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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ c : OrderedFinpartition n
⊢ Subsingleton (Fin 1)

Tactic:
infer_instance

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : 𝕜 → F
s : Set 𝕜
⊢ iteratedFDerivWithin 𝕜 n f s =
    (⇑(ContinuousMultilinearMap.piFieldEquiv 𝕜 (Fin n) F) ∘ ⇑(ContinuousMultilinearMap.piFieldEquiv 𝕜 (Fin n) F).symm) ∘
      iteratedFDerivWithin 𝕜 n f s

Tactic:
LinearIsometryEquiv.self_comp_symm,

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : 𝕜 → F
s : Set 𝕜
⊢ iteratedFDerivWithin 𝕜 n f s = id ∘ iteratedFDerivWithin 𝕜 n f s
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Defs

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : 𝕜 → F
s : Set 𝕜
x : 𝕜
hxs : AccPt x (𝓟 s)
⊢ ((iteratedFDerivWithin 𝕜 (n + 1) f s x) fun x ↦ 1) = derivWithin (iteratedDerivWithin n f s) s x

Tactic:
iteratedFDerivWithin_succ_apply_left,

Goal state after:
case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : 𝕜 → F
s : Set 𝕜
x : 𝕜
hxs : AccPt x (𝓟 s)
⊢ ((fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s x) 1) (Fin.tail fun x ↦ 1) =
    derivWithin (iteratedDerivWithin n f s) s x
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : 𝕜 → F
⊢ iteratedFDeriv 𝕜 n f =
    (⇑(ContinuousMultilinearMap.piFieldEquiv 𝕜 (Fin n) F) ∘ ⇑(ContinuousMultilinearMap.piFieldEquiv 𝕜 (Fin n) F).symm) ∘
      iteratedFDeriv 𝕜 n f

Tactic:
LinearIsometryEquiv.self_comp_symm,

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : 𝕜 → F
⊢ iteratedFDeriv 𝕜 n f = id ∘ iteratedFDeriv 𝕜 n f
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: n : ℕ
c : OrderedFinpartition n
⊢ Nonempty (Fin 1)

Tactic:
infer_instance

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

Goal state before:
pp: case e'_3.inl
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ : OrderedFinpartition n
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) ≠ {0}
j : Fin c.length
hij : c.index 0 < j
A : 1 < c.partSize (c.index 0)
⊢ 1 ≤ ↑(c.emb (c.index 0) ⟨c.partSize (c.index 0) - 1 - 1 + 1, ⋯⟩)

Tactic:
Order.one_le_iff_pos

Goal state after:
case e'_3.inl
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ : OrderedFinpartition n
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) ≠ {0}
j : Fin c.length
hij : c.index 0 < j
A : 1 < c.partSize (c.index 0)
⊢ 0 < ↑(c.emb (c.index 0) ⟨c.partSize (c.index 0) - 1 - 1 + 1, ⋯⟩)
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: case e'_4.inl
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ : OrderedFinpartition n
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) ≠ {0}
i : Fin c.length
hij : i < c.index 0
A : 1 < c.partSize (c.index 0)
⊢ 1 ≤ ↑(c.emb (c.index 0) ⟨c.partSize (c.index 0) - 1 - 1 + 1, ⋯⟩)

Tactic:
Order.one_le_iff_pos

Goal state after:
case e'_4.inl
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ : OrderedFinpartition n
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) ≠ {0}
i : Fin c.length
hij : i < c.index 0
A : 1 < c.partSize (c.index 0)
⊢ 0 < ↑(c.emb (c.index 0) ⟨c.partSize (c.index 0) - 1 - 1 + 1, ⋯⟩)
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: case inl.refine_1
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ : OrderedFinpartition n
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) ≠ {0}
x : Fin n
j : Fin (c.partSize (c.index 0))
hij : c.emb (c.index 0) j = x.succ
j_ne : j ≠ 0
je_ne' : ↑j ≠ 0
A : c.partSize (c.index 0) - 1 + 1 = c.partSize (c.index 0)
B : update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) (c.index 0) = c.partSize (c.index 0) - 1
⊢ Fin.cast ⋯ j ≠ 0

Tactic:
simpa using j_ne

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

Goal state before:
pp: e : ℂ → ℂ
e' : ℂ
z : ℝ
h : HasStrictDerivAt e e' ↑z
⊢ HasStrictDerivAt (fun x ↦ (e ↑x).re) e'.re z

Tactic:
have A : HasStrictFDerivAt ((↑) : ℝ → ℂ) ofRealCLM z := ofRealCLM.hasStrictFDerivAt
  have B :
    HasStrictFDerivAt e ((ContinuousLinearMap.smulRight 1 e' : ℂ →L[ℂ] ℂ).restrictScalars ℝ)
      (ofRealCLM z) :=
    h.hasStrictFDerivAt.restrictScalars ℝ
  have C : HasStrictFDerivAt re reCLM (e (ofRealCLM z)) := reCLM.hasStrictFDerivAt
  simpa using (C.comp z (B.comp z A)).hasStrictDerivAt

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

Goal state before:
pp: case neg.emb.refine_1
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n✝ : ℕ
c✝ : OrderedFinpartition n✝
n : ℕ
c : OrderedFinpartition (n + 1)
h : ¬range (c.emb 0) = {0}
B : c.partSize (c.index 0) - 1 + 1 = c.partSize (c.index 0)
i : Fin c.length
⊢ c.partSize i =
    update (update c.partSize (c.index 0) (c.partSize (c.index 0) - 1)) (c.index 0)
      (update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) (c.index 0) + 1) i

pp: case neg.emb.refine_2
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n✝ : ℕ
c✝ : OrderedFinpartition n✝
n : ℕ
c : OrderedFinpartition (n + 1)
h : ¬range (c.emb 0) = {0}
B : c.partSize (c.index 0) - 1 + 1 = c.partSize (c.index 0)
i : Fin c.length
⊢ ∀ (i_1 : Fin (c.partSize i)),
    c.emb i i_1 =
      (if h_1 : i = c.index 0 then
          (fun i ↦ cases 0 (succ ∘ fun j ↦ (c.emb (c.index 0) (Fin.cast ⋯ j.succ)).pred ⋯) i) ∘ Fin.cast ⋯
        else
          succ ∘
            (fun j ↦
                if h : i = c.index 0 then (c.emb i (Fin.cast ⋯ j.succ)).pred ⋯ else (c.emb i (Fin.cast ⋯ j)).pred ⋯) ∘
              Fin.cast ⋯)
        ⟨↑i_1, ⋯⟩

Tactic:
· simp only [update_self, B, update_idem, update_eq_self]

Goal state after:
case neg.emb.refine_2
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n✝ : ℕ
c✝ : OrderedFinpartition n✝
n : ℕ
c : OrderedFinpartition (n + 1)
h : ¬range (c.emb 0) = {0}
B : c.partSize (c.index 0) - 1 + 1 = c.partSize (c.index 0)
i : Fin c.length
⊢ ∀ (i_1 : Fin (c.partSize i)),
    c.emb i i_1 =
      (if h_1 : i = c.index 0 then
          (fun i ↦ cases 0 (succ ∘ fun j ↦ (c.emb (c.index 0) (Fin.cast ⋯ j.succ)).pred ⋯) i) ∘ Fin.cast ⋯
        else
          succ ∘
            (fun j ↦
                if h : i = c.index 0 then (c.emb i (Fin.cast ⋯ j.succ)).pred ⋯ else (c.emb i (Fin.cast ⋯ j)).pred ⋯) ∘
              Fin.cast ⋯)
        ⟨↑i_1, ⋯⟩
Module: Mathlib.Analysis.Normed.Algebra.Exponential

Goal state before:
pp: 𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕂
inst✝³ : CharZero 𝕂
inst✝² : ContinuousSMul ℚ 𝕂
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕂 𝔸
this : ∀ {n : ℕ}, ↑n ! ≠ 0
⊢ Tendsto (fun n ↦ (↑n)⁻¹) atTop (𝓝 0)

Tactic:
exact tendsto_inv_atTop_nhds_zero_nat

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

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
n : ℕ
c : OrderedFinpartition n
p : (i : Fin c.length) → E [×c.partSize i]→L[𝕜] F
j : Fin n
v : Fin n → E
z : E
⊢ c.applyOrderedFinpartition p (update v j z) =
    update (c.applyOrderedFinpartition p v) (c.index j)
      ((p (c.index j)) (update (v ∘ c.emb (c.index j)) (c.invEmbedding j) z))

Tactic:
ext m
  by_cases h : m = c.index j
  · rw [h]
    simp only [applyOrderedFinpartition, update_self]
    congr
    rw [← Function.update_comp_eq_of_injective]
    · simp
    · exact (c.emb_strictMono (c.index j)).injective
  · simp only [applyOrderedFinpartition, ne_eq, h, not_false_eq_true,
      update_of_ne]
    congr 1
    apply Function.update_comp_eq_of_notMem_range
    have A : Disjoint (range (c.emb m)) (range (c.emb (c.index j))) :=
      c.disjoint (mem_univ m) (mem_univ (c.index j)) h
    have : j ∈ range (c.emb (c.index j)) := mem_range.2 ⟨c.invEmbedding j, by simp⟩
    exact Set.disjoint_right.1 A this

Goal state after:
Module: Mathlib.Analysis.Normed.Algebra.Exponential

Goal state before:
pp: 𝔸 : Type u_1
inst✝² : NormedDivisionRing 𝔸
inst✝¹ : NormedAlgebra ℚ 𝔸
inst✝ : CompleteSpace 𝔸
z : ℤ
x : 𝔸
⊢ exp (z • x) = exp x ^ z

Tactic:
obtain ⟨n, rfl | rfl⟩ := z.eq_nat_or_neg
  · rw [zpow_natCast, natCast_zsmul, exp_nsmul]
  · rw [zpow_neg, zpow_natCast, neg_smul, exp_neg, natCast_zsmul, exp_nsmul]

Goal state after:
Module: Mathlib.Topology.MetricSpace.CauSeqFilter

Goal state before:
pp: β : Type v
inst✝ : NormedField β
f : CauSeq β norm
⊢ (map (↑f) atTop).NeBot

Tactic:
infer_instance

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

Goal state before:
pp: case h.refine_1
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'
h_nhds : {y | ∃ e, ↑e = f' y} ∈ 𝓝 (↑f.symm a)
t : Set E
htu : t ⊆ u ∩ {y | ∃ e, ↑e = f' y}
ht : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
htf : ↑f.symm a ∈ t
⊢ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f.target ∩ ↑f.symm ⁻¹' t)

Tactic:
exact f.isOpen_inter_preimage_symm ht

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)
hf : ContDiffAt 𝕜 n (↑f) (↑f.symm a)
⊢ ContDiffAt 𝕜 n (↑f.symm) a

Tactic:
by
  match n with
  | ω =>
    apply AnalyticAt.contDiffAt
    exact f.analyticAt_symm ha hf.analyticAt hf₀'.fderiv
  | (n : ℕ∞) =>
    -- We prove this by induction on `n`
    induction n using ENat.nat_induction with
    | zero =>
      apply contDiffAt_zero.2
      exact ⟨f.target, IsOpen.mem_nhds f.open_target ha, f.continuousOn_invFun⟩
    | succ n IH =>
      obtain ⟨f', ⟨u, hu, hff'⟩, hf'⟩ := contDiffAt_succ_iff_hasFDerivAt.mp hf
      apply contDiffAt_succ_iff_hasFDerivAt.2
      -- For showing `n.succ` times continuous differentiability (the main inductive step), it
      -- suffices to produce the derivative and show that it is `n` times continuously
      -- differentiable
      have eq_f₀' : f' (f.symm a) = f₀' := (hff' (f.symm a) (mem_of_mem_nhds hu)).unique hf₀'
      -- This follows by a bootstrapping formula expressing the derivative as a
      -- function of `f` itself
      refine ⟨inverse ∘ f' ∘ f.symm, ?_, ?_⟩
      · -- We first check that the derivative of `f` is that formula
        have h_nhds : { y : E | ∃ e : E ≃L[𝕜] F, ↑e = f' y } ∈ 𝓝 (f.symm a) := by
          have hf₀' := f₀'.nhds
          rw [← eq_f₀'] at hf₀'
          exact hf'.continuousAt.preimage_mem_nhds hf₀'
        obtain ⟨t, htu, ht, htf⟩ := mem_nhds_iff.mp (Filter.inter_mem hu h_nhds)
        use f.target ∩ f.symm ⁻¹' t
        refine ⟨IsOpen.mem_nhds ?_ ?_, ?_⟩
        · exact f.isOpen_inter_preimage_symm ht
        · exact mem_inter ha (mem_preimage.mpr htf)
        intro x hx
        obtain ⟨hxu, e, he⟩ := htu hx.2
        have h_deriv : HasFDerivAt f (e : E →L[𝕜] F) (f.symm x) := by
          rw [he]
          exact hff' (f.symm x) hxu
        convert! f.hasFDerivAt_symm hx.1 h_deriv
        simp [← he]
      · -- Then we check that the formula, being a composition of `ContDiff` pieces, is
        -- itself `ContDiff`
        have h_deriv₁ : ContDiffAt 𝕜 n inverse (f' (f.symm a)) := by
          rw [eq_f₀']
          exact contDiffAt_map_inverse _
        have h_deriv₂ : ContDiffAt 𝕜 n f.symm a := by
          refine IH (hf.of_le ?_)
          norm_cast
          exact Nat.le_succ n
        exact (h_deriv₁.comp _ hf').comp _ h_deriv₂
    | top Itop => exact contDiffAt_infty.mpr fun n ↦ Itop n (contDiffAt_infty.mp hf n)

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
fmeas : Measurable f
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
w : α → ℝ≥0
wpos : ∀ (x : α), 0 < w x
wmeas : Measurable w
wint : ∫⁻ (x : α), ↑(w x) ∂μ < ε / 2
f' : α → ℝ≥0∞ := fun x ↦ ↑(f x + w x)
g : α → ℝ≥0∞
le_g : ∀ (x : α), f' x ≤ g x
gcont : LowerSemicontinuous g
gint : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), f' x ∂μ + ε / 2
⊢ ∫⁻ (x : α), ↑(f x) + ↑(w x) ∂μ + ε / 2 = ∫⁻ (x : α), ↑(f x) ∂μ + ∫⁻ (x : α), ↑(w x) ∂μ + ε / 2

Tactic:
lintegral_add_right _ wmeas.coe_nnreal_ennreal

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≥0
fmeas : Measurable f
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
w : α → ℝ≥0
wpos : ∀ (x : α), 0 < w x
wmeas : Measurable w
wint : ∫⁻ (x : α), ↑(w x) ∂μ < ε / 2
f' : α → ℝ≥0∞ := fun x ↦ ↑(f x + w x)
g : α → ℝ≥0∞
le_g : ∀ (x : α), f' x ≤ g x
gcont : LowerSemicontinuous g
gint : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), f' x ∂μ + ε / 2
⊢ ∫⁻ (a : α), ↑(f a) ∂μ + ∫⁻ (a : α), ↑(w a) ∂μ + ε / 2 = ∫⁻ (x : α), ↑(f x) ∂μ + ∫⁻ (x : α), ↑(w x) ∂μ + ε / 2
Module: Mathlib.Geometry.Convex.Cone.Pointed

Goal state before:
pp: R : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁷ : Semiring R
inst✝⁶ : PartialOrder R
inst✝⁵ : IsOrderedRing R
inst✝⁴ : AddCommMonoid E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : Module R E
inst✝ : PosSMulMono R E
c : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
x✝ : E
hx : x✝ ∈ (AddSubmonoid.nonneg E).carrier
⊢ c • x✝ ∈ (AddSubmonoid.nonneg E).carrier

Tactic:
by simpa using smul_nonneg c.property hx

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
⊢ ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e

Tactic:
have J1 : ‖L e p q - L e p r‖ ≤ 4 * ‖c‖ * (1 / 2) ^ e := by
      have I1 : x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) := (hn e p q hp hq).2.1
      have I2 : x ∈ A f (L e p r) ((1 / 2) ^ p) ((1 / 2) ^ e) := (hn e p r hp (le_max_left _ _)).2.1
      exact norm_sub_le_of_mem_A hc P P I1 I2

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
⊢ ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
a : ℝ
hs₁ : Convex ℝ s
hs₀ : 0 ∈ s
hs₂ : Absorbent ℝ s
ha : 0 ≤ a
x : E
h : gauge s x ≤ a
r : ℝ
hr : a < r
hr' : 0 < r
δ : ℝ
δ_pos : 0 < δ
hδr : δ < r
hδ : x ∈ δ • s
this : (r⁻¹ * δ) • δ⁻¹ • x ∈ s
⊢ r⁻¹ • x ∈ s

Tactic:
by rwa [smul_smul, mul_inv_cancel_right₀ δ_pos.ne'] at this

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)
⊢ ∃ 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:
have ae_g : ∀ᵐ x ∂μ, (g x).toReal = (gp x : EReal).toReal - (gm x : EReal).toReal := by
    filter_upwards [gp_lt_top] with _ hx
    rw [EReal.toReal_sub] <;> simp [hx.ne]

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
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
⊢ ∃ 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.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:
div_zero,

Goal state after:
case inl
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
hr : 0 ≤ 0
⊢ gauge (closedBall 0 0) x = 0
Module: Mathlib.Analysis.LocallyConvex.SeparatingDual

Goal state before:
pp: case inl
R : Type u_1
V : Type u_2
inst✝⁶ : Field R
inst✝⁵ : AddCommGroup V
inst✝⁴ : TopologicalSpace R
inst✝³ : TopologicalSpace V
inst✝² : IsTopologicalRing R
inst✝¹ : Module R V
inst✝ : SeparatingDual R V
x y : V
hx : x ≠ 0
hy : y ≠ 0
u : StrongDual R V
ux : u x = 1
uy : u y ≠ 0
⊢ ∃ f, f x = 1 ∧ f y ≠ 0

Tactic:
exact ⟨u, ux, uy⟩

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

Goal state before:
pp: E : Type u_2
inst✝⁶ : TopologicalSpace E
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℝ E
x y : E
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul ℝ E
inst✝¹ : LocallyConvexSpace ℝ E
inst✝ : T1Space E
hxy : x ≠ y
f : StrongDual ℝ E
s t : ℝ
hs : ∀ a ∈ {x}, f a < s
st : s < t
ht : ∀ b ∈ {y}, t < f b
⊢ ∃ f, f x < f y

Tactic:
exact ⟨f, by linarith [hs x rfl, ht y rfl]⟩

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
x y : 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✝ : T1Space E
hxy : x ≠ y
f : StrongDual 𝕜 E
s t : ℝ
hs : ∀ a ∈ {x}, re (f a) < s
st : s < t
ht : ∀ b ∈ {y}, t < re (f b)
⊢ ∃ f, re (f x) < re (f y)

Tactic:
exact ⟨f, by linarith [hs x rfl, ht y rfl]⟩

Goal state after: