mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.MetricSpace.Algebra

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : PseudoMetricSpace β
inst✝³ : Zero α
inst✝² : Zero β
inst✝¹ : SMul α β
inst✝ : IsBoundedSMul α β
s : Set (α × β)
hs : IsBounded s
C : ℝ
hC₀ : 0 < C
hC : s ⊆ Metric.ball 0 C
ε : ℝ
hε : ε > 0
x✝¹ x✝ : α × β
a : α
b : β
hab : max (dist (a, b).1 0.1) (dist (a, b).2 0.2) < C
x : α
y : β
hxy : max (dist (x, y).1 0.1) (dist (x, y).2 0.2) < C
h : dist (a, b) (x, y) ≤ ε / (2 * C)
⊢ dist (Function.uncurry (fun x1 x2 ↦ x1 • x2) (a, b)) (Function.uncurry (fun x1 x2 ↦ x1 • x2) (x, y)) ≤ ε

Tactic:
max_lt_iff

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : PseudoMetricSpace β
inst✝³ : Zero α
inst✝² : Zero β
inst✝¹ : SMul α β
inst✝ : IsBoundedSMul α β
s : Set (α × β)
hs : IsBounded s
C : ℝ
hC₀ : 0 < C
hC : s ⊆ Metric.ball 0 C
ε : ℝ
hε : ε > 0
x✝¹ x✝ : α × β
a : α
b : β
hab : dist (a, b).1 0.1 < C ∧ dist (a, b).2 0.2 < C
x : α
y : β
hxy : dist (x, y).1 0.1 < C ∧ dist (x, y).2 0.2 < C
h : dist (a, b) (x, y) ≤ ε / (2 * C)
⊢ dist (Function.uncurry (fun x1 x2 ↦ x1 • x2) (a, b)) (Function.uncurry (fun x1 x2 ↦ x1 • x2) (x, y)) ≤ ε
Module: Mathlib.Analysis.Normed.Group.Uniform

Goal state before:
pp: α : Type u_4
E : Type u_5
inst✝¹ : SeminormedCommGroup E
inst✝ : PseudoEMetricSpace α
Kf Kg : ℝ≥0
f g : α → E
hf : AntilipschitzWith Kf f
hg : LipschitzWith Kg g
hK : Kg < Kf⁻¹
⊢ AntilipschitzWith (Kf⁻¹ - Kg)⁻¹ fun x ↦ f x * g x

Tactic:
let : PseudoMetricSpace α := PseudoEMetricSpace.toPseudoMetricSpace hf.edist_ne_top

Goal state after:
α : Type u_4
E : Type u_5
inst✝¹ : SeminormedCommGroup E
inst✝ : PseudoEMetricSpace α
Kf Kg : ℝ≥0
f g : α → E
hf : AntilipschitzWith Kf f
hg : LipschitzWith Kg g
hK : Kg < Kf⁻¹
this : PseudoMetricSpace α := PseudoEMetricSpace.toPseudoMetricSpace ⋯
⊢ AntilipschitzWith (Kf⁻¹ - Kg)⁻¹ fun x ↦ f x * g x
Module: Mathlib.Topology.MetricSpace.Dilation

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
F : Type u_4
inst✝³ : PseudoEMetricSpace α
inst✝² : PseudoEMetricSpace β
inst✝¹ : FunLike F α β
inst✝ : DilationClass F α β
f : F
x y : α
key : ∀ (x y : α), edist x y = 0 ∨ edist x y = ∞
r : ℝ≥0
hne : r ≠ 0
hr : edist (f x) (f y) = ↑r * edist x y
⊢ edist (f x) (f y) = ↑1 * edist x y

Tactic:
rcases key x y with h | h

Goal state after:
case pos.inl
α : Type u_1
β : Type u_2
F : Type u_4
inst✝³ : PseudoEMetricSpace α
inst✝² : PseudoEMetricSpace β
inst✝¹ : FunLike F α β
inst✝ : DilationClass F α β
f : F
x y : α
key : ∀ (x y : α), edist x y = 0 ∨ edist x y = ∞
r : ℝ≥0
hne : r ≠ 0
hr : edist (f x) (f y) = ↑r * edist x y
h : edist x y = 0
⊢ edist (f x) (f y) = ↑1 * edist x y

case pos.inr
α : Type u_1
β : Type u_2
F : Type u_4
inst✝³ : PseudoEMetricSpace α
inst✝² : PseudoEMetricSpace β
inst✝¹ : FunLike F α β
inst✝ : DilationClass F α β
f : F
x y : α
key : ∀ (x y : α), edist x y = 0 ∨ edist x y = ∞
r : ℝ≥0
hne : r ≠ 0
hr : edist (f x) (f y) = ↑r * edist x y
h : edist x y = ∞
⊢ edist (f x) (f y) = ↑1 * edist x y
Module: Mathlib.Analysis.Normed.Group.Bounded

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝¹ : NormedAddGroup α
f : α → E
inst✝ : One E
hf : HasCompactMulSupport f
K : Set α
hK1 : IsCompact K
hK2 : ∀ x ∉ K, f x = 1
S : ℝ
hS : S > 0
hS' : ∀ x ∈ K, ‖x‖ ≤ S
⊢ ∃ R, 0 < R ∧ ∀ (x : α), R ≤ ‖x‖ → f x = 1

Tactic:
refine ⟨S + 1, by positivity, fun x hx => hK2 x ((mt <| hS' x) ?_)⟩

Goal state after:
α : Type u_1
E : Type u_2
inst✝¹ : NormedAddGroup α
f : α → E
inst✝ : One E
hf : HasCompactMulSupport f
K : Set α
hK1 : IsCompact K
hK2 : ∀ x ∉ K, f x = 1
S : ℝ
hS : S > 0
hS' : ∀ x ∈ K, ‖x‖ ≤ S
x : α
hx : S + 1 ≤ ‖x‖
⊢ ¬‖x‖ ≤ S
Module: Mathlib.Analysis.Normed.MulAction

Goal state before:
pp: case mpr
α : Type u_1
β : Type u_2
inst✝³ : NormedDivisionRing α
inst✝² : SeminormedAddCommGroup β
inst✝¹ : Module α β
inst✝ : NormSMulClass α β
s : α
hs : s ≠ 0
x : β
ε : ℝ
p : β
h : dist p (s • x) ≤ ‖s‖ * ε
⊢ dist (s⁻¹ • p) x ≤ ε

Tactic:
refine le_of_mul_le_mul_of_pos_left ?_ (norm_pos_iff.mpr hs)

Goal state after:
case mpr
α : Type u_1
β : Type u_2
inst✝³ : NormedDivisionRing α
inst✝² : SeminormedAddCommGroup β
inst✝¹ : Module α β
inst✝ : NormSMulClass α β
s : α
hs : s ≠ 0
x : β
ε : ℝ
p : β
h : dist p (s • x) ≤ ‖s‖ * ε
⊢ ‖s‖ * dist (s⁻¹ • p) x ≤ ‖s‖ * ε
Module: Mathlib.Data.Finset.Update

Goal state before:
pp: ι : Type u_1
π : ι → Sort u_2
x : (i : ι) → π i
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
y : (i : ↥univ) → π ↑i
⊢ updateFinset x univ y = fun i ↦ y ⟨i, ⋯⟩

Tactic:
by
  simp [updateFinset_def]

Goal state after:
Module: Mathlib.Data.Finset.Update

Goal state before:
pp: ι : Type u_1
π : ι → Sort u_2
x : (i : ι) → π i
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
y : (i : ↥univ) → π ↑i
i : ι
⊢ updateFinset x univ y i = y ⟨i, ⋯⟩

Tactic:
by
  simp [updateFinset_def]

Goal state after:
Module: Mathlib.MeasureTheory.OuterMeasure.Operations

Goal state before:
pp: α : Type u_1
a : ℝ≥0∞
b : α
s : Set α
⊢ (a • dirac b) s = s.indicator (fun x ↦ a) b

Tactic:
by
  simp only [smul_apply, smul_eq_mul, dirac_apply, ← indicator_mul_right _ fun _ => a, mul_one]

Goal state after:
Module: Mathlib.MeasureTheory.MeasurableSpace.Constructions

Goal state before:
pp: δ : Type u_4
X : δ → Type u_6
inst✝¹ : (a : δ) → MeasurableSpace (X a)
inst✝ : DecidableEq δ
s : Finset δ
⊢ Measurable fun p ↦ updateFinset p.1 s p.2

Tactic:
by
  simp only [updateFinset, measurable_pi_iff]
  intro i
  by_cases h : i ∈ s <;> simp [h, Measurable.eval, measurable_fst, measurable_snd]

Goal state after:
Module: Mathlib.MeasureTheory.MeasurableSpace.Constructions

Goal state before:
pp: α : Type u_1
β : Type u_2
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSingletonClass β
f : α → β
hf : Measurable f
a : β
| Measurable fun x ↦ a = f x

Tactic:
conv =>

Goal state after:
α : Type u_1
β : Type u_2
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSingletonClass β
f : α → β
hf : Measurable f
a : β
x : α
| f x = a
Module: Mathlib.Data.Sign.Defs

Goal state before:
pp: case zero
b : SignType
⊢ Decidable (zero ≤ b)

Tactic:
cases b

Goal state after:
case zero.zero
⊢ Decidable (zero ≤ zero)

case zero.neg
⊢ Decidable (zero ≤ neg)

case zero.pos
⊢ Decidable (zero ≤ pos)
Module: Mathlib.Data.Sign.Defs

Goal state before:
pp: case neg
b : SignType
⊢ Decidable (neg ≤ b)

Tactic:
cases b

Goal state after:
case neg.zero
⊢ Decidable (neg ≤ zero)

case neg.neg
⊢ Decidable (neg ≤ neg)

case neg.pos
⊢ Decidable (neg ≤ pos)
Module: Mathlib.Data.Sign.Defs

Goal state before:
pp: case pos
b : SignType
⊢ Decidable (pos ≤ b)

Tactic:
cases b

Goal state after:
case pos.zero
⊢ Decidable (pos ≤ zero)

case pos.neg
⊢ Decidable (pos ≤ neg)

case pos.pos
⊢ Decidable (pos ≤ pos)
Module: Mathlib.Data.Sign.Defs

Goal state before:
pp: case zero
b : SignType
⊢ (match zero * b with
    | zero => ⟨0, fin3Equiv._proof_7⟩
    | pos => ⟨1, fin3Equiv._proof_8⟩
    | neg => ⟨2, fin3Equiv._proof_9⟩) =
    (match zero with
      | zero => ⟨0, fin3Equiv._proof_7⟩
      | pos => ⟨1, fin3Equiv._proof_8⟩
      | neg => ⟨2, fin3Equiv._proof_9⟩) *
      match b with
      | zero => ⟨0, fin3Equiv._proof_7⟩
      | pos => ⟨1, fin3Equiv._proof_8⟩
      | neg => ⟨2, fin3Equiv._proof_9⟩

Tactic:
cases b

Goal state after:
case zero.zero
⊢ (match zero * zero with
    | zero => ⟨0, fin3Equiv._proof_7⟩
    | pos => ⟨1, fin3Equiv._proof_8⟩
    | neg => ⟨2, fin3Equiv._proof_9⟩) =
    (match zero with
      | zero => ⟨0, fin3Equiv._proof_7⟩
      | pos => ⟨1, fin3Equiv._proof_8⟩
      | neg => ⟨2, fin3Equiv._proof_9⟩) *
      match zero with
      | zero => ⟨0, fin3Equiv._proof_7⟩
      | pos => ⟨1, fin3Equiv._proof_8⟩
      | neg => ⟨2, fin3Equiv._proof_9⟩

case zero.neg
⊢ (match zero * neg with
    | zero => ⟨0, fin3Equiv._proof_7⟩
    | pos => ⟨1, fin3Equiv._proof_8⟩
    | neg => ⟨2, fin3Equiv._proof_9⟩) =
    (match zero with
      | zero => ⟨0, fin3Equiv._proof_7⟩
      | pos => ⟨1, fin3Equiv._proof_8⟩
      | neg => ⟨2, fin3Equiv._proof_9⟩) *
      match neg with
      | zero => ⟨0, fin3Equiv._proof_7⟩
      | pos => ⟨1, fin3Equiv._proof_8⟩
      | neg => ⟨2, fin3Equiv._proof_9⟩

case zero.pos
⊢ (match zero * pos with
    | zero => ⟨0, fin3Equiv._proof_7⟩
    | pos => ⟨1, fin3Equiv._proof_8⟩
    | neg => ⟨2, fin3Equiv._proof_9⟩) =
    (match zero with
      | zero => ⟨0, fin3Equiv._proof_7⟩
      | pos => ⟨1, fin3Equiv._proof_8⟩
      | neg => ⟨2, fin3Equiv._proof_9⟩) *
      match pos with
      | zero => ⟨0, fin3Equiv._proof_7⟩
      | pos => ⟨1, fin3Equiv._proof_8⟩
      | neg => ⟨2, fin3Equiv._proof_9⟩
Module: Mathlib.Data.Sign.Defs

Goal state before:
pp: case neg
b : SignType
⊢ (match neg * b with
    | zero => ⟨0, fin3Equiv._proof_7⟩
    | pos => ⟨1, fin3Equiv._proof_8⟩
    | neg => ⟨2, fin3Equiv._proof_9⟩) =
    (match neg with
      | zero => ⟨0, fin3Equiv._proof_7⟩
      | pos => ⟨1, fin3Equiv._proof_8⟩
      | neg => ⟨2, fin3Equiv._proof_9⟩) *
      match b with
      | zero => ⟨0, fin3Equiv._proof_7⟩
      | pos => ⟨1, fin3Equiv._proof_8⟩
      | neg => ⟨2, fin3Equiv._proof_9⟩

Tactic:
cases b

Goal state after:
case neg.zero
⊢ (match neg * zero with
    | zero => ⟨0, fin3Equiv._proof_7⟩
    | pos => ⟨1, fin3Equiv._proof_8⟩
    | neg => ⟨2, fin3Equiv._proof_9⟩) =
    (match neg with
      | zero => ⟨0, fin3Equiv._proof_7⟩
      | pos => ⟨1, fin3Equiv._proof_8⟩
      | neg => ⟨2, fin3Equiv._proof_9⟩) *
      match zero with
      | zero => ⟨0, fin3Equiv._proof_7⟩
      | pos => ⟨1, fin3Equiv._proof_8⟩
      | neg => ⟨2, fin3Equiv._proof_9⟩

case neg.neg
⊢ (match neg * neg with
    | zero => ⟨0, fin3Equiv._proof_7⟩
    | pos => ⟨1, fin3Equiv._proof_8⟩
    | neg => ⟨2, fin3Equiv._proof_9⟩) =
    (match neg with
      | zero => ⟨0, fin3Equiv._proof_7⟩
      | pos => ⟨1, fin3Equiv._proof_8⟩
      | neg => ⟨2, fin3Equiv._proof_9⟩) *
      match neg with
      | zero => ⟨0, fin3Equiv._proof_7⟩
      | pos => ⟨1, fin3Equiv._proof_8⟩
      | neg => ⟨2, fin3Equiv._proof_9⟩

case neg.pos
⊢ (match neg * pos with
    | zero => ⟨0, fin3Equiv._proof_7⟩
    | pos => ⟨1, fin3Equiv._proof_8⟩
    | neg => ⟨2, fin3Equiv._proof_9⟩) =
    (match neg with
      | zero => ⟨0, fin3Equiv._proof_7⟩
      | pos => ⟨1, fin3Equiv._proof_8⟩
      | neg => ⟨2, fin3Equiv._proof_9⟩) *
      match pos with
      | zero => ⟨0, fin3Equiv._proof_7⟩
      | pos => ⟨1, fin3Equiv._proof_8⟩
      | neg => ⟨2, fin3Equiv._proof_9⟩
Module: Mathlib.Data.Sign.Defs

Goal state before:
pp: case pos
b : SignType
⊢ (match pos * b with
    | zero => ⟨0, fin3Equiv._proof_7⟩
    | pos => ⟨1, fin3Equiv._proof_8⟩
    | neg => ⟨2, fin3Equiv._proof_9⟩) =
    (match pos with
      | zero => ⟨0, fin3Equiv._proof_7⟩
      | pos => ⟨1, fin3Equiv._proof_8⟩
      | neg => ⟨2, fin3Equiv._proof_9⟩) *
      match b with
      | zero => ⟨0, fin3Equiv._proof_7⟩
      | pos => ⟨1, fin3Equiv._proof_8⟩
      | neg => ⟨2, fin3Equiv._proof_9⟩

Tactic:
cases b

Goal state after:
case pos.zero
⊢ (match pos * zero with
    | zero => ⟨0, fin3Equiv._proof_7⟩
    | pos => ⟨1, fin3Equiv._proof_8⟩
    | neg => ⟨2, fin3Equiv._proof_9⟩) =
    (match pos with
      | zero => ⟨0, fin3Equiv._proof_7⟩
      | pos => ⟨1, fin3Equiv._proof_8⟩
      | neg => ⟨2, fin3Equiv._proof_9⟩) *
      match zero with
      | zero => ⟨0, fin3Equiv._proof_7⟩
      | pos => ⟨1, fin3Equiv._proof_8⟩
      | neg => ⟨2, fin3Equiv._proof_9⟩

case pos.neg
⊢ (match pos * neg with
    | zero => ⟨0, fin3Equiv._proof_7⟩
    | pos => ⟨1, fin3Equiv._proof_8⟩
    | neg => ⟨2, fin3Equiv._proof_9⟩) =
    (match pos with
      | zero => ⟨0, fin3Equiv._proof_7⟩
      | pos => ⟨1, fin3Equiv._proof_8⟩
      | neg => ⟨2, fin3Equiv._proof_9⟩) *
      match neg with
      | zero => ⟨0, fin3Equiv._proof_7⟩
      | pos => ⟨1, fin3Equiv._proof_8⟩
      | neg => ⟨2, fin3Equiv._proof_9⟩

case pos.pos
⊢ (match pos * pos with
    | zero => ⟨0, fin3Equiv._proof_7⟩
    | pos => ⟨1, fin3Equiv._proof_8⟩
    | neg => ⟨2, fin3Equiv._proof_9⟩) =
    (match pos with
      | zero => ⟨0, fin3Equiv._proof_7⟩
      | pos => ⟨1, fin3Equiv._proof_8⟩
      | neg => ⟨2, fin3Equiv._proof_9⟩) *
      match pos with
      | zero => ⟨0, fin3Equiv._proof_7⟩
      | pos => ⟨1, fin3Equiv._proof_8⟩
      | neg => ⟨2, fin3Equiv._proof_9⟩
Module: Mathlib.Data.Sign.Basic

Goal state before:
pp: case inl
α : Type u
inst✝¹ : Nonempty α
inst✝ : DecidableEq α
s : Finset α
f : α → ℤ
n : ℕ
h : ∑ i ∈ s, (f i).natAbs ≤ n
β : Type u
w✝ : Fintype β
sgn : β → SignType
g : β → α
hg : ∀ (b : β), g b ∈ s
hβ : Fintype.card β = ∑ a ∈ s, (f a).natAbs
hf : ∀ a ∈ s, (∑ b, if g b = a then ↑(sgn b) else 0) = f a
b : β
hb : Sum.elim g (Classical.arbitrary (Fin (n - ∑ i ∈ s, (f i).natAbs) → α)) (Sum.inl b) ∉ s
⊢ Sum.elim sgn 0 (Sum.inl b) = 0

Tactic:
cases hb (hg _)

Goal state after:
Module: Mathlib.Data.EReal.Inv

Goal state before:
pp: case mp
x y : EReal
h : ↑(sign x) * ↑x.abs ≤ ↑(sign y) * ↑y.abs
hs : sign x = sign y
⊢ sign x = neg ∧ sign y = neg ∧ y.abs ≤ x.abs ∨
    sign x = zero ∧ sign y = zero ∨ sign x = pos ∧ sign y = pos ∧ x.abs ≤ y.abs

Tactic:
cases hy : sign y

Goal state after:
case mp.zero
x y : EReal
h : ↑(sign x) * ↑x.abs ≤ ↑(sign y) * ↑y.abs
hs : sign x = sign y
hy : sign y = zero
⊢ sign x = neg ∧ zero = neg ∧ y.abs ≤ x.abs ∨ sign x = zero ∧ zero = zero ∨ sign x = pos ∧ zero = pos ∧ x.abs ≤ y.abs

case mp.neg
x y : EReal
h : ↑(sign x) * ↑x.abs ≤ ↑(sign y) * ↑y.abs
hs : sign x = sign y
hy : sign y = neg
⊢ sign x = neg ∧ neg = neg ∧ y.abs ≤ x.abs ∨ sign x = zero ∧ neg = zero ∨ sign x = pos ∧ neg = pos ∧ x.abs ≤ y.abs

case mp.pos
x y : EReal
h : ↑(sign x) * ↑x.abs ≤ ↑(sign y) * ↑y.abs
hs : sign x = sign y
hy : sign y = pos
⊢ sign x = neg ∧ pos = neg ∧ y.abs ≤ x.abs ∨ sign x = zero ∧ pos = zero ∨ sign x = pos ∧ pos = pos ∧ x.abs ≤ y.abs
Module: Mathlib.Topology.Semicontinuity.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : TopologicalSpace α
r : α → β → Prop
s : Set α
⊢ Semicontinuous (s.domRestrict r) ↔ SemicontinuousOn r s

Tactic:
SemicontinuousOn,

Goal state after:
α : Type u_1
β : Type u_2
inst✝ : TopologicalSpace α
r : α → β → Prop
s : Set α
⊢ Semicontinuous (s.domRestrict r) ↔ ∀ x ∈ s, SemicontinuousWithinAt r s x
Module: Mathlib.Topology.Semicontinuity.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : TopologicalSpace α
r : α → β → Prop
⊢ Semicontinuous r ↔ ∀ (b : β), IsOpen[inst✝] {x | r x b}

Tactic:
exact ⟨fun h b ↦ by simpa [isOpen_iff_mem_nhds, Filter.Eventually] using fun x hx ↦ h x b hx,
    fun h x b hbx ↦ (h b).mem_nhds hbx⟩

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.ENNReal

Goal state before:
pp: α : Type u_1
f g : α → ℝ≥0
sf sg : ℝ≥0
i : α
h : ∀ (a : α), f a ≤ g a
hi : f i < g i
hf : HasSum f sf
hg : HasSum g sg
⊢ sf < sg

Tactic:
have A : ∀ a : α, (f a : ℝ) ≤ g a := fun a => NNReal.coe_le_coe.2 (h a)

Goal state after:
α : Type u_1
f g : α → ℝ≥0
sf sg : ℝ≥0
i : α
h : ∀ (a : α), f a ≤ g a
hi : f i < g i
hf : HasSum f sf
hg : HasSum g sg
A : ∀ (a : α), ↑(f a) ≤ ↑(g a)
⊢ sf < sg
Module: Mathlib.Topology.Semicontinuity.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → Set β
⊢ LowerHemicontinuous f ↔ ∀ (x : α) (t : Set β), IsClosed[inst✝] t → (∃ᶠ (x' : α) in 𝓝 x, f x' ⊆ t) → f x ⊆ t

Tactic:
lowerHemicontinuous_iff,

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → Set β
⊢ (∀ (x : α), LowerHemicontinuousAt f x) ↔
    ∀ (x : α) (t : Set β), IsClosed[inst✝] t → (∃ᶠ (x' : α) in 𝓝 x, f x' ⊆ t) → f x ⊆ t
Module: Mathlib.Data.EReal.Inv

Goal state before:
pp: a : EReal
a_0 : a ∈ Ioi 0
b : EReal
b_0 : b ∈ Ioi 0
a_b : a < b
⊢ (fun x ↦ x⁻¹) b < (fun x ↦ x⁻¹) a

Tactic:
push _ ∈ _ at *

Goal state after:
a b : EReal
a_b : a < b
b_0 : 0 < b
a_0 : 0 < a
⊢ b⁻¹ < a⁻¹
Module: Mathlib.Data.EReal.Inv

Goal state before:
pp: b : EReal
h : b ≤ 0
a a' : EReal
h' : a ≤ a'
⊢ a * (-b)⁻¹ ≤ a' * (-b)⁻¹

Tactic:
have : 0 ≤ -b := by apply EReal.le_neg_of_le_neg; simp [h]

Goal state after:
b : EReal
h : b ≤ 0
a a' : EReal
h' : a ≤ a'
this : 0 ≤ -b
⊢ a * (-b)⁻¹ ≤ a' * (-b)⁻¹
Module: Mathlib.Topology.Algebra.InfiniteSum.ENNReal

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
f : ℕ → α
d : ℕ → ℝ≥0
hf : ∀ (n : ℕ), edist (f n) (f n.succ) ≤ ↑(d n)
ε : ℝ≥0
εpos : 0 < ε
hd : CauchySeq fun n ↦ ∑ x ∈ Finset.range n, d x
N n : ℕ
hN :
  max (∑ k ∈ Finset.Ico N n, d k)
      (∑ x ∈ Finset.range N, d x - (∑ k ∈ Finset.range N, d k + ∑ k ∈ Finset.Ico N n, d k)) <
    ε
hn : N ≤ n
⊢ edist (f n) (f N) < ↑ε

Tactic:
max_lt_iff

Goal state after:
α : Type u_1
inst✝ : PseudoEMetricSpace α
f : ℕ → α
d : ℕ → ℝ≥0
hf : ∀ (n : ℕ), edist (f n) (f n.succ) ≤ ↑(d n)
ε : ℝ≥0
εpos : 0 < ε
hd : CauchySeq fun n ↦ ∑ x ∈ Finset.range n, d x
N n : ℕ
hN :
  ∑ k ∈ Finset.Ico N n, d k < ε ∧
    ∑ x ∈ Finset.range N, d x - (∑ k ∈ Finset.range N, d k + ∑ k ∈ Finset.Ico N n, d k) < ε
hn : N ≤ n
⊢ edist (f n) (f N) < ↑ε
Module: Mathlib.Topology.Semicontinuity.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
f : α → β
s : Set α
inst✝ : Preorder β
⊢ LowerSemicontinuousOn f s ↔ ∀ (b : β), ∃ u, IsOpen[inst✝¹] u ∧ s ∩ f ⁻¹' Ioi b = s ∩ u

Tactic:
simp only [← lowerSemicontinuous_restrict_iff, domRestrict_eq,
    lowerSemicontinuous_iff_isOpen_preimage, preimage_comp, isOpen_induced_iff,
    Subtype.preimage_coe_eq_preimage_coe_iff, eq_comm]

Goal state after:
Module: Mathlib.Topology.Semicontinuity.Basic

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
s : Set α
γ : Type u_4
inst✝ : LinearOrder γ
f : α → γ
hfs : LowerSemicontinuousOn f s
ks : IsCompact s
c : γ
⊢ IsCompact (s ∩ f ⁻¹' Iic c)

Tactic:
by
  rw [lowerSemicontinuousOn_iff_preimage_Iic] at hfs
  obtain ⟨v, hv, hv'⟩ := hfs c
  exact hv' ▸ ks.inter_right hv

Goal state after:
Module: Mathlib.Topology.Instances.EReal.Lemmas

Goal state before:
pp: a : ℝ
⊢ ∀ (x : ℝ), ∀ᶠ (a : EReal × EReal) in 𝓝 (⊤, ↑a), ↑x < a.1 + a.2

Tactic:
refine fun r ↦ ((lt_mem_nhds (coe_lt_top (r - (a - 1)))).prod_nhds
    (lt_mem_nhds <| EReal.coe_lt_coe_iff.2 <| sub_one_lt _)).mono fun _ h ↦ ?_

Goal state after:
a r : ℝ
x✝ : EReal × EReal
h : ↑(r - (a - 1)) < x✝.1 ∧ ↑(a - 1) < x✝.2
⊢ ↑r < x✝.1 + x✝.2
Module: Mathlib.Analysis.SpecificLimits.Basic

Goal state before:
pp: u : ℕ → ℝ
c : ℝ
hc : 0 ≤ c
n : ℕ
h : ∀ k < n, u (k + 1) ≤ c * u k
⊢ u n ≤ c ^ n * u 0

Tactic:
by
  apply (monotone_mul_left_of_nonneg hc).seq_le_seq n _ h _ <;>
    simp [_root_.pow_succ', mul_assoc]

Goal state after:
Module: Mathlib.MeasureTheory.OuterMeasure.OfFunction

Goal state before:
pp: α : Type u_1
m : Set α → ℝ≥0∞
m_empty : m ∅ = 0
c : ℝ≥0∞
hc : c ≠ ∞
⊢ c • OuterMeasure.ofFunction m m_empty = OuterMeasure.ofFunction (c • m) ⋯

Tactic:
ext1 s

Goal state after:
α : Type u_1
m : Set α → ℝ≥0∞
m_empty : m ∅ = 0
c : ℝ≥0∞
hc : c ≠ ∞
s : Set α
⊢ (c • OuterMeasure.ofFunction m m_empty) s = (OuterMeasure.ofFunction (c • m) ⋯) s
Module: Mathlib.Topology.Instances.EReal.Lemmas

Goal state before:
pp: a : ℝ
h : 0 < a
x : ℝ
p : EReal × EReal
p1_gt : ↑(2 * max (x + 1) 0 / a) < p.1
p2_gt : ↑(a / 2) < p.2
⊢ 0 ≤ ↑(2 * max (x + 1) 0 / a)

Tactic:
EReal.coe_nonneg

Goal state after:
a : ℝ
h : 0 < a
x : ℝ
p : EReal × EReal
p1_gt : ↑(2 * max (x + 1) 0 / a) < p.1
p2_gt : ↑(a / 2) < p.2
⊢ 0 ≤ 2 * max (x + 1) 0 / a
Module: Mathlib.MeasureTheory.OuterMeasure.OfFunction

Goal state before:
pp: α : Type u_1
m : Set (OuterMeasure α)
h : m.Nonempty
t : Set α
⊢ ⨆ (_ : t.Nonempty), sInfGen m t = ⨅ μ ∈ m, μ t

Tactic:
by
  rcases t.eq_empty_or_nonempty with (rfl | ht)
  · simp [biInf_const h]
  · simp [ht, sInfGen_def]

Goal state after:
Module: Mathlib.MeasureTheory.PiSystem

Goal state before:
pp: α : Type u_3
ι : Type u_4
π : ι → Set (Set α)
i : ι
⊢ piiUnionInter π {i} = π i ∪ {univ}

Tactic:
ext1 s

Goal state after:
α : Type u_3
ι : Type u_4
π : ι → Set (Set α)
i : ι
s : Set α
⊢ s ∈ piiUnionInter π {i} ↔ s ∈ π i ∪ {univ}
Module: Mathlib.MeasureTheory.OuterMeasure.Induced

Goal state before:
pp: α : Type u_1
P : Set α → Prop
m : (s : Set α) → P s → ℝ≥0∞
P0 : P ∅
m0 : m ∅ P0 = 0
PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)
msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯
m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂
s : Set α
hs : (inducedOuterMeasure m P0 m0) s ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
h : (inducedOuterMeasure m P0 m0) s < (inducedOuterMeasure m P0 m0) s + ε
| (inducedOuterMeasure m P0 m0) s < (inducedOuterMeasure m P0 m0) s + ε

Tactic:
conv =>

Goal state after:
α : Type u_1
P : Set α → Prop
m : (s : Set α) → P s → ℝ≥0∞
P0 : P ∅
m0 : m ∅ P0 = 0
PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)
msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯
m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂
s : Set α
hs : (inducedOuterMeasure m P0 m0) s ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
h : (inducedOuterMeasure m P0 m0) s < (inducedOuterMeasure m P0 m0) s + ε
| ⨅ t, ⨅ (ht : P t), ⨅ (_ : s ⊆ t), m t ht
Module: Mathlib.MeasureTheory.PiSystem

Goal state before:
pp: α✝ : Type u_1
β : Type u_2
α : Type u_3
d : DynkinSystem α
h_inter : ∀ (s₁ s₂ : Set α), d.Has s₁ → d.Has s₂ → d.Has (s₁ ∩ s₂)
f : ℕ → Set α
hf : ∀ (i : ℕ), d.Has (f i)
⊢ d.Has (⋃ i, f i)

Tactic:
rw [← iUnion_disjointed]
    exact
      d.has_iUnion (disjoint_disjointed _) fun n =>
        disjointedRec (fun (t : Set α) i h => h_inter _ _ h <| d.has_compl <| hf i) (hf n)

Goal state after:
Module: Mathlib.MeasureTheory.OuterMeasure.Induced

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
ι : Sort u_2
inst✝ : Countable ι
μ : ι → OuterMeasure α
⊢ (⨆ i, μ i.down).trim = ⨆ i, (μ i.down).trim

Tactic:
ext1 s

Goal state after:
α : Type u_1
inst✝¹ : MeasurableSpace α
ι : Sort u_2
inst✝ : Countable ι
μ : ι → OuterMeasure α
s : Set α
⊢ (⨆ i, μ i.down).trim s = (⨆ i, (μ i.down).trim) s
Module: Mathlib.MeasureTheory.MeasurableSpace.Embedding

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s✝ t u✝ : Set α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
f : α → β
g : β → α
s : Set α
hf : MeasurableEmbedding f
u : Set α
hu : MeasurableSet u
⊢ MeasurableSet (⇑(Equiv.Set.image f s ⋯).symm ⁻¹' Subtype.val ⁻¹' u)

Tactic:
simpa [preimage_preimage, Set.image_symm_preimage hf.injective]
      using measurable_subtype_coe (hf.measurableSet_image' hu)

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

Goal state before:
pp: α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝² : MeasurableSingletonClass α
mβ : MeasurableSpace β
inst✝¹ : MeasurableSingletonClass β
inst✝ : Countable α
μ : Measure (α × β)
y : β
⊢ μ (Prod.snd ⁻¹' {y}) = ∑' (x : α), μ {(x, y)}

Tactic:
have : Prod.snd ⁻¹' {y} = ⋃ x : α, {(x, y)} := by ext y; simp [Prod.ext_iff, eq_comm]
  rw [this, measure_iUnion] <;> simp [Pairwise]

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
h : ae μ ≤ ae ν
s : Set α
⊢ ν s = 0 → μ s = 0

Tactic:
measure_eq_zero_iff_ae_notMem,

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
h : ae μ ≤ ae ν
s : Set α
⊢ (∀ᵐ (a : α) ∂ν, a ∉ s) → μ s = 0
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : Set α
hs : NullMeasurableSet s μ
t : Set α
⊢ μ s + μ (t \ s) = μ (s ∪ t)

Tactic:
← measure_union₀' hs disjoint_sdiff_right.aedisjoint,

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : Set α
hs : NullMeasurableSet s μ
t : Set α
⊢ μ (s ∪ t \ s) = μ (s ∪ t)