mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.GiryMonad

Goal state before:
pp: case refine_2
α : Type u_1
mα : MeasurableSpace α
m : Measure (Measure α)
f : α → ℝ≥0∞
hfm : Measurable f
this :
  ∀ (s : ℕ → Finset ℝ≥0∞) (f : ℕ → ℝ≥0∞ → Measure α → ℝ≥0∞),
    (∀ (n : ℕ) (r : ℝ≥0∞), Measurable (f n r)) →
      (Monotone fun n μ ↦ ∑ r ∈ s n, r * f n r μ) →
        ⨆ n, ∑ r ∈ s n, r * ∫⁻ (μ : Measure α), f n r μ ∂m = ∫⁻ (μ : Measure α), ⨆ n, ∑ r ∈ s n, r * f n r μ ∂m
⊢ Monotone fun n μ ↦ ∑ r ∈ (SimpleFunc.eapprox f n).range, r * μ (⇑(SimpleFunc.eapprox f n) ⁻¹' {r})

Tactic:
exact fun n m h μ => SimpleFunc.lintegral_mono (SimpleFunc.monotone_eapprox _ h) le_rfl

Goal state after:
Module: Mathlib.MeasureTheory.Measure.GiryMonad

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
m : Measure α
⊢ m.bind dirac = m

Tactic:
ext1 s hs
  simp only [bind_apply hs measurable_dirac.aemeasurable, dirac_apply' _ hs, lintegral_indicator hs,
    Pi.one_apply, lintegral_one, restrict_apply, MeasurableSet.univ, univ_inter]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
⊢ Measurable fun x ↦ map (Prod.mk x) ν

Tactic:
apply measurable_of_measurable_coe

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
⊢ ∀ (s : Set (α × β)), MeasurableSet s → Measurable fun b ↦ (map (Prod.mk b) ν) s
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
inst✝ : SFinite μ
⊢ Measurable fun y ↦ map (fun x ↦ (x, y)) μ

Tactic:
apply measurable_of_measurable_coe

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
inst✝ : SFinite μ
⊢ ∀ (s : Set (α × β)), MeasurableSet s → Measurable fun b ↦ (map (fun x ↦ (x, b)) μ) s
Module: Mathlib.MeasureTheory.Group.Convolution

Goal state before:
pp: M : Type u_1
inst✝⁴ : Monoid M
inst✝³ : MeasurableSpace M
inst✝² : MeasurableMul₂ M
μ ν ρ : Measure M
inst✝¹ : ρ.IsMulLeftInvariant
inst✝ : SFinite ν
hν : ν ≪ ρ
s : Set M
hs : MeasurableSet s
h : ρ s = 0
⊢ (μ ∗ₘ ν) s = 0

Tactic:
rw [← lintegral_indicator_one hs, lintegral_mconv (by measurability)]

Goal state after:
M : Type u_1
inst✝⁴ : Monoid M
inst✝³ : MeasurableSpace M
inst✝² : MeasurableMul₂ M
μ ν ρ : Measure M
inst✝¹ : ρ.IsMulLeftInvariant
inst✝ : SFinite ν
hν : ν ≪ ρ
s : Set M
hs : MeasurableSet s
h : ρ s = 0
⊢ ∫⁻ (x : M), ∫⁻ (y : M), s.indicator 1 (x * y) ∂ν ∂μ = 0
Module: Mathlib.MeasureTheory.Group.Convolution

Goal state before:
pp: M : Type u_1
inst✝⁴ : Monoid M
inst✝³ : MeasurableSpace M
inst✝² : MeasurableMul₂ M
μ ν ρ : Measure M
inst✝¹ : ρ.IsMulLeftInvariant
inst✝ : SFinite ν
hν : ν ≪ ρ
s : Set M
hs : MeasurableSet s
h : ρ s = 0
x : M
⊢ (map (HMul.hMul x) ρ) s = 0

Tactic:
IsMulLeftInvariant.map_mul_left_eq_self,

Goal state after:
M : Type u_1
inst✝⁴ : Monoid M
inst✝³ : MeasurableSpace M
inst✝² : MeasurableMul₂ M
μ ν ρ : Measure M
inst✝¹ : ρ.IsMulLeftInvariant
inst✝ : SFinite ν
hν : ν ≪ ρ
s : Set M
hs : MeasurableSet s
h : ρ s = 0
x : M
⊢ ρ s = 0
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: G : Type u_1
inst✝ : Group G
S : Set G
g : G
h : IsComplement S {g}
x : G
x✝ : x ∈ ⊤
y : ↑S × ↑{g}
hy : (fun x ↦ ↑x.1 * ↑x.2) y = x * ↑y.2
⊢ ↑y.1 ∈ S

Tactic:
exact y.1.2

Goal state after:
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H : Subgroup G
S : Set G
⊢ IsComplement S ↑H ↔ ∀ (q : G ⧸ H), ∃! s, ↑↑s = q

Tactic:
simp_rw [isComplement_iff_existsUnique_inv_mul_mem, SetLike.mem_coe, ← QuotientGroup.eq,
    QuotientGroup.forall_mk]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: case refine_1
α : Type u_2
β : Type u_3
inst✝ : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure β
pa qa : Set α → Prop
pb qb : Set β → Prop
H : μ.InnerRegularWRT pb qb
f : α → β
hf : MeasurableEmbedding f
hAB : ∀ (U : Set α), qa U → qb (f '' U)
hAB' : ∀ K ⊆ range f, pb K → pa (f ⁻¹' K)
U : Set α
hU : qa U
r : ℝ≥0∞
hr : r < μ (f '' U)
K : Set β
hKU : K ⊆ f '' U
hK : pb K
hμU : r < μ K
hKrange : K ⊆ range f
⊢ f ⁻¹' K ⊆ U

Tactic:
rw [← hf.injective.preimage_image U]

Goal state after:
case refine_1
α : Type u_2
β : Type u_3
inst✝ : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure β
pa qa : Set α → Prop
pb qb : Set β → Prop
H : μ.InnerRegularWRT pb qb
f : α → β
hf : MeasurableEmbedding f
hAB : ∀ (U : Set α), qa U → qb (f '' U)
hAB' : ∀ K ⊆ range f, pb K → pa (f ⁻¹' K)
U : Set α
hU : qa U
r : ℝ≥0∞
hr : r < μ (f '' U)
K : Set β
hKU : K ⊆ f '' U
hK : pb K
hμU : r < μ K
hKrange : K ⊆ range f
⊢ f ⁻¹' K ⊆ f ⁻¹' f '' U
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OpensMeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : TopologicalSpace β
inst✝¹ : BorelSpace β
f : α ≃ₜ β
μ : Measure α
inst✝ : μ.OuterRegular
A : Set β
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > μ (⇑f.symm '' A)
U : Set α
hAU : U ⊇ ⇑f.symm '' A
hUo : IsOpen[inst✝⁵] U
hU : μ U < r
this : IsOpen[inst✝²] (⇑f.symm ⁻¹' U)
⊢ μ (⇑f ⁻¹' ⇑f.symm ⁻¹' U) < r

Tactic:
f.preimage_symm,

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OpensMeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : TopologicalSpace β
inst✝¹ : BorelSpace β
f : α ≃ₜ β
μ : Measure α
inst✝ : μ.OuterRegular
A : Set β
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > μ (⇑f.symm '' A)
U : Set α
hAU : U ⊇ ⇑f.symm '' A
hUo : IsOpen[inst✝⁵] U
hU : μ U < r
this : IsOpen[inst✝²] (⇑f.symm ⁻¹' U)
⊢ μ (⇑f ⁻¹' ⇑f '' U) < r
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
H : Subgroup G
S : Set G
inst✝ : H.FiniteIndex
h : IsComplement S ↑H
⊢ S.encard = ↑H.index

Tactic:
rw [← h.finite_left.cast_ncard_eq, h.ncard_left]

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

Goal state before:
pp: case inr.inr
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : SFinite ν
inst✝ : SFinite μ
s : Set α
t : Set β
hs : μ s ≠ 0
ht : ν t ≠ 0
⊢ NullMeasurableSet (s ×ˢ t) (μ.prod ν) ↔ NullMeasurableSet s μ ∧ NullMeasurableSet t ν ∨ μ s = 0 ∨ ν t = 0

Tactic:
simp [*, nullMeasurableSet_prod_of_ne_zero]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_1
inst✝² : MeasurableSpace α
μ : Measure α
p : Set α → Prop
inst✝¹ : TopologicalSpace α
inst✝ : μ.OuterRegular
H : μ.InnerRegularWRT p IsOpen[inst✝¹]
hd : ∀ ⦃s U : Set α⦄, p s → IsOpen[inst✝¹] U → p (s \ U)
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
h0 : p ∅
ε : ℝ≥0∞
hε : ε ≠ 0
hεs : ε + ε ≤ μ s
hr : μ s - (ε + ε) < μ s
U : Set α
hsU : U ⊇ s
hUo : IsOpen[inst✝¹] U
hUt : μ U < ∞
hμU : μ (U \ s) < ε
U' : Set α
hU'o : IsOpen[inst✝¹] U'
hμU' : μ U' < ε
hsU' : U \ U' ⊆ s
⊢ ∃ K ⊆ s, p K ∧ μ s - (ε + ε) < μ K

Tactic:
rcases H.exists_subset_lt_add h0 hUo hUt.ne hε with ⟨K, hKU, hKc, hKr⟩

Goal state after:
α : Type u_1
inst✝² : MeasurableSpace α
μ : Measure α
p : Set α → Prop
inst✝¹ : TopologicalSpace α
inst✝ : μ.OuterRegular
H : μ.InnerRegularWRT p IsOpen[inst✝¹]
hd : ∀ ⦃s U : Set α⦄, p s → IsOpen[inst✝¹] U → p (s \ U)
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
h0 : p ∅
ε : ℝ≥0∞
hε : ε ≠ 0
hεs : ε + ε ≤ μ s
hr : μ s - (ε + ε) < μ s
U : Set α
hsU : U ⊇ s
hUo : IsOpen[inst✝¹] U
hUt : μ U < ∞
hμU : μ (U \ s) < ε
U' : Set α
hU'o : IsOpen[inst✝¹] U'
hμU' : μ U' < ε
hsU' : U \ U' ⊆ s
K : Set α
hKU : K ⊆ U
hKc : p K
hKr : μ U < μ K + ε
⊢ ∃ K ⊆ s, p K ∧ μ s - (ε + ε) < μ K
Module: Mathlib.Topology.ContinuousMap.CocompactMap

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f g : CocompactMap α β
h : (fun f ↦ f.toFun) f = (fun f ↦ f.toFun) g
⊢ f = g

Tactic:
by
    obtain ⟨⟨_, _⟩, _⟩ := f
    obtain ⟨⟨_, _⟩, _⟩ := g
    congr

Goal state after:
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: G : Type u_1
inst✝² : MeasurableSpace G
inst✝¹ : Mul G
inst✝ : MeasurableMul G
μ : Measure G
g : G
A : Set G
hA : MeasurableSet A
⊢ μ ((fun h ↦ g * h) ⁻¹' A) = μ A ↔ (Measure.map (fun x ↦ g * x) μ) A = μ A

Tactic:
rw [map_apply (measurable_const_mul g) hA]

Goal state after:
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: G : Type u_1
inst✝³ : MeasurableSpace G
inst✝² : Group G
inst✝¹ : MeasurableMul G
μ : Measure G
inst✝ : μ.IsMulRightInvariant
t : G
p : G → Prop
⊢ (∀ᵐ (x : G) ∂μ, p (x * t)) ↔ ∀ᵐ (x : G) ∂μ, p x

Tactic:
conv_rhs => rw [Filter.Eventually, ← map_mul_right_ae μ t]

Goal state after:
G : Type u_1
inst✝³ : MeasurableSpace G
inst✝² : Group G
inst✝¹ : MeasurableMul G
μ : Measure G
inst✝ : μ.IsMulRightInvariant
t : G
p : G → Prop
⊢ (∀ᵐ (x : G) ∂μ, p (x * t)) ↔ {x | p x} ∈ Filter.map (fun h ↦ h * t) (ae μ)
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: case inr
ι : Type u_2
γ : Type u_3
β : Type u_5
inst✝⁶ : MeasurableSpace β
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : IsCompletelyPseudoMetrizableSpace γ
inst✝³ : SecondCountableTopology γ
inst✝² : MeasurableSpace γ
hγ : OpensMeasurableSpace γ
inst✝¹ : Countable ι
l : Filter ι
inst✝ : l.IsCountablyGenerated
f : ι → β → γ
hf : ∀ (i : ι), Measurable (f i)
hl : l.NeBot
this : UpgradedIsCompletelyPseudoMetrizableSpace γ := upgradeIsCompletelyPseudoMetrizable γ
u : ℕ → Set ι
hu : l.HasAntitoneBasis u
⊢ MeasurableSet {x | ∃ c, Tendsto (fun n ↦ f n x) l (𝓝 c)}

Tactic:
← cauchy_map_iff_exists_tendsto

Goal state after:
case inr
ι : Type u_2
γ : Type u_3
β : Type u_5
inst✝⁶ : MeasurableSpace β
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : IsCompletelyPseudoMetrizableSpace γ
inst✝³ : SecondCountableTopology γ
inst✝² : MeasurableSpace γ
hγ : OpensMeasurableSpace γ
inst✝¹ : Countable ι
l : Filter ι
inst✝ : l.IsCountablyGenerated
f : ι → β → γ
hf : ∀ (i : ι), Measurable (f i)
hl : l.NeBot
this : UpgradedIsCompletelyPseudoMetrizableSpace γ := upgradeIsCompletelyPseudoMetrizable γ
u : ℕ → Set ι
hu : l.HasAntitoneBasis u
⊢ MeasurableSet {x | Cauchy (map (fun n ↦ f n x) l)}
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : SFinite ν
inst✝ : SFinite μ
s : Set α
t : Set β
f : α × β → ℝ≥0∞
hf : AEMeasurable f ((μ.prod ν).restrict (s ×ˢ t))
⊢ map Prod.swap ((ν.restrict t).prod (μ.restrict s)) = (μ.prod ν).restrict (s ×ˢ t)

Tactic:
Measure.prod_swap,

Goal state after:
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : SFinite ν
inst✝ : SFinite μ
s : Set α
t : Set β
f : α × β → ℝ≥0∞
hf : AEMeasurable f ((μ.prod ν).restrict (s ×ˢ t))
⊢ (μ.restrict s).prod (ν.restrict t) = (μ.prod ν).restrict (s ×ˢ t)
Module: Mathlib.MeasureTheory.Group.LIntegral

Goal state before:
pp: G : Type u_1
inst✝⁵ : MeasurableSpace G
μ : Measure G
inst✝⁴ : Group G
inst✝³ : MeasurableMul G
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : MeasurableInv G
inst✝ : μ.IsInvInvariant
f : G → ℝ≥0∞
g : G
⊢ ∫⁻ (x : G), f (g * x) ∂μ = ∫⁻ (x : G), f x ∂μ

Tactic:
lintegral_mul_left_eq_self

Goal state after:
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: G : Type u_1
inst✝⁸ : MeasurableSpace G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : WeaklyLocallyCompactSpace G
inst✝² : NoncompactSpace G
μ : Measure G
inst✝¹ : μ.IsOpenPosMeasure
inst✝ : μ.IsMulLeftInvariant
K : Set G
K1 : K ∈ 𝓝 1
hK : IsCompact K
Kclosed : IsClosed[inst✝⁷] K
K_pos : 0 < μ K
g : Set G → G
hg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)
L : ℕ → Set G := fun n ↦ (fun T ↦ T ∪ g T • K)^[n] K
Lcompact : ∀ (n : ℕ), IsCompact (L n)
n : ℕ
⊢ IsClosed[inst✝⁷] (L n)

Tactic:
by
    induction n with
    | zero => exact Kclosed
    | succ n IH =>
      simp_rw [L, iterate_succ']
      apply IsClosed.union IH (Kclosed.smul (g (L n)))

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

Goal state before:
pp: G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul₂ G
μ : Measure G
inst✝² : SFinite μ
inst✝¹ : MeasurableInv G
inst✝ : μ.IsMulLeftInvariant
g : G
⊢ μ ≪ map (fun x ↦ x * g) μ

Tactic:
refine AbsolutelyContinuous.mk fun s hs => ?_

Goal state after:
G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul₂ G
μ : Measure G
inst✝² : SFinite μ
inst✝¹ : MeasurableInv G
inst✝ : μ.IsMulLeftInvariant
g : G
s : Set G
hs : MeasurableSet s
⊢ (map (fun x ↦ x * g) μ) s = 0 → μ s = 0
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_1
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace α
inst✝¹ : OpensMeasurableSpace α
inst✝ : μ.WeaklyRegular
A : Set α
hA : MeasurableSet A
h'A : μ A ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
⊢ ∃ F ⊆ A, IsClosed[inst✝²] F ∧ μ (A \ F) < ε

Tactic:
rcases hA.exists_isClosed_lt_add h'A hε with ⟨F, hFA, hFc, hF⟩

Goal state after:
α : Type u_1
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace α
inst✝¹ : OpensMeasurableSpace α
inst✝ : μ.WeaklyRegular
A : Set α
hA : MeasurableSet A
h'A : μ A ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
F : Set α
hFA : F ⊆ A
hFc : IsClosed[inst✝²] F
hF : μ A < μ F + ε
⊢ ∃ F ⊆ A, IsClosed[inst✝²] F ∧ μ (A \ F) < ε
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x y : ℝ
hy : 0 ≤ y
⊢ ↑(NNReal.sqrt x.toNNReal) ≤ ↑(NNReal.sqrt y.toNNReal) ↔ x ≤ y

Tactic:
NNReal.coe_le_coe,

Goal state after:
x y : ℝ
hy : 0 ≤ y
⊢ NNReal.sqrt x.toNNReal ≤ NNReal.sqrt y.toNNReal ↔ x ≤ y
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x y : ℝ
h : x ≤ y
⊢ ↑(NNReal.sqrt x.toNNReal) ≤ ↑(NNReal.sqrt y.toNNReal)

Tactic:
NNReal.coe_le_coe,

Goal state after:
x y : ℝ
h : x ≤ y
⊢ NNReal.sqrt x.toNNReal ≤ NNReal.sqrt y.toNNReal
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x : ℝ
⊢ ↑(NNReal.sqrt x⁻¹.toNNReal) = (√x)⁻¹

Tactic:
Real.toNNReal_inv,

Goal state after:
x : ℝ
⊢ ↑(NNReal.sqrt x.toNNReal⁻¹) = (√x)⁻¹
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
hf_ne_zero✝ : ∀ᵐ (x : α) ∂μ, f x ≠ 0
hf_ne_top✝ : ∀ᵐ (x : α) ∂μ, f x ≠ ∞
x : α
hf_ne_zero : f x ≠ 0
hf_ne_top : f x ≠ ∞
⊢ f x * (f x)⁻¹ = 1 x

Tactic:
rw [ENNReal.mul_inv_cancel hf_ne_zero hf_ne_top, Pi.one_apply]

Goal state after:
Module: Mathlib.Data.Complex.Basic

Goal state before:
pp: x y : ℝ
⊢ normSq (↑x + ↑y * I) = x ^ 2 + y ^ 2

Tactic:
← mk_eq_add_mul_I,

Goal state after:
x y : ℝ
⊢ normSq { re := x, im := y } = x ^ 2 + y ^ 2
Module: Mathlib.Data.Complex.Basic

Goal state before:
pp: z w : ℂ
⊢ normSq (z + w) = normSq z + normSq w + 2 * (z * (starRingEnd ℂ) w).re

Tactic:
by
  simp [normSq]; ring

Goal state after:
Module: Mathlib.Algebra.Star.Unitary

Goal state before:
pp: R : Type u_1
inst✝¹ : GroupWithZero R
inst✝ : StarMul R
U : ↥(unitary R)
z : ℤ
⊢ ↑(U ^ z) = ↑U ^ z

Tactic:
cases z
  · simp [SubmonoidClass.coe_pow]
  · simp [coe_inv]

Goal state after:
Module: Mathlib.Algebra.Star.Unitary

Goal state before:
pp: R : Type u_1
inst✝³ : Ring R
inst✝² : StarRing R
inst✝¹ : TrivialStar R
inst✝ : NoZeroDivisors R
a : R
⊢ a ∈ unitary R ↔ a = 1 ∨ a = -1

Tactic:
by
  simp [mem_iff, mul_self_eq_one_iff]

Goal state after:
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
z : K
⊢ (starRingEnd K) ↑(re z) + (starRingEnd K) ↑(im z) * -I = ↑(re z) - ↑(im z) * I

Tactic:
conj_ofReal,

Goal state after:
K : Type u_1
inst✝ : RCLike K
z : K
⊢ ↑(re z) + (starRingEnd K) ↑(im z) * -I = ↑(re z) - ↑(im z) * I
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
z : K
⊢ ↑(re z) + (starRingEnd K) ↑(im z) * -I = ↑(re z) - ↑(im z) * I

Tactic:
conj_ofReal,

Goal state after:
K : Type u_1
inst✝ : RCLike K
z : K
⊢ ↑(re z) + ↑(im z) * -I = ↑(re z) - ↑(im z) * I
Module: Mathlib.Analysis.Complex.Basic

Goal state before:
pp: x : ℂ
y : ℝ
hx : IsSelfAdjoint x
⊢ y = x.re ↔ ↑y = x

Tactic:
by
  rw [← RCLike.re_eq_complex_re]
  exact RCLike.ofReal_eq_re_of_isSelfAdjoint hx

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

Goal state before:
pp: z : ℂ
⊢ z ∈ Metric.ball 1 1 → z ∈ slitPlane

Tactic:
hz

Goal state after:
z : ℂ
hz : z ∈ Metric.ball 1 1
⊢ z ∈ slitPlane
Module: Mathlib.LinearAlgebra.Complex.Module

Goal state before:
pp: r : ℝ
⊢ 2⁻¹ • (↑r + (starRingEnd ℂ) ↑r) = ↑r

Tactic:
conj_ofReal,

Goal state after:
r : ℝ
⊢ 2⁻¹ • (↑r + ↑r) = ↑r
Module: Mathlib.Topology.PartialHomeomorph.Basic

Goal state before:
pp: X : Type u_1
X' : Type u_2
Y : Type u_3
Y' : Type u_4
Z : Type u_5
Z' : Type u_6
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace X'
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Y'
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace Z'
e : PartialHomeomorph X Y
f : PartialEquiv X Y
h : IsHomeomorph ⇑f.toEquiv
⊢ ContinuousOn f.invFun f.target

Tactic:
rw [continuousOn_iff_continuous_domRestrict,
      ← continuous_codRestrict_iff (s := f.source) (by simp)]
    exact ((Equiv.isHomeomorph_iff _).1 h).2

Goal state after:
Module: Mathlib.Analysis.Asymptotics.Lemmas

Goal state before:
pp: α : Type u_1
F' : Type u_7
E'' : Type u_9
inst✝¹ : SeminormedAddCommGroup F'
inst✝ : NormedAddCommGroup E''
g' : α → F'
f'' : α → E''
s : Set α
h : ∀ x ∈ s, f'' x = 0
⊢ (fun _x ↦ 0) =ᶠ[𝓟 s] f''

Tactic:
exact fun x hx ↦ (h x hx).symm

Goal state after:
Module: Mathlib.Analysis.Asymptotics.Lemmas

Goal state before:
pp: α : Type u_1
𝕜 : Type u_15
inst✝ : NormedDivisionRing 𝕜
c : ℝ
l : Filter α
u v : α → 𝕜
hc : 0 ≤ c
⊢ IsBigOWith c l u v ↔ ∃ φ, (∀ᶠ (x : α) in l, ‖φ x‖ ≤ c) ∧ u =ᶠ[l] φ * v

Tactic:
by
  constructor
  · intro h
    use fun x => u x / v x
    refine ⟨Eventually.mono h.bound fun y hy => ?_, h.eventually_mul_div_cancel.symm⟩
    simpa using div_le_of_le_mul₀ (norm_nonneg _) hc hy
  · rintro ⟨φ, hφ, h⟩
    exact isBigOWith_of_eq_mul φ hφ h

Goal state after:
Module: Mathlib.Analysis.Asymptotics.Lemmas

Goal state before:
pp: α : Type u_1
E' : Type u_6
𝕜 : Type u_15
inst✝¹ : SeminormedAddCommGroup E'
inst✝ : NormedDivisionRing 𝕜
u : α → E'
v : α → 𝕜
l : Filter α
y : 𝕜
huv : u =o[l] v
hv : Tendsto v l (𝓝 y)
⊢ Tendsto u l (𝓝 0)

Tactic:
by
  suffices h : u =o[l] fun _x => (1 : 𝕜) by
    rwa [isLittleO_one_iff] at h
  exact huv.trans_isBigO (hv.isBigO_one 𝕜)

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

Goal state before:
pp: x : ℝ
⊢ |sin x| = √(1 - cos x ^ 2)

Tactic:
by
  rw [← sin_sq, sqrt_sq_eq_abs]

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

Goal state before:
pp: x : ℝ
hx : 0 < cos x
⊢ (√(1 + tan x ^ 2))⁻¹ = cos x

Tactic:
rw [← sqrt_sq hx.le, ← sqrt_inv, inv_one_add_tan_sq hx.ne']

Goal state after:
Module: Mathlib.Algebra.Order.CauSeq.BigOperators

Goal state before:
pp: case hbc
α : Type u_1
β : Type u_2
inst✝⁴ : Field α
inst✝³ : LinearOrder α
inst✝² : IsStrictOrderedRing α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : ℕ → β
ha : IsCauSeq abs fun m ↦ ∑ n ∈ range m, abv (f n)
hb : IsCauSeq abv fun m ↦ ∑ n ∈ range m, g n
ε : α
ε0 : 0 < ε
P : α
hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P
Q : α
hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q
hP0 : 0 < P
hPε0 : 0 < ε / (2 * P)
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, abv (∑ n ∈ range j, g n - ∑ n ∈ range k, g n) < ε / (2 * P)
hQε0 : 0 < ε / (4 * Q)
M : ℕ
hM : ∀ j ≥ M, ∀ k ≥ M, |∑ n ∈ range j, abv (f n) - ∑ n ∈ range k, abv (f n)| < ε / (4 * Q)
K : ℕ
hK : K ≥ 2 * (max N M + 1)
h₁ : ∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k) = ∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n
h₂ : (fun i ↦ ∑ k ∈ range (K - i), f i * g k) = fun i ↦ f i * ∑ k ∈ range (K - i), g k
h₃ :
  ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k =
    ∑ i ∈ range K, f i * (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) + ∑ i ∈ range K, f i * ∑ k ∈ range K, g k
two_mul_two : 4 = 2 * 2
hQ0 : Q ≠ 0
h2Q0 : 2 * Q ≠ 0
hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε
hNMK : max N M + 1 < K
hKN : N < K
hsumlesum :
  ∑ i ∈ range (max N M + 1), abv (f i) * abv (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) ≤
    ∑ i ∈ range (max N M + 1), abv (f i) * (ε / (2 * P))
hsumltP : ∑ n ∈ range (max N M + 1), abv (f n) < P
i : ℕ
a✝ : i ∈ {i ∈ range K | max N M + 1 ≤ i}
⊢ abv (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) ≤ 2 * Q

Tactic:
rw [sub_eq_add_neg]

Goal state after:
case hbc
α : Type u_1
β : Type u_2
inst✝⁴ : Field α
inst✝³ : LinearOrder α
inst✝² : IsStrictOrderedRing α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : ℕ → β
ha : IsCauSeq abs fun m ↦ ∑ n ∈ range m, abv (f n)
hb : IsCauSeq abv fun m ↦ ∑ n ∈ range m, g n
ε : α
ε0 : 0 < ε
P : α
hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P
Q : α
hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q
hP0 : 0 < P
hPε0 : 0 < ε / (2 * P)
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, abv (∑ n ∈ range j, g n - ∑ n ∈ range k, g n) < ε / (2 * P)
hQε0 : 0 < ε / (4 * Q)
M : ℕ
hM : ∀ j ≥ M, ∀ k ≥ M, |∑ n ∈ range j, abv (f n) - ∑ n ∈ range k, abv (f n)| < ε / (4 * Q)
K : ℕ
hK : K ≥ 2 * (max N M + 1)
h₁ : ∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k) = ∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n
h₂ : (fun i ↦ ∑ k ∈ range (K - i), f i * g k) = fun i ↦ f i * ∑ k ∈ range (K - i), g k
h₃ :
  ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k =
    ∑ i ∈ range K, f i * (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) + ∑ i ∈ range K, f i * ∑ k ∈ range K, g k
two_mul_two : 4 = 2 * 2
hQ0 : Q ≠ 0
h2Q0 : 2 * Q ≠ 0
hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε
hNMK : max N M + 1 < K
hKN : N < K
hsumlesum :
  ∑ i ∈ range (max N M + 1), abv (f i) * abv (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) ≤
    ∑ i ∈ range (max N M + 1), abv (f i) * (ε / (2 * P))
hsumltP : ∑ n ∈ range (max N M + 1), abv (f n) < P
i : ℕ
a✝ : i ∈ {i ∈ range K | max N M + 1 ≤ i}
⊢ abv (∑ k ∈ range (K - i), g k + -∑ k ∈ range K, g k) ≤ 2 * Q
Module: Mathlib.Analysis.SpecialFunctions.Exp

Goal state before:
pp: r : ℝ
hr_nonneg : 0 ≤ r
hr_le : r ≤ 1
x y : ℂ
hyx : ‖y - x‖ < r
hy_eq : y = x + (y - x)
hyx_sq_le : ‖y - x‖ ^ 2 ≤ r * ‖y - x‖
z : ℂ
⊢ ‖z‖ ≤ 1 → ‖cexp (x + z) - cexp x‖ ≤ ‖z‖ * ‖cexp x‖ + ‖cexp x‖ * ‖z‖ ^ 2

Tactic:
hz

Goal state after:
r : ℝ
hr_nonneg : 0 ≤ r
hr_le : r ≤ 1
x y : ℂ
hyx : ‖y - x‖ < r
hy_eq : y = x + (y - x)
hyx_sq_le : ‖y - x‖ ^ 2 ≤ r * ‖y - x‖
z : ℂ
hz : ‖z‖ ≤ 1
⊢ ‖cexp (x + z) - cexp x‖ ≤ ‖z‖ * ‖cexp x‖ + ‖cexp x‖ * ‖z‖ ^ 2
Module: Mathlib.Algebra.Order.CauSeq.BigOperators

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : Field α
inst✝⁴ : LinearOrder α
inst✝³ : IsStrictOrderedRing α
inst✝² : Ring β
abv : β → α
inst✝¹ : IsAbsoluteValue abv
inst✝ : Archimedean α
f : ℕ → β
n m : ℕ
hmn : n.succ ≤ m
hr0 : 0 ≤ 0
hr1 : 0 < 1
h : ∀ (m : ℕ), n ≤ m → abv (f m.succ) ≤ 0 * abv (f m)
har1 : |0| < 1
m_pos : 0 < m
⊢ n.succ ≤ m.pred.succ

Tactic:
by rwa [Nat.succ_pred_eq_of_pos m_pos]

Goal state after: