mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : T2Space α
inst✝¹ : MeasurableSpace α
inst✝ : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous[_, inst✝³] f
hg : Continuous[_, inst✝³] g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
I :
  ∀ (n : ℕ) (x y : ℕ → ℕ),
    ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →
      ∃ x' y',
        x' ∈ cylinder x n ∧
          y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))
A : Type := { p // ¬MeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }
F : A → A
hFn : ∀ (p : A), (↑(F p)).1 = (↑p).1 + 1
hFx : ∀ (p : A), (↑(F p)).2.1 ∈ cylinder (↑p).2.1 (↑p).1
hFy : ∀ (p : A), (↑(F p)).2.2 ∈ cylinder (↑p).2.2 (↑p).1
p0 : A := ⟨(0, fun x ↦ 0, fun x ↦ 0), ⋯⟩
p : ℕ → A := fun n ↦ F^[n] p0
prec : ∀ (n : ℕ), p (n + 1) = F (p n)
pn_fst : ∀ (n : ℕ), (↑(p n)).1 = n
Ix : ∀ (m n : ℕ), m + 1 ≤ n → (↑(p n)).2.1 m = (↑(p (m + 1))).2.1 m
Iy : ∀ (m n : ℕ), m + 1 ≤ n → (↑(p n)).2.2 m = (↑(p (m + 1))).2.2 m
x : ℕ → ℕ := fun n ↦ (↑(p (n + 1))).2.1 n
hx : x = fun n ↦ (↑(p (n + 1))).2.1 n
y : ℕ → ℕ := fun n ↦ (↑(p (n + 1))).2.2 n
hy : y = fun n ↦ (↑(p (n + 1))).2.2 n
M : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)
u v : Set α
u_open : IsOpen[inst✝³] u
v_open : IsOpen[inst✝³] v
xu : f x ∈ u
yv : g y ∈ v
huv : Disjoint u v
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
⊢ ∃ εx > 0, ball x εx ⊆ f ⁻¹' u

Tactic:
apply Metric.mem_nhds_iff.1
    exact hf.continuousAt.preimage_mem_nhds (u_open.mem_nhds xu)

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

Goal state before:
pp: α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α✝
inst✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
μ✝ μ' : Measure α✝
ν✝ ν' : Measure β✝
τ : Measure γ
inst✝² : SFinite ν✝
α : Type u_4
β : Type u_5
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
⊢ (μ.prod ν) (univ ×ˢ univ) < ∞

Tactic:
prod_prod

Goal state after:
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α✝
inst✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
μ✝ μ' : Measure α✝
ν✝ ν' : Measure β✝
τ : Measure γ
inst✝² : SFinite ν✝
α : Type u_4
β : Type u_5
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
⊢ μ univ * ν univ < ∞
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹⁰ : MeasurableSpace α
inst✝⁹ : MeasurableSpace β
inst✝⁸ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝⁷ : SFinite ν✝
X : Type u_4
Y : Type u_5
inst✝⁶ : PseudoMetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : PseudoMetricSpace Y
inst✝³ : MeasurableSpace Y
μ : Measure X
ν : Measure Y
inst✝² : SFinite ν
inst✝¹ : IsUnifLocDoublingMeasure μ
inst✝ : IsUnifLocDoublingMeasure ν
r : ℝ
hμr : ∀ (x : X), μ (Metric.closedBall x (2 * r)) ≤ ↑(doublingConstant μ) * μ (Metric.closedBall x r)
hνr : ∀ (x : Y), ν (Metric.closedBall x (2 * r)) ≤ ↑(doublingConstant ν) * ν (Metric.closedBall x r)
x : X × Y
⊢ μ (Metric.closedBall x.1 (2 * r)) * ν (Metric.closedBall x.2 (2 * r)) ≤
    ↑(doublingConstant μ * doublingConstant ν) * (μ.prod ν) (Metric.closedBall x.1 r ×ˢ Metric.closedBall x.2 r)

Tactic:
prod_prod

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹⁰ : MeasurableSpace α
inst✝⁹ : MeasurableSpace β
inst✝⁸ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝⁷ : SFinite ν✝
X : Type u_4
Y : Type u_5
inst✝⁶ : PseudoMetricSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : PseudoMetricSpace Y
inst✝³ : MeasurableSpace Y
μ : Measure X
ν : Measure Y
inst✝² : SFinite ν
inst✝¹ : IsUnifLocDoublingMeasure μ
inst✝ : IsUnifLocDoublingMeasure ν
r : ℝ
hμr : ∀ (x : X), μ (Metric.closedBall x (2 * r)) ≤ ↑(doublingConstant μ) * μ (Metric.closedBall x r)
hνr : ∀ (x : Y), ν (Metric.closedBall x (2 * r)) ≤ ↑(doublingConstant ν) * ν (Metric.closedBall x r)
x : X × Y
⊢ μ (Metric.closedBall x.1 (2 * r)) * ν (Metric.closedBall x.2 (2 * r)) ≤
    ↑(doublingConstant μ * doublingConstant ν) * (μ (Metric.closedBall x.1 r) * ν (Metric.closedBall x.2 r))
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
ν : Measure β
inst✝ : SFinite ν
s : Set α
t : Set β
s_ae_univ : μ sᶜ = 0
t_ae_univ : ν tᶜ = 0
⊢ (μ.prod ν) (sᶜ ×ˢ univ ∪ univ ×ˢ tᶜ) = 0

Tactic:
measure_union_null_iff

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝ : SFinite ν
s : Set α
t : Set β
s_ae_univ : μ sᶜ = 0
t_ae_univ : ν tᶜ = 0
⊢ (μ.prod ν) (sᶜ ×ˢ univ) = 0 ∧ (μ.prod ν) (univ ×ˢ tᶜ) = 0
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace γ
μ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝ : SFinite ν✝
ν : Measure β
C : Set (Set α)
D : Set (Set β)
hμ : μ.FiniteSpanningSetsIn C
hν : ν.FiniteSpanningSetsIn D
this : SigmaFinite ν
n : ℕ
⊢ (μ.prod ν) (hμ.set (Nat.unpair n).1 ×ˢ hν.set (Nat.unpair n).2) < ∞

Tactic:
prod_prod

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace γ
μ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝ : SFinite ν✝
ν : Measure β
C : Set (Set α)
D : Set (Set β)
hμ : μ.FiniteSpanningSetsIn C
hν : ν.FiniteSpanningSetsIn D
this : SigmaFinite ν
n : ℕ
⊢ μ (hμ.set (Nat.unpair n).1) * ν (hν.set (Nat.unpair n).2) < ∞
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
ν : Measure β
C : Set (Set α)
D : Set (Set β)
hC : generateFrom C = inst✝¹
hD : generateFrom D = inst✝
h2C : IsPiSystem C
h2D : IsPiSystem D
h3C : μ.FiniteSpanningSetsIn C
h3D : ν.FiniteSpanningSetsIn D
μν : Measure (α × β)
h₁ : ∀ s ∈ C, ∀ t ∈ D, μν (s ×ˢ t) = μ s * ν t
s : Set α
hs : s ∈ C
t : Set β
ht : t ∈ D
this : SigmaFinite ν
⊢ (μ.prod ν) ((fun x1 x2 ↦ x1 ×ˢ x2) s t) = μ s * ν t

Tactic:
prod_prod

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
ν : Measure β
C : Set (Set α)
D : Set (Set β)
hC : generateFrom C = inst✝¹
hD : generateFrom D = inst✝
h2C : IsPiSystem C
h2D : IsPiSystem D
h3C : μ.FiniteSpanningSetsIn C
h3D : ν.FiniteSpanningSetsIn D
μν : Measure (α × β)
h₁ : ∀ s ∈ C, ∀ t ∈ D, μν (s ×ˢ t) = μ s * ν t
s : Set α
hs : s ∈ C
t : Set β
ht : t ∈ D
this : SigmaFinite ν
⊢ μ s * ν t = μ s * ν t
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H : Subgroup G
T : Set G
hHT : IsComplement (↑H) T
g : G
⊢ MulOpposite.op g • ↑H = MulOpposite.op ↑(hHT.equiv g).2 • ↑H

Tactic:
rightCoset_eq_iff,

Goal state after:
G : Type u_1
inst✝ : Group G
H : Subgroup G
T : Set G
hHT : IsComplement (↑H) T
g : G
⊢ ↑(hHT.equiv g).2 * g⁻¹ ∈ H
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: G : Type u_1
inst✝ : Group G
S T : Set G
hST : IsComplement S T
g : G
h1 : 1 ∈ T
hg : g ∈ S
⊢ (Equiv.ofBijective (fun x ↦ ↑x.1 * ↑x.2) hST).symm.symm (⟨g, hg⟩, ⟨1, h1⟩) = g

Tactic:
Equiv.ofBijective

Goal state after:
G : Type u_1
inst✝ : Group G
S T : Set G
hST : IsComplement S T
g : G
h1 : 1 ∈ T
hg : g ∈ S
⊢ { toFun := fun x ↦ ↑x.1 * ↑x.2, invFun := surjInv ⋯, left_inv := ⋯, right_inv := ⋯ }.symm.symm (⟨g, hg⟩, ⟨1, h1⟩) = g
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: G : Type u_1
inst✝ : Group G
S T : Set G
hST : IsComplement S T
g : G
h1 : 1 ∈ S
hg : g ∈ T
⊢ (Equiv.ofBijective (fun x ↦ ↑x.1 * ↑x.2) hST).symm.symm (⟨1, h1⟩, ⟨g, hg⟩) = g

Tactic:
Equiv.ofBijective

Goal state after:
G : Type u_1
inst✝ : Group G
S T : Set G
hST : IsComplement S T
g : G
h1 : 1 ∈ S
hg : g ∈ T
⊢ { toFun := fun x ↦ ↑x.1 * ↑x.2, invFun := surjInv ⋯, left_inv := ⋯, right_inv := ⋯ }.symm.symm (⟨1, h1⟩, ⟨g, hg⟩) = g
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: case refine_3
α : Type u_4
β : Type u_5
γ : Type u_6
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ ν : Measure (α × β × γ)
inst✝ : IsFiniteMeasure μ
h :
  ∀ {s : Set α} {t : Set β} {u : Set γ},
    MeasurableSet s → MeasurableSet t → MeasurableSet u → μ (s ×ˢ t ×ˢ u) = ν (s ×ˢ t ×ˢ u)
s : Set (α × β × γ)
hs : MeasurableSet s
h_univ : μ univ = ν univ
this : IsFiniteMeasure ν
C₂ : Set (Set (β × γ)) := image2 (fun x1 x2 ↦ x1 ×ˢ x2) {t | MeasurableSet t} {u | MeasurableSet u}
C : Set (Set (α × β × γ)) := image2 (fun x1 x2 ↦ x1 ×ˢ x2) {s | MeasurableSet s} C₂
⊢ ∀ t ∈ C, μ t = ν t

Tactic:
rintro - ⟨s, hs, -, ⟨t, ht, u, hu, rfl⟩, rfl⟩
    exact h hs ht hu

Goal state after:
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
i : ℕ × ℕ
s' : Set α
t' : Set β
hs' : MeasurableSet s'
ht' : MeasurableSet t'
⊢ (((sfiniteSeq μ i.1).prod (sfiniteSeq ν i.2)).restrict (s ×ˢ t)) (s' ×ˢ t') =
    ((sfiniteSeq μ i.1).restrict s) s' * ((sfiniteSeq ν i.2).restrict t) t'

Tactic:
restrict_apply (hs'.prod ht'),

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
i : ℕ × ℕ
s' : Set α
t' : Set β
hs' : MeasurableSet s'
ht' : MeasurableSet t'
⊢ ((sfiniteSeq μ i.1).prod (sfiniteSeq ν i.2)) (s' ×ˢ t' ∩ s ×ˢ t) =
    ((sfiniteSeq μ i.1).restrict s) s' * ((sfiniteSeq ν i.2).restrict t) t'
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
δ : Type u_4
inst✝² : MeasurableSpace δ
f : α → β
g : γ → δ
μa : Measure α
μc : Measure γ
inst✝¹ : SFinite μa
inst✝ : SFinite μc
hf : Measurable f
hg : Measurable g
⊢ (sum fun i ↦ map f (sfiniteSeq μa i)).prod (map g (sum (sfiniteSeq μc))) =
    map (Prod.map f g) ((sum (sfiniteSeq μa)).prod (sum (sfiniteSeq μc)))

Tactic:
map_sum hg.aemeasurable,

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
δ : Type u_4
inst✝² : MeasurableSpace δ
f : α → β
g : γ → δ
μa : Measure α
μc : Measure γ
inst✝¹ : SFinite μa
inst✝ : SFinite μc
hf : Measurable f
hg : Measurable g
⊢ (sum fun i ↦ map f (sfiniteSeq μa i)).prod (sum fun i ↦ map g (sfiniteSeq μc i)) =
    map (Prod.map f g) ((sum (sfiniteSeq μa)).prod (sum (sfiniteSeq μc)))
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ι : Type u_2
inst✝ : Countable ι
f : ι → α → ℝ≥0∞
h : ∀ (i : ι), Measurable (f i)
s : Set α
hs : MeasurableSet s
⊢ ∫⁻ (x : α) in s, (∑' (n : ι), f n) x ∂μ = ∑' (i : ι), ∫⁻ (x : α) in s, f i x ∂μ

Tactic:
← lintegral_tsum fun i => (h i).aemeasurable

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ι : Type u_2
inst✝ : Countable ι
f : ι → α → ℝ≥0∞
h : ∀ (i : ι), Measurable (f i)
s : Set α
hs : MeasurableSet s
⊢ ∫⁻ (x : α) in s, (∑' (n : ι), f n) x ∂μ = ∫⁻ (a : α) in s, ∑' (i : ι), f i a ∂μ
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: G : Type u_1
inst✝⁴ : MeasurableSpace G
inst✝³ : TopologicalSpace G
inst✝² : BorelSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
s : Set G
μ : Measure G
| μ s

Tactic:
rw [measure_eq_iInf]

Goal state after:
G : Type u_1
inst✝⁴ : MeasurableSpace G
inst✝³ : TopologicalSpace G
inst✝² : BorelSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
s : Set G
μ : Measure G
| ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), μ t
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: case refine_1
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Measurable f
S : Set α := {x | f x < 0}
hS : MeasurableSet S
⊢ (μ.withDensity fun x ↦ ENNReal.ofReal (f x)) S = 0

Tactic:
rw [withDensity_apply _ hS, lintegral_eq_zero_iff hf.ennreal_ofReal, EventuallyEq]
    exact (ae_restrict_mem hS).mono fun x hx => ENNReal.ofReal_eq_zero.2 (le_of_lt hx)

Goal state after:
Module: Mathlib.MeasureTheory.Function.Egorov

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
inst✝² : PseudoEMetricSpace β
μ : Measure α
s : Set α
f : ι → α → β
g : α → β
inst✝¹ : SemilatticeSup ι
inst✝ : Nonempty ι
n : ℕ
hfg : μ {a | ¬(a ∈ s → ∀ ε > 0, ∃ N, ∀ n ≥ N, edist (f n a) (g a) < ε)} = 0
x : α
hmem : x ∈ s
hx : ∀ (i : ι), ∃ k ≥ i, (↑n)⁻¹ < edist (f k x) (g x)
N : ι
⊢ ∃ n_1 ≥ N, (↑n)⁻¹ ≤ edist (f n_1 x) (g x)

Tactic:
obtain ⟨n, hn₁, hn₂⟩ := hx N

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
inst✝² : PseudoEMetricSpace β
μ : Measure α
s : Set α
f : ι → α → β
g : α → β
inst✝¹ : SemilatticeSup ι
inst✝ : Nonempty ι
n✝ : ℕ
hfg : μ {a | ¬(a ∈ s → ∀ ε > 0, ∃ N, ∀ n ≥ N, edist (f n a) (g a) < ε)} = 0
x : α
hmem : x ∈ s
hx : ∀ (i : ι), ∃ k ≥ i, (↑n✝)⁻¹ < edist (f k x) (g x)
N n : ι
hn₁ : n ≥ N
hn₂ : (↑n✝)⁻¹ < edist (f n x) (g x)
⊢ ∃ n ≥ N, (↑n✝)⁻¹ ≤ edist (f n x) (g x)
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x y : ℝ
h : 0 < y
⊢ √x = y ↔ y * y = x

Tactic:
by
  simp [sqrt_eq_cases, h.ne', h.le]

Goal state after:
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: a : ℕ
⊢ ↑↑a.sqrt ≤ √↑a ∧ √↑a < ↑↑a.sqrt + 1

Tactic:
exact ⟨nat_sqrt_le_real_sqrt, real_sqrt_lt_nat_sqrt_succ⟩

Goal state after:
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: a : ℕ
⊢ ↑a.sqrt ≤ √↑a ∧ √↑a < ↑a.sqrt + 1

Tactic:
exact ⟨nat_sqrt_le_real_sqrt, real_sqrt_lt_nat_sqrt_succ⟩

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

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝⁴ : Fintype ι
inst✝³ : AddCommMonoid α
inst✝² : PartialOrder α
inst✝¹ : IsOrderedAddMonoid α
inst✝ : Module ℚ≥0 α
f : ι → α
hf : f ≤ 0
⊢ 𝔼 i, f i = 0 ↔ f = 0

Tactic:
rw [Finset.expect_eq_zero_iff_of_nonpos (by aesop)]

Goal state after:
ι : Type u_1
α : Type u_2
inst✝⁴ : Fintype ι
inst✝³ : AddCommMonoid α
inst✝² : PartialOrder α
inst✝¹ : IsOrderedAddMonoid α
inst✝ : Module ℚ≥0 α
f : ι → α
hf : f ≤ 0
⊢ (∀ i ∈ univ, f i = 0) ↔ f = 0
Module: Mathlib.Algebra.Star.MonoidHom

Goal state before:
pp: F : Type u_1
A : Type u_2
B : Type u_3
C : Type u_4
D : Type u_5
inst✝⁷ : Monoid A
inst✝⁶ : Star A
inst✝⁵ : Monoid B
inst✝⁴ : Star B
inst✝³ : Monoid C
inst✝² : Star C
inst✝¹ : Monoid D
inst✝ : Star D
f : B →⋆* C
g : A →⋆* B
a : A
⊢ (↑(f.comp g.toMonoidHom)).toFun (star a) = star ((↑(f.comp g.toMonoidHom)).toFun a)

Tactic:
simp [map_star]

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

Goal state before:
pp: E : Type u_2
inst✝² : NonUnitalNormedRing E
inst✝¹ : StarRing E
inst✝ : CStarRing E
x : E
⊢ x * x⋆ ≠ 0 ↔ x ≠ 0

Tactic:
by
  simp only [Ne, mul_star_self_eq_zero_iff]

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.ContinuousLinearMap

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
inst✝⁴ : SeminormedRing 𝕜
inst✝³ : Ring 𝕜₂
inst✝² : SeminormedAddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
f : 𝕜 →ₗ[𝕜] E
x : 𝕜
| ‖f x‖

Tactic:
← mul_one x

Goal state after:
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
inst✝⁴ : SeminormedRing 𝕜
inst✝³ : Ring 𝕜₂
inst✝² : SeminormedAddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
f : 𝕜 →ₗ[𝕜] E
x : 𝕜
| ‖f (x * 1)‖
Module: Mathlib.LinearAlgebra.Complex.Module

Goal state before:
pp: R : Type u_1
S : Type u_2
c : Fin 2 → ℝ
i : Fin (Nat.succ 1)
⊢ (fun z ↦ ![z.re, z.im]) ((fun c ↦ ↑(c 0) + c 1 • I) c) i = c i

Tactic:
fin_cases i

Goal state after:
case «0»
R : Type u_1
S : Type u_2
c : Fin 2 → ℝ
⊢ (fun z ↦ ![z.re, z.im]) ((fun c ↦ ↑(c 0) + c 1 • I) c) ((fun i ↦ i) ⟨0, ⋯⟩) = c ((fun i ↦ i) ⟨0, ⋯⟩)

case «1»
R : Type u_1
S : Type u_2
c : Fin 2 → ℝ
⊢ (fun z ↦ ![z.re, z.im]) ((fun c ↦ ↑(c 0) + c 1 • I) c) ((fun i ↦ i) ⟨1, ⋯⟩) = c ((fun i ↦ i) ⟨1, ⋯⟩)
Module: Mathlib.LinearAlgebra.Complex.Module

Goal state before:
pp: i j : Fin 2
⊢ (basisOneI.repr (![1, I] i)) j = (Finsupp.single i 1) j

Tactic:
fin_cases i

Goal state after:
case «0»
j : Fin 2
⊢ (basisOneI.repr (![1, I] ((fun i ↦ i) ⟨0, ⋯⟩))) j = (Finsupp.single ((fun i ↦ i) ⟨0, ⋯⟩) 1) j

case «1»
j : Fin 2
⊢ (basisOneI.repr (![1, I] ((fun i ↦ i) ⟨1, ⋯⟩))) j = (Finsupp.single ((fun i ↦ i) ⟨1, ⋯⟩) 1) j
Module: Mathlib.LinearAlgebra.Complex.Module

Goal state before:
pp: i j : Fin 2
⊢ (LinearMap.toMatrix basisOneI basisOneI) (↑↑conjAe) i j = !![1, 0; 0, -1] i j

Tactic:
fin_cases i

Goal state after:
case «0»
j : Fin 2
⊢ (LinearMap.toMatrix basisOneI basisOneI) (↑↑conjAe) ((fun i ↦ i) ⟨0, ⋯⟩) j = !![1, 0; 0, -1] ((fun i ↦ i) ⟨0, ⋯⟩) j

case «1»
j : Fin 2
⊢ (LinearMap.toMatrix basisOneI basisOneI) (↑↑conjAe) ((fun i ↦ i) ⟨1, ⋯⟩) j = !![1, 0; 0, -1] ((fun i ↦ i) ⟨1, ⋯⟩) j
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝² : RCLike K
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace K E
n : ℕ
x : E
⊢ ‖n • x‖₊ = n • ‖x‖₊

Tactic:
simpa [Nat.cast_smul_eq_nsmul] using nnnorm_smul (n : K) x

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

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
a : { x // 0 < x }
b c : K
h : (fun x1 x2 ↦ x1 ≤ x2) ((fun x y ↦ ↑x * y) a b) ((fun x y ↦ ↑x * y) a c)
a' : ℝ
ha1 : a' > 0
ha2 : ↑a' = ↑a
⊢ 0 ≤ c - b

Tactic:
#adaptation_note /-- 2025-03-29 need beta reduce for https://github.com/leanprover/lean4/issues/7717 -/

Goal state after:
K : Type u_1
inst✝ : RCLike K
a : { x // 0 < x }
b c : K
h : (fun x1 x2 ↦ x1 ≤ x2) ((fun x y ↦ ↑x * y) a b) ((fun x y ↦ ↑x * y) a c)
a' : ℝ
ha1 : a' > 0
ha2 : ↑a' = ↑a
⊢ 0 ≤ c - b
Module: Mathlib.Analysis.Asymptotics.Defs

Goal state before:
pp: α : Type u_1
E' : Type u_6
inst✝ : SeminormedAddCommGroup E'
f' : α → E'
l : Filter α
h : ∃ᶠ (x : α) in l, ‖f' x‖ ≠ 0
ho : f' =o[l] f'
⊢ False

Tactic:
rcases ((ho.bound one_half_pos).and_frequently h).exists with ⟨x, hle, hne⟩

Goal state after:
α : Type u_1
E' : Type u_6
inst✝ : SeminormedAddCommGroup E'
f' : α → E'
l : Filter α
h : ∃ᶠ (x : α) in l, ‖f' x‖ ≠ 0
ho : f' =o[l] f'
x : α
hle : ‖f' x‖ ≤ 1 / 2 * ‖f' x‖
hne : ‖f' x‖ ≠ 0
⊢ False
Module: Mathlib.Analysis.Asymptotics.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝¹ : Norm E
inst✝ : Norm F
f : α → E
g : α → F
l : Filter α
l' : Filter β
⊢ (∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g x‖) →
    ∀ ⦃c : ℝ⦄,
      0 < c →
        ∃ pa,
          (∀ᶠ (x : α) in l, pa x) ∧
            ∃ pb,
              (∀ᶠ (y : β) in l', pb y) ∧
                ∀ {x : α}, pa x → ∀ {y : β}, pb y → ‖(f ∘ Prod.fst) (x, y)‖ ≤ c * ‖(g ∘ Prod.fst) (x, y)‖

Tactic:
exact fun h _ hc ↦ ⟨_, h hc, fun _ ↦ True, eventually_true l', fun {_} h {_} _ ↦ h⟩

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

Goal state before:
pp: α : Type u_1
R : Type u_13
inst✝³ : SeminormedRing R
S : Type u_17
inst✝² : NormedRing S
inst✝¹ : NormMulClass S
c : ℝ
l : Filter α
inst✝ : NormOneClass S
f : α → R
g : α → S
h : IsBigOWith c l f g
n : ℕ
⊢ IsBigOWith (Nat.casesOn (n + 2) ‖1‖ fun n ↦ c ^ (n + 1)) l (fun x ↦ f x ^ (n + 2)) fun x ↦ g x ^ (n + 2)

Tactic:
by simpa [pow_succ] using (IsBigOWith.pow' h (n + 1)).mul h

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

Goal state before:
pp: α : Type u_1
R : Type u_13
𝕜 : Type u_15
inst✝¹ : SeminormedRing R
inst✝ : NormedDivisionRing 𝕜
l : Filter α
ι : Type u_17
L : List ι
f : ι → α → R
g : ι → α → 𝕜
hf : ∀ i ∈ L, f i =O[l] g i
⊢ (fun x ↦ (List.map (fun x_1 ↦ f x_1 x) L).prod) =O[l] fun x ↦ (List.map (fun x_1 ↦ g x_1 x) L).prod

Tactic:
induction L with

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

Goal state before:
pp: α : Type u_1
R : Type u_13
𝕜 : Type u_15
inst✝¹ : SeminormedRing R
inst✝ : NormedDivisionRing 𝕜
l : Filter α
ι : Type u_17
L : List ι
f : ι → α → R
g : ι → α → 𝕜
h₁ : ∀ i ∈ L, f i =O[l] g i
h₂ : ∃ i ∈ L, f i =o[l] g i
⊢ (fun x ↦ (List.map (fun x_1 ↦ f x_1 x) L).prod) =o[l] fun x ↦ (List.map (fun x_1 ↦ g x_1 x) L).prod

Tactic:
induction L with

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

Goal state before:
pp: x : ℝ
n : ℕ
⊢ ↑(rexp (↑n * x)) = ↑(rexp x ^ n)

Tactic:
by simp [exp_nat_mul]

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

Goal state before:
pp: x : ℂ
⊢ tanh (x * I) = tan x * I

Tactic:
by
  rw [tanh_eq_sinh_div_cosh, cosh_mul_I, sinh_mul_I, mul_div_right_comm, tan]

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

Goal state before:
pp: 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
⊢ abv (-∑ i ∈ range K, f i * (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k)) < ε

Tactic:
abv_neg abv

Goal state after:
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
⊢ abv (∑ i ∈ range K, f i * (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k)) < ε
Module: Mathlib.Analysis.Complex.Trigonometric

Goal state before:
pp: x : ℂ
h : cos x ≠ 0
| sin x ^ 2 + cos x ^ 2

Tactic:
← tan_mul_cos h

Goal state after:
x : ℂ
h : cos x ≠ 0
| (tan x * cos x) ^ 2 + cos x ^ 2