Zero X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace R
inst✝³ : CommSemiring R
inst✝² : IsTopologicalSemiring R
inst✝¹ : StarRing R
inst✝ : ContinuousStar R
f : C(X, R)₀
⊢ (star ↑f) 0 = 0

Tactic:
simp

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

Goal state before:
pp: X : Type u_1
R : Type u_2
inst✝⁴ : Zero X
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace R
inst✝¹ : CommSemiring R
inst✝ : IsTopologicalSemiring R
f g : C(X, R)₀
⊢ ⇑(f + g) = ⇑f + ⇑g

Tactic:
simp

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
M : Type u_3
R : Type u_4
S : Type u_5
inst✝¹⁸ : Zero X
inst✝¹⁷ : Zero Y
inst✝¹⁶ : CommSemiring M
inst✝¹⁵ : TopologicalSpace X
inst✝¹⁴ : TopologicalSpace Y
inst✝¹³ : TopologicalSpace R
inst✝¹² : TopologicalSpace S
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : StarRing R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : CommSemiring S
inst✝⁶ : StarRing S
inst✝⁵ : IsTopologicalSemiring S
inst✝⁴ : ContinuousStar S
inst✝³ : Module M R
inst✝² : Module M S
inst✝¹ : ContinuousConstSMul M R
inst✝ : ContinuousConstSMul M S
φ : R →⋆ₙₐ[M] S
hφ : Continuous[inst✝¹³, inst✝¹²] ⇑φ
⊢ { toFun := ⇑φ, continuous_toFun := hφ } 0 = 0

Tactic:
simp

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
M : Type u_3
R : Type u_4
S : Type u_5
inst✝¹⁸ : Zero X
inst✝¹⁷ : Zero Y
inst✝¹⁶ : CommSemiring M
inst✝¹⁵ : TopologicalSpace X
inst✝¹⁴ : TopologicalSpace Y
inst✝¹³ : TopologicalSpace R
inst✝¹² : TopologicalSpace S
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : StarRing R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : CommSemiring S
inst✝⁶ : StarRing S
inst✝⁵ : IsTopologicalSemiring S
inst✝⁴ : ContinuousStar S
inst✝³ : Module M R
inst✝² : Module M S
inst✝¹ : ContinuousConstSMul M R
inst✝ : ContinuousConstSMul M S
φ : R →⋆ₙₐ[M] S
hφ : Continuous[inst✝¹³, inst✝¹²] ⇑φ
r : M
f : C(X, R)₀
⊢ ∀ (x : X),
    ({ toFun := ⇑φ, continuous_toFun := hφ, map_zero' := ⋯ }.comp (r • f)) x =
      ((MonoidHom.id M) r • { toFun := ⇑φ, continuous_toFun := hφ, map_zero' := ⋯ }.comp f) x

Tactic:
simp

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
M : Type u_3
R : Type u_4
S : Type u_5
inst✝¹⁸ : Zero X
inst✝¹⁷ : Zero Y
inst✝¹⁶ : CommSemiring M
inst✝¹⁵ : TopologicalSpace X
inst✝¹⁴ : TopologicalSpace Y
inst✝¹³ : TopologicalSpace R
inst✝¹² : TopologicalSpace S
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : StarRing R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : CommSemiring S
inst✝⁶ : StarRing S
inst✝⁵ : IsTopologicalSemiring S
inst✝⁴ : ContinuousStar S
inst✝³ : Module M R
inst✝² : Module M S
inst✝¹ : ContinuousConstSMul M R
inst✝ : ContinuousConstSMul M S
φ : R →⋆ₙₐ[M] S
hφ : Continuous[inst✝¹³, inst✝¹²] ⇑φ
⊢ ∀ (x : X), ({ toFun := ⇑φ, continuous_toFun := hφ, map_zero' := ⋯ }.comp 0) x = 0 x

Tactic:
simp

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
M : Type u_3
R : Type u_4
S : Type u_5
inst✝¹⁸ : Zero X
inst✝¹⁷ : Zero Y
inst✝¹⁶ : CommSemiring M
inst✝¹⁵ : TopologicalSpace X
inst✝¹⁴ : TopologicalSpace Y
inst✝¹³ : TopologicalSpace R
inst✝¹² : TopologicalSpace S
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : StarRing R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : CommSemiring S
inst✝⁶ : StarRing S
inst✝⁵ : IsTopologicalSemiring S
inst✝⁴ : ContinuousStar S
inst✝³ : Module M R
inst✝² : Module M S
inst✝¹ : ContinuousConstSMul M R
inst✝ : ContinuousConstSMul M S
φ : R →⋆ₙₐ[M] S
hφ : Continuous[inst✝¹³, inst✝¹²] ⇑φ
x✝¹ x✝ : C(X, R)₀
⊢ ∀ (x : X),
    ({ toFun := ⇑φ, continuous_toFun := hφ, map_zero' := ⋯ }.comp (x✝¹ + x✝)) x =
      ({ toFun := ⇑φ, continuous_toFun := hφ, map_zero' := ⋯ }.comp x✝¹ +
          { toFun := ⇑φ, continuous_toFun := hφ, map_zero' := ⋯ }.comp x✝)
        x

Tactic:
simp

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
M : Type u_3
R : Type u_4
S : Type u_5
inst✝¹⁸ : Zero X
inst✝¹⁷ : Zero Y
inst✝¹⁶ : CommSemiring M
inst✝¹⁵ : TopologicalSpace X
inst✝¹⁴ : TopologicalSpace Y
inst✝¹³ : TopologicalSpace R
inst✝¹² : TopologicalSpace S
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : StarRing R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : CommSemiring S
inst✝⁶ : StarRing S
inst✝⁵ : IsTopologicalSemiring S
inst✝⁴ : ContinuousStar S
inst✝³ : Module M R
inst✝² : Module M S
inst✝¹ : ContinuousConstSMul M R
inst✝ : ContinuousConstSMul M S
φ : R →⋆ₙₐ[M] S
hφ : Continuous[inst✝¹³, inst✝¹²] ⇑φ
x✝¹ x✝ : C(X, R)₀
⊢ ∀ (x : X),
    ({ toFun := ⇑φ, continuous_toFun := hφ, map_zero' := ⋯ }.comp (x✝¹ * x✝)) x =
      ({ toFun := ⇑φ, continuous_toFun := hφ, map_zero' := ⋯ }.comp x✝¹ *
          { toFun := ⇑φ, continuous_toFun := hφ, map_zero' := ⋯ }.comp x✝)
        x

Tactic:
simp

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

Goal state before:
pp: case h_add
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
a✝ : Disjoint (Function.support f) (Function.support g)
f_int : Integrable f μ
g_int : Integrable g 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) → ∫ (x : X), L ((f + g) x) ∂μ = L (∫ (x : X), (f + g) x ∂μ)

Tactic:
hf

Goal state after:
case h_add
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
a✝ : Disjoint (Function.support f) (Function.support g)
f_int : Integrable f μ
g_int : Integrable g μ
hf : ∫ (x : X), L (f x) ∂μ = L (∫ (x : X), f x ∂μ)
⊢ ∫ (x : X), L (g x) ∂μ = L (∫ (x : X), g x ∂μ) → ∫ (x : X), L ((f + g) x) ∂μ = L (∫ (x : X), (f + g) x ∂μ)
Module: Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

Goal state before:
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
hfg : f =ᵐ[μ] g
a✝ : 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:
hf

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
hfg : f =ᵐ[μ] g
a✝ : Integrable f μ
hf : ∫ (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.Bochner.ContinuousLinearMap

Goal state before:
pp: X : Type u_1
E : Type u_3
Fₗ : Type u_5
inst✝⁹ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_6
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup Fₗ
inst✝⁴ : NormedSpace 𝕜 Fₗ
inst✝³ : NormedSpace ℝ Fₗ
inst✝² : CompleteSpace Fₗ
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
L : E →L[𝕜] Fₗ
φ : X → E
φ_int : Integrable φ μ
⊢ ∀ (r : ℝ) (x : 𝕜), (RingHom.id 𝕜) (r • x) = r • (RingHom.id 𝕜) x

Tactic:
simp

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

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
x : ℝ
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ), n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * (1 / 2) ^ e
L0 : ℕ → F := fun e ↦ L e (n e) (n e)
this : CauchySeq L0
f' : F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
e : ℕ
he : (1 / 2) ^ e < ε / 16
⊢ x < x + (1 / 2) ^ (n e + 1)

Tactic:
simp

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

Goal state before:
pp: case pos.refine_4
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
f_meas : Measurable f
g : X → E
hE : CompleteSpace E
hg : Integrable g (μ.withDensity fun x ↦ ↑(f x))
u v : X → E
huv : u =ᵐ[μ.withDensity fun x ↦ ↑(f x)] v
a✝ : Integrable u (μ.withDensity fun x ↦ ↑(f x))
hu : (∫ (x : X), u x ∂μ.withDensity fun x ↦ ↑(f x)) = ∫ (x : X), f x • u x ∂μ
⊢ (∫ (x : X), v x ∂μ.withDensity fun x ↦ ↑(f x)) = ∫ (x : X), f x • v x ∂μ

Tactic:
← integral_congr_ae huv,

Goal state after:
case pos.refine_4
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
f_meas : Measurable f
g : X → E
hE : CompleteSpace E
hg : Integrable g (μ.withDensity fun x ↦ ↑(f x))
u v : X → E
huv : u =ᵐ[μ.withDensity fun x ↦ ↑(f x)] v
a✝ : Integrable u (μ.withDensity fun x ↦ ↑(f x))
hu : (∫ (x : X), u x ∂μ.withDensity fun x ↦ ↑(f x)) = ∫ (x : X), f x • u x ∂μ
⊢ (∫ (a : X), u a ∂μ.withDensity fun x ↦ ↑(f x)) = ∫ (x : X), f x • v x ∂μ
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: case inl
ι : Type u_1
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
F : ι → ℝ → E
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : ⊥.IsCountablyGenerated
hF : ∀ᶠ (i : ι) in ⊥, ContinuousOn (F i) [[a, b]]
h_lim : TendstoUniformlyOn F f ⊥ [[a, b]]
⊢ Tendsto (fun n ↦ ∫ (x : ℝ) in a..b, F n x ∂μ) ⊥ (𝓝 (∫ (x : ℝ) in a..b, f x ∂μ))

Tactic:
simp

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

Goal state before:
pp: ι : Type u_1
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
l : Filter ι
inst✝¹ : l.IsCountablyGenerated
F : ι → ℝ → E
inst✝ : IsLocallyFiniteMeasure μ
hF : ∀ᶠ (i : ι) in l, ContinuousOn (F i) [[a, b]]
h_lim : TendstoUniformlyOn F f l [[a, b]]
hl : l.NeBot
C : ℝ
hC : C ∈ upperBounds ((fun x ↦ ‖f x‖) '' [[a, b]])
⊢ C < C + 1

Tactic:
simp

Goal state after: