mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Order.Rolle

Goal state before:
pp: case pos
X : Type u_1
Y : Type u_2
inst✝⁶ : ConditionallyCompleteLinearOrder X
inst✝⁵ : DenselyOrdered X
inst✝⁴ : TopologicalSpace X
inst✝³ : OrderTopology X
inst✝² : LinearOrder Y
inst✝¹ : TopologicalSpace Y
inst✝ : OrderTopology Y
f : X → Y
a b : X
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : (Icc a b).Nonempty
c : X
cmem : c ∈ Icc a b
cle : ∀ x ∈ Icc a b, f c ≤ f x
C : X
Cmem : C ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
hc : f c = f a
hC : f C = f a
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c

pp: case neg
X : Type u_1
Y : Type u_2
inst✝⁶ : ConditionallyCompleteLinearOrder X
inst✝⁵ : DenselyOrdered X
inst✝⁴ : TopologicalSpace X
inst✝³ : OrderTopology X
inst✝² : LinearOrder Y
inst✝¹ : TopologicalSpace Y
inst✝ : OrderTopology Y
f : X → Y
a b : X
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : (Icc a b).Nonempty
c : X
cmem : c ∈ Icc a b
cle : ∀ x ∈ Icc a b, f c ≤ f x
C : X
Cmem : C ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
hc : f c = f a
hC : ¬f C = f a
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c

Tactic:
· have : ∀ x ∈ Icc a b, f x = f a := fun x hx => le_antisymm (hC ▸ Cge x hx) (hc ▸ cle x hx)
      -- `f` is a constant, so we can take any point in `Ioo a b`
      rcases nonempty_Ioo.2 hab with ⟨c', hc'⟩
      refine ⟨c', hc', Or.inl fun x hx ↦ ?_⟩
      simp only [mem_ofPred_eq, this x hx, this c' (Ioo_subset_Icc_self hc'), le_rfl]

Goal state after:
case neg
X : Type u_1
Y : Type u_2
inst✝⁶ : ConditionallyCompleteLinearOrder X
inst✝⁵ : DenselyOrdered X
inst✝⁴ : TopologicalSpace X
inst✝³ : OrderTopology X
inst✝² : LinearOrder Y
inst✝¹ : TopologicalSpace Y
inst✝ : OrderTopology Y
f : X → Y
a b : X
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : (Icc a b).Nonempty
c : X
cmem : c ∈ Icc a b
cle : ∀ x ∈ Icc a b, f c ≤ f x
C : X
Cmem : C ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
hc : f c = f a
hC : ¬f C = f a
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type uE
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
x : E
f g : E → 𝕜
n : ℕ∞ω
hf : ContDiffWithinAt 𝕜 n f s x
hg : ContDiffWithinAt 𝕜 n g s x
hx : g x ≠ 0
⊢ ContDiffWithinAt 𝕜 n (f / g) s x

Tactic:
by
  change ContDiffWithinAt 𝕜 n (fun x => f x / g x) s x
  simpa only [div_eq_mul_inv] using hf.mul (hg.fun_inv hx)

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

Goal state before:
pp: case zero
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)
hf : ContDiffAt 𝕜 (↑0) (↑f) (↑f.symm a)
⊢ ∃ u ∈ 𝓝 a, ContinuousOn (↑f.symm) u

Tactic:
exact ⟨f.target, IsOpen.mem_nhds f.open_target ha, f.continuousOn_invFun⟩

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

Goal state before:
pp: 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)
n : ℕ
IH : ContDiffAt 𝕜 (↑↑n) (↑f) (↑f.symm a) → ContDiffAt 𝕜 (↑↑n) (↑f.symm) a
hf : ContDiffAt 𝕜 (↑↑n.succ) (↑f) (↑f.symm a)
f' : E → E →L[𝕜] F
hf' : ContDiffAt 𝕜 (↑n) f' (↑f.symm a)
u : Set E
hu : u ∈ 𝓝 (↑f.symm a)
hff' : ∀ x ∈ u, HasFDerivAt (↑f) (f' x) x
eq_f₀' : f' (↑f.symm a) = ↑f₀'
hf₀' : range ContinuousLinearEquiv.toContinuousLinearMap ∈ 𝓝 ↑f₀'
⊢ {y | ∃ e, ↑e = f' y} ∈ 𝓝 (↑f.symm a)

Tactic:
← eq_f₀'

Goal state after:
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)
n : ℕ
IH : ContDiffAt 𝕜 (↑↑n) (↑f) (↑f.symm a) → ContDiffAt 𝕜 (↑↑n) (↑f.symm) a
hf : ContDiffAt 𝕜 (↑↑n.succ) (↑f) (↑f.symm a)
f' : E → E →L[𝕜] F
hf' : ContDiffAt 𝕜 (↑n) f' (↑f.symm a)
u : Set E
hu : u ∈ 𝓝 (↑f.symm a)
hff' : ∀ x ∈ u, HasFDerivAt (↑f) (f' x) x
eq_f₀' : f' (↑f.symm a) = ↑f₀'
hf₀' : range ContinuousLinearEquiv.toContinuousLinearMap ∈ 𝓝 (f' (↑f.symm a))
⊢ {y | ∃ e, ↑e = f' y} ∈ 𝓝 (↑f.symm a)
Module: Mathlib.Analysis.Calculus.Deriv.MeanValue

Goal state before:
pp: f : ℝ → ℝ
a : ℝ
hf : Tendsto (derivWithin f (Ioi a)) (𝓝[>] a) atTop
hcont_at_a : ContinuousWithinAt f (Ici a) a
hdiff : Tendsto (slope f a) (𝓝[>] a) (𝓝 (derivWithin f (Ioi a) a))
h₀ : ∀ᶠ (b : ℝ) in 𝓝[>] a, ∀ x ∈ Ioc a b, max (derivWithin f (Ioi a) a + 1) 0 < derivWithin f (Ioi a) x
h₁ : ∀ᶠ (b : ℝ) in 𝓝[>] a, slope f a b < derivWithin f (Ioi a) a + 1
b : ℝ
hb : ∀ x ∈ Ioc a b, max (derivWithin f (Ioi a) a + 1) 0 < derivWithin f (Ioi a) x
hslope : slope f a b < derivWithin f (Ioi a) a + 1
hab : a < b
z : ℝ
hz : z ∈ Ioc a b
⊢ DifferentiableWithinAt ℝ f (Ioi a) z

Tactic:
have : derivWithin f (Ioi a) z ≠ 0 := ne_of_gt <| by
          simp_all only [and_imp, mem_Ioc, max_lt_iff]

Goal state after:
f : ℝ → ℝ
a : ℝ
hf : Tendsto (derivWithin f (Ioi a)) (𝓝[>] a) atTop
hcont_at_a : ContinuousWithinAt f (Ici a) a
hdiff : Tendsto (slope f a) (𝓝[>] a) (𝓝 (derivWithin f (Ioi a) a))
h₀ : ∀ᶠ (b : ℝ) in 𝓝[>] a, ∀ x ∈ Ioc a b, max (derivWithin f (Ioi a) a + 1) 0 < derivWithin f (Ioi a) x
h₁ : ∀ᶠ (b : ℝ) in 𝓝[>] a, slope f a b < derivWithin f (Ioi a) a + 1
b : ℝ
hb : ∀ x ∈ Ioc a b, max (derivWithin f (Ioi a) a + 1) 0 < derivWithin f (Ioi a) x
hslope : slope f a b < derivWithin f (Ioi a) a + 1
hab : a < b
z : ℝ
hz : z ∈ Ioc a b
this : derivWithin f (Ioi a) z ≠ 0
⊢ DifferentiableWithinAt ℝ f (Ioi a) z
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: 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
L : E →L[𝕜] F
r ε : ℝ
⊢ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (A f L r ε)

Tactic:
by
  rw [Metric.isOpen_iff]
  rintro x ⟨r', r'_mem, hr'⟩
  obtain ⟨s, s_gt, s_lt⟩ : ∃ s : ℝ, r / 2 < s ∧ s < r' := exists_between r'_mem.1
  have : s ∈ Ioc (r / 2) r := ⟨s_gt, le_of_lt (s_lt.trans_le r'_mem.2)⟩
  refine ⟨r' - s, by linarith, fun x' hx' => ⟨s, this, ?_⟩⟩
  have B : ball x' s ⊆ ball x r' := ball_subset (le_of_lt hx')
  intro y hy z hz
  exact hr' y (B hy) z (B hz)

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Pointed

Goal state before:
pp: R : Type u_1
E : Type u_2
inst✝⁴ : Semiring R
inst✝³ : PartialOrder R
inst✝² : IsOrderedRing R
inst✝¹ : AddCommMonoid E
inst✝ : Module R E
s : Set (Submodule R E)
⊢ ↑(⨆ S ∈ s, S) = ⨆ S ∈ s, ↑S

Tactic:
by
  rw [← sSup_eq_iSup, ofSubmodule_sSup, sSup_eq_iSup, iSup_image]

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

Goal state before:
pp: 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
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → E →L[𝕜] 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)
e p q e' p' q' : ℕ
hp : n e ≤ p
hq : n e ≤ q
hp' : n e' ≤ p'
hq' : n e' ≤ q'
he' : e ≤ e'
r : ℕ := max (n e) (n e')
I : (1 / 2) ^ e' ≤ (1 / 2) ^ e
J1 : ‖L e p q - L e p r‖ ≤ 4 * ‖c‖ * (1 / 2) ^ e
I1 : x ∈ A f (L e p r) ((1 / 2) ^ r) ((1 / 2) ^ e)
⊢ ‖L e p r - L e' p' r‖ ≤ 4 * ‖c‖ * (1 / 2) ^ e

Tactic:
have I2 : x ∈ A f (L e' p' r) ((1 / 2) ^ r) ((1 / 2) ^ e') :=
        (hn e' p' r hp' (le_max_right _ _)).2.2

Goal state after:
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
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → E →L[𝕜] 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)
e p q e' p' q' : ℕ
hp : n e ≤ p
hq : n e ≤ q
hp' : n e' ≤ p'
hq' : n e' ≤ q'
he' : e ≤ e'
r : ℕ := max (n e) (n e')
I : (1 / 2) ^ e' ≤ (1 / 2) ^ e
J1 : ‖L e p q - L e p r‖ ≤ 4 * ‖c‖ * (1 / 2) ^ e
I1 : x ∈ A f (L e p r) ((1 / 2) ^ r) ((1 / 2) ^ e)
I2 : x ∈ A f (L e' p' r) ((1 / 2) ^ r) ((1 / 2) ^ e')
⊢ ‖L e p r - L e' p' r‖ ≤ 4 * ‖c‖ * (1 / 2) ^ e
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: a : ℝ
h : 0 < a
⊢ 0 < 1 / (2 * a + 1)

Tactic:
· exact div_pos one_pos (by linarith)

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

Goal state before:
pp: case refine_2.inr.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
⊢ (RingHom.id ℝ) r • c ≤ ↑f ⟨x, hx⟩

Tactic:
subst r
      simp only [zero_smul, add_zero] at hzs ⊢
      rw [RingHom.id_apply, zero_smul]
      apply nonneg
      exact hzs

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

Goal state before:
pp: case refine_2
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
a : ℝ
hs₀ : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃a : ℝ⦄, 0 ≤ a → a ≤ 1 → a • x ∈ s
r : ℝ
hr : r > 0
b : ℝ
hb : 0 < b
x : E
hx' : x ∈ s
hs₂ : Absorbs ℝ s {(fun x ↦ b • x) x}
hx : (fun x ↦ b • x) x ∉ a • s
h : ∀ (c : ℝ), r ≤ ‖c‖ → {(fun x ↦ b • x) x} ⊆ c • s
hba : b < a
ha : 0 < a
⊢ (fun x ↦ a • x) ((a⁻¹ * b) • x) = (fun x ↦ b • x) x

Tactic:
· dsimp only
    rw [← mul_smul, mul_inv_cancel_left₀ ha.ne']

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

Goal state before:
pp: E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
f : E →ₗ.[ℝ] ℝ
N : E → ℝ
N_hom : ∀ (c : ℝ), 0 < c → ∀ (x : E), N (c • x) = c * N x
N_add : ∀ (x y : E), N (x + y) ≤ N x + N y
hf : ∀ (x : ↥f.domain), ↑f x ≤ N ↑x
⊢ ∃ g, (∀ (x : ↥f.domain), g ↑x = ↑f x) ∧ ∀ (x : E), g x ≤ N x

Tactic:
by
  have N_0 : N 0 = 0 := by grind [N_hom 2 (by norm_num) 0, smul_zero]
  let s : PointedCone ℝ (E × ℝ) :=
    { carrier := { p : E × ℝ | N p.1 ≤ p.2 }
      zero_mem' := by simp [N_0]
      smul_mem' := fun ⟨_, hc⟩ _ _ => by rcases eq_or_lt_of_le' hc <;> simp_all
      add_mem' := fun hx hy => (N_add _ _).trans (add_le_add hx hy) }
  set f' := (-f).coprod (LinearMap.id.toPMap ⊤)
  have hf'_nonneg : ∀ x : f'.domain, x.1 ∈ s → 0 ≤ f' x := fun x (hx : N x.1.1 ≤ x.1.2) ↦ by
    simpa [f'] using le_trans (hf ⟨x.1.1, x.2.1⟩) hx
  have hf'_dense : ∀ y : E × ℝ, ∃ x : f'.domain, ↑x + y ∈ s := by
    rintro ⟨x, y⟩
    exact ⟨⟨(0, N x - y), ⟨f.domain.zero_mem, trivial⟩⟩, by simp [s]⟩
  obtain ⟨g, g_eq, g_nonneg⟩ := riesz_extension s f' hf'_nonneg hf'_dense
  replace g_eq : ∀ (x : f.domain) (y : ℝ), g (x, y) = y - f x := fun x y ↦
    (g_eq ⟨(x, y), ⟨x.2, trivial⟩⟩).trans (sub_eq_neg_add _ _).symm
  refine ⟨-g.comp (inl ℝ E ℝ), fun x ↦ ?_, fun x ↦ ?_⟩
  · simp [g_eq x 0]
  · calc -g (x, 0) = g (0, N x) - g (x, N x) := by simp [← map_sub, ← map_neg]
      _ = N x - g (x, N x) := by simpa using g_eq 0 (N x)
      _ ≤ N x := by simpa using g_nonneg ⟨x, N x⟩ (le_refl (N 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
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
⊢ ∃ g,
    (∀ (x : α), ↑(f x) < g x) ∧
      LowerSemicontinuous g ∧
        Integrable (fun x ↦ (g x).toReal) μ ∧
          (∀ᵐ (x : α) ∂μ, g x < ⊤) ∧ ∫ (x : α), (g x).toReal ∂μ < ∫ (x : α), f x ∂μ + ε

Tactic:
rcases exists_lt_lowerSemicontinuous_integral_gt_nnreal fp int_fp δpos with
    ⟨gp, fp_lt_gp, gpcont, gp_lt_top, gp_integrable, gpint⟩

Goal state after:
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
⊢ ∃ g,
    (∀ (x : α), ↑(f x) < g x) ∧
      LowerSemicontinuous g ∧
        Integrable (fun x ↦ (g x).toReal) μ ∧
          (∀ᵐ (x : α) ∂μ, g x < ⊤) ∧ ∫ (x : α), (g x).toReal ∂μ < ∫ (x : α), f x ∂μ + ε
Module: Mathlib.Analysis.LocallyConvex.Separation

Goal state before:
pp: case inr.inr.refine_1.refine_1
E : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s t : Set E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul ℝ E
hs₁ : Convex ℝ s
hs₂ : IsOpen s
ht : Convex ℝ t
disj : Disjoint s t
a₀ : E
ha₀ : a₀ ∈ s
b₀ : E
hb₀ : b₀ ∈ t
x₀ : E := b₀ - a₀
C : Set E := x₀ +ᵥ s - t
this✝¹ : 0 ∈ C
this✝ : Convex ℝ C
this : x₀ ∉ C
f : StrongDual ℝ E
hf₁ : f x₀ = 1
hf₂ : ∀ x ∈ C, f x < 1
forall_le : ∀ a ∈ s, ∀ b ∈ t, f a ≤ f b
a : E
ha : a ∈ s
⊢ a ∈ ⇑f ⁻¹' Iic (sInf (⇑f '' t))

pp: case inr.inr.refine_1.refine_2
E : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s t : Set E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul ℝ E
hs₁ : Convex ℝ s
hs₂ : IsOpen s
ht : Convex ℝ t
disj : Disjoint s t
a₀ : E
ha₀ : a₀ ∈ s
b₀ : E
hb₀ : b₀ ∈ t
x₀ : E := b₀ - a₀
C : Set E := x₀ +ᵥ s - t
this✝¹ : 0 ∈ C
this✝ : Convex ℝ C
this : x₀ ∉ C
f : StrongDual ℝ E
hf₁ : f x₀ = 1
hf₂ : ∀ x ∈ C, f x < 1
forall_le : ∀ a ∈ s, ∀ b ∈ t, f a ≤ f b
⊢ f ≠ 0

Tactic:
· exact le_csInf (Nonempty.image _ ⟨_, hb₀⟩) (forall_mem_image.2 <| forall_le _ ha)

Goal state after:
case inr.inr.refine_1.refine_2
E : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s t : Set E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul ℝ E
hs₁ : Convex ℝ s
hs₂ : IsOpen s
ht : Convex ℝ t
disj : Disjoint s t
a₀ : E
ha₀ : a₀ ∈ s
b₀ : E
hb₀ : b₀ ∈ t
x₀ : E := b₀ - a₀
C : Set E := x₀ +ᵥ s - t
this✝¹ : 0 ∈ C
this✝ : Convex ℝ C
this : x₀ ∉ C
f : StrongDual ℝ E
hf₁ : f x₀ = 1
hf₂ : ∀ x ∈ C, f x < 1
forall_le : ∀ a ∈ s, ∀ b ∈ t, f a ≤ f b
⊢ f ≠ 0
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: case inl
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
hr : 0 ≤ 0
⊢ gauge (closedBall 0 0) x = ‖x‖ / 0

Tactic:
rw [div_zero, closedBall_zero', singleton_zero, gauge_closure_zero]

Goal state after:
case inl
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
hr : 0 ≤ 0
⊢ 0 x = 0
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)).toReal ∂μ - ∫ (x : α), ↑(gm x) ∂μ < ∫ (x : α), ↑(fp x) ∂μ + ↑δ - ∫ (x : α), ↑(gm x) ∂μ

Tactic:
by
          apply sub_lt_sub_right
          convert! gpint
          simp only [EReal.toReal_coe_ennreal]

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

Goal state before:
pp: R : Type u_1
V : Type u_2
inst✝¹⁵ : Field R
inst✝¹⁴ : AddCommGroup V
inst✝¹³ : TopologicalSpace R
inst✝¹² : TopologicalSpace V
inst✝¹¹ : IsTopologicalRing R
inst✝¹⁰ : Module R V
inst✝⁹ : SeparatingDual R V
inst✝⁸ : IsTopologicalAddGroup V
inst✝⁷ : ContinuousSMul R V
S : Type u_3
inst✝⁶ : CommSemiring S
inst✝⁵ : Module S V
inst✝⁴ : SMulCommClass R S V
inst✝³ : Algebra S R
inst✝² : IsScalarTower S R V
inst✝¹ : ContinuousConstSMul S V
inst✝ : Algebra.IsCentral S R
f : V →L[R] V
hf : f ∈ Subalgebra.center S (V →L[R] V)
a✝ : Nontrivial V
x : V
hx : x ≠ 0
⊢ ∃ α ∈ Subalgebra.center S R, f = α • ContinuousLinearMap.id R V

Tactic:
obtain ⟨g, hg⟩ := exists_eq_one (R := R) hx

Goal state after:
R : Type u_1
V : Type u_2
inst✝¹⁵ : Field R
inst✝¹⁴ : AddCommGroup V
inst✝¹³ : TopologicalSpace R
inst✝¹² : TopologicalSpace V
inst✝¹¹ : IsTopologicalRing R
inst✝¹⁰ : Module R V
inst✝⁹ : SeparatingDual R V
inst✝⁸ : IsTopologicalAddGroup V
inst✝⁷ : ContinuousSMul R V
S : Type u_3
inst✝⁶ : CommSemiring S
inst✝⁵ : Module S V
inst✝⁴ : SMulCommClass R S V
inst✝³ : Algebra S R
inst✝² : IsScalarTower S R V
inst✝¹ : ContinuousConstSMul S V
inst✝ : Algebra.IsCentral S R
f : V →L[R] V
hf : f ∈ Subalgebra.center S (V →L[R] V)
a✝ : Nontrivial V
x : V
hx : x ≠ 0
g : StrongDual R V
hg : g x = 1
⊢ ∃ α ∈ Subalgebra.center S R, f = α • ContinuousLinearMap.id R V
Module: Mathlib.Topology.Algebra.IsUniformGroup.Order

Goal state before:
pp: α : Type u_1
ι : Type u_2
β : Type u_3
inst✝⁶ : UniformSpace β
inst✝⁵ : AddGroup β
inst✝⁴ : IsUniformAddGroup β
inst✝³ : PartialOrder β
inst✝² : OrderTopology β
inst✝¹ : AddLeftMono β
inst✝ : AddRightMono β
f : ι → α → β
g : α → β
K : Set α
p : Filter ι
u v : β
huv : u < v
hg : ∀ x ∈ K, g x ≤ u
hf :
  ∀ ⦃p_1 : β × β → Prop⦄,
    (∀ᶠ (b : β) in 𝓝 0, ∀ (a b_1 : β), -a + b_1 = b → p_1 (a, b_1)) → ∀ᶠ (x : ι × α) in p ×ˢ 𝓟 K, p_1 (g x.2, f x.1 x.2)
p✝ : β × β → Prop
| (∀ᶠ (b : β) in 𝓝 0, ∀ (a b_1 : β), -a + b_1 = b → p✝ (a, b_1)) → ∀ᶠ (x : ι × α) in p ×ˢ 𝓟 K, p✝ (g x.2, f x.1 x.2)

Tactic:
rw [eventually_iff_exists_mem]

Goal state after:
α : Type u_1
ι : Type u_2
β : Type u_3
inst✝⁶ : UniformSpace β
inst✝⁵ : AddGroup β
inst✝⁴ : IsUniformAddGroup β
inst✝³ : PartialOrder β
inst✝² : OrderTopology β
inst✝¹ : AddLeftMono β
inst✝ : AddRightMono β
f : ι → α → β
g : α → β
K : Set α
p : Filter ι
u v : β
huv : u < v
hg : ∀ x ∈ K, g x ≤ u
hf :
  ∀ ⦃p_1 : β × β → Prop⦄,
    (∀ᶠ (b : β) in 𝓝 0, ∀ (a b_1 : β), -a + b_1 = b → p_1 (a, b_1)) → ∀ᶠ (x : ι × α) in p ×ˢ 𝓟 K, p_1 (g x.2, f x.1 x.2)
p✝ : β × β → Prop
| (∃ v ∈ 𝓝 0, ∀ y ∈ v, ∀ (a b : β), -a + b = y → p✝ (a, b)) → ∀ᶠ (x : ι × α) in p ×ˢ 𝓟 K, p✝ (g x.2, f x.1 x.2)
Module: Mathlib.Topology.Algebra.IsUniformGroup.Order

Goal state before:
pp: α : Type u_1
ι : Type u_2
β : Type u_3
inst✝⁶ : UniformSpace β
inst✝⁵ : AddGroup β
inst✝⁴ : IsUniformAddGroup β
inst✝³ : PartialOrder β
inst✝² : OrderTopology β
inst✝¹ : AddLeftMono β
inst✝ : AddRightMono β
f : ι → α → β
g : α → β
K : Set α
p : Filter ι
u v : β
huv : u < v
hg : ∀ x ∈ K, g x ≤ u
hf :
  ∀ (a : β × β → Prop),
    (∃ v ∈ 𝓝 0, ∀ y ∈ v, ∀ (a_2 b : β), -a_2 + b = y → a (a_2, b)) → ∀ᶠ (x : ι × α) in p ×ˢ 𝓟 K, a (g x.2, f x.1 x.2)
⊢ 0 ∈ {b | b < -u + v}

Tactic:
by simp [huv]

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)
e p q e' p' q' : ℕ
hp : n e ≤ p
hq : n e ≤ q
hp' : n e' ≤ p'
hq' : n e' ≤ q'
he' : e ≤ e'
r : ℕ := max (n e) (n e')
I : (1 / 2) ^ e' ≤ (1 / 2) ^ e
J1 : ‖L e p q - L e p r‖ ≤ 4 * (1 / 2) ^ e
I1 : x ∈ A f (L e p r) ((1 / 2) ^ r) ((1 / 2) ^ e)
⊢ ‖L e p r - L e' p' r‖ ≤ 4 * (1 / 2) ^ e

Tactic:
have I2 : x ∈ A f (L e' p' r) ((1 / 2) ^ r) ((1 / 2) ^ e') :=
        (hn e' p' r hp' (le_max_right _ _)).2.2

Goal state after:
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)
e p q e' p' q' : ℕ
hp : n e ≤ p
hq : n e ≤ q
hp' : n e' ≤ p'
hq' : n e' ≤ q'
he' : e ≤ e'
r : ℕ := max (n e) (n e')
I : (1 / 2) ^ e' ≤ (1 / 2) ^ e
J1 : ‖L e p q - L e p r‖ ≤ 4 * (1 / 2) ^ e
I1 : x ∈ A f (L e p r) ((1 / 2) ^ r) ((1 / 2) ^ e)
I2 : x ∈ A f (L e' p' r) ((1 / 2) ^ r) ((1 / 2) ^ e')
⊢ ‖L e p r - L e' p' r‖ ≤ 4 * (1 / 2) ^ e
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
⊢ (∫ (x : X), φ x ∂μ) m = ∫ (x : X), (φ x) m ∂μ

Tactic:
by_cases hE : CompleteSpace E
  · exact ((ContinuousMultilinearMap.apply 𝕜 M E m).integral_comp_comm φ_int).symm
  · by_cases! hm : ∀ i, m i ≠ 0
    · have : ¬ CompleteSpace (ContinuousMultilinearMap 𝕜 M E) := by
        rwa [SeparatingDual.completeSpace_continuousMultilinearMap_iff _ _ hm]
      simp [integral, hE, this]
    · rcases hm with ⟨i, hi⟩
      simp [ContinuousMultilinearMap.map_coord_zero _ i hi]

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

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
ι : Type u_4
inst✝ : Countable ι
F : ι → α → E
hF_int : ∀ (i : ι), Integrable (F i) μ
hF_sum : Summable fun i ↦ ∫ (a : α), ‖F i a‖ ∂μ
hE : CompleteSpace E
i : ι
⊢ ENNReal.ofReal (∫ (a : α), ↑‖F i a‖₊ ∂μ) = ENNReal.ofReal ‖∫ (a : α), ‖F i a‖ ∂μ‖

Tactic:
Real.norm_of_nonneg (integral_nonneg (fun a ↦ norm_nonneg (F i a)))

Goal state after:
α : Type u_1
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
ι : Type u_4
inst✝ : Countable ι
F : ι → α → E
hF_int : ∀ (i : ι), Integrable (F i) μ
hF_sum : Summable fun i ↦ ∫ (a : α), ‖F i a‖ ∂μ
hE : CompleteSpace E
i : ι
⊢ ENNReal.ofReal (∫ (a : α), ↑‖F i a‖₊ ∂μ) = ENNReal.ofReal (∫ (x : α), ‖F i x‖ ∂μ)