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

Goal state before:
pp: case inl
X : Type u_3
inst✝¹ : MetricSpace X
inst✝ : SeparableSpace X
x : X
C : Set X
hxC : C ∈ 𝓝 x
ε : ℝ := min (infDist x (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ)) 1
hC : closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ = ∅
⊢ ∃ n, C ∈ comap (distDenseSeq X n) (𝓝 (distDenseSeq X n x))

pp: case inr
X : Type u_3
inst✝¹ : MetricSpace X
inst✝ : SeparableSpace X
x : X
C : Set X
hxC : C ∈ 𝓝 x
ε : ℝ := min (infDist x (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ)) 1
hC : (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ).Nonempty
⊢ ∃ n, C ∈ comap (distDenseSeq X n) (𝓝 (distDenseSeq X n x))

Tactic:
· simp_all

Goal state after:
case inr
X : Type u_3
inst✝¹ : MetricSpace X
inst✝ : SeparableSpace X
x : X
C : Set X
hxC : C ∈ 𝓝 x
ε : ℝ := min (infDist x (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ)) 1
hC : (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ).Nonempty
⊢ ∃ n, C ∈ comap (distDenseSeq X n) (𝓝 (distDenseSeq X n x))
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
p q : Set α → Prop
ν : Measure α
hμ : μ.InnerRegularWRT p q
hν : ν.InnerRegularWRT p q
hμν : ∀ (U : Set α), p U → μ U = ν U
U : Set α
hU : q U
⊢ μ U = ν U

Tactic:
by
  rw [hμ.measure_eq_iSup hU, hν.measure_eq_iSup hU]
  congr! 4 with t _ ht2
  exact hμν t ht2

Goal state after:
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: case mpr.py₂
G : Type u_1
inst✝ : Group G
K : Subgroup G
S : Set G
hSK : IsComplement S ↑K
g₁ g₂ : G
h : g₁⁻¹ * g₂ ∈ K
⊢ (↑(hSK.equiv g₂).1)⁻¹ * g₁ ∈ ↑K

Tactic:
rw [SetLike.mem_coe, ← mul_mem_cancel_right h]
      -- This used to be `simp [...]` before https://github.com/leanprover/lean4/pull/2644
      rw [equiv_fst_eq_mul_inv]; simp [← mul_assoc]

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
⊢ (μ.restrict s).prod (ν.restrict t) = (μ.prod ν).restrict (s ×ˢ t)

Tactic:
rw [← sum_sfiniteSeq μ, ← sum_sfiniteSeq ν, restrict_sum_of_countable, restrict_sum_of_countable,
    prod_sum, prod_sum, restrict_sum_of_countable]

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
⊢ (sum fun p ↦ ((sfiniteSeq μ p.1).restrict s).prod ((sfiniteSeq ν p.2).restrict t)) =
    sum fun i ↦ ((sfiniteSeq μ i.1).prod (sfiniteSeq ν i.2)).restrict (s ×ˢ t)
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
this : ν = ν.restrict univ
⊢ (μ.restrict s).prod ν = (μ.prod ν).restrict (s ×ˢ univ)

Tactic:
rw [this, Measure.prod_restrict, ← this]

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

Goal state before:
pp: α : Type u_1
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
H✝ : μ.InnerRegularWRT IsClosed[inst✝²] IsOpen[inst✝²]
hfin : ∀ {s : Set α}, μ s ≠ ∞
s : ℕ → Set α
hsd : Pairwise (Function.onFun Disjoint s)
hsm : ∀ (i : ℕ), MeasurableSet (s i)
H :
  ∀ (i : ℕ) (ε : ℝ≥0∞),
    ε ≠ 0 → ∃ F ⊆ s i, ∃ U ⊇ s i, IsClosed[inst✝²] F ∧ IsOpen[inst✝²] U ∧ μ (s i) ≤ μ F + ε ∧ μ U ≤ μ (s i) + ε
ε : ℝ≥0∞
ε0 : ε ≠ 0
ε0' : ε / 2 ≠ 0
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : ∑' (i : ℕ), δ i < ε / 2
F : ℕ → Set α
hFs : ∀ (n : ℕ), F n ⊆ s n
U : ℕ → Set α
hsU : ∀ (n : ℕ), U n ⊇ s n
hFc : ∀ (n : ℕ), IsClosed[inst✝²] (F n)
hUo : ∀ (n : ℕ), IsOpen[inst✝²] (U n)
hF : ∀ (n : ℕ), μ (s n) ≤ μ (F n) + δ n
hU : ∀ (n : ℕ), μ (U n) ≤ μ (s n) + δ n
this : Tendsto (fun t ↦ ∑ k ∈ t, μ (s k) + ε / 2) atTop (𝓝 (μ (⋃ n, s n) + ε / 2))
t : Finset ℕ
ht : μ (⋃ n, s n) < ∑ k ∈ t, μ (s k) + ε / 2
⊢ ∑ k ∈ t, μ (s k) + ε / 2 ≤ ∑ k ∈ t, μ (F k) + ∑ k ∈ t, δ k + ε / 2

Tactic:
rw [← sum_add_distrib]
          gcongr
          apply hF

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:
by
  conv_rhs => rw [Filter.Eventually, ← map_div_right_ae μ t]
  rfl

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

Goal state before:
pp: G : Type u_1
H : Type u_2
inst✝³ : MeasurableSpace G
inst✝² : MeasurableSpace H
inst✝¹ : Group G
μ : Measure G
inst✝ : MeasurableMul G
g : G
s : Set G
hs : MeasurableSet s
⊢ ↑((fun x ↦ g * x) ⁻¹' s).encard = ↑s.encard

Tactic:
encard_preimage_of_bijective (Group.mulLeft_bijective _)

Goal state after:
G : Type u_1
H : Type u_2
inst✝³ : MeasurableSpace G
inst✝² : MeasurableSpace H
inst✝¹ : Group G
μ : Measure G
inst✝ : MeasurableMul G
g : G
s : Set G
hs : MeasurableSet s
⊢ ↑s.encard = ↑s.encard
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: G : Type u_1
H : Type u_2
inst✝³ : MeasurableSpace G
inst✝² : MeasurableSpace H
inst✝¹ : Group G
μ : Measure G
inst✝ : MeasurableMul G
g : G
⊢ map (fun x ↦ x * g) count = count

Tactic:
ext s hs
    rw [count_apply hs, map_apply (measurable_mul_const _) hs,
      count_apply (measurable_mul_const _ hs),
      encard_preimage_of_bijective (Group.mulRight_bijective _)]

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✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : SFinite ν
ρ : Measure (α × β)
inst✝ : SFinite ρ
⊢ SFinite ρ.fst

Tactic:
rw [fst]
  infer_instance

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✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : SFinite ν
ρ : Measure (α × β)
inst✝ : IsFiniteMeasure ρ
⊢ IsFiniteMeasure ρ.fst

Tactic:
rw [fst]
  infer_instance

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

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

Tactic:
simpa using (lintegral_map_equiv f (μ := μ) <| MeasurableEquiv.inv G).symm

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hfg : f ≤ᵐ[μ] g
⊢ μ.withDensity f ≤ μ.withDensity g

Tactic:
refine le_iff.2 fun s hs ↦ ?_

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hfg : f ≤ᵐ[μ] g
s : Set α
hs : MeasurableSet s
⊢ (μ.withDensity f) s ≤ (μ.withDensity g) s
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
μ : Measure G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
h : μ.InnerRegularCompactLTTop
⊢ μ.InnerRegularWRT (fun s ↦ IsCompact s ∧ IsClosed[inst✝³] s) fun s ↦ MeasurableSet s ∧ μ s ≠ ∞

Tactic:
intro s ⟨s_meas, μs⟩ r hr
  rcases h.innerRegular ⟨s_meas, μs⟩ r hr with ⟨K, Ks, K_comp, hK⟩
  refine ⟨closure K, ?_, ⟨K_comp.closure, isClosed_closure⟩, ?_⟩
  · exact IsCompact.closure_subset_measurableSet K_comp s_meas Ks
  · rwa [K_comp.measure_closure]

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
p : α → Prop
f : α → ℝ≥0∞
hf : AEMeasurable f μ
⊢ μ ({x | f x ≠ 0} ∩ {a | ¬p a}) = 0 ↔ μ {a | ¬(f a ≠ 0 → p a)} = 0

Tactic:
iff_iff_eq

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
p : α → Prop
f : α → ℝ≥0∞
hf : AEMeasurable f μ
⊢ (μ ({x | f x ≠ 0} ∩ {a | ¬p a}) = 0) = (μ {a | ¬(f a ≠ 0 → p a)} = 0)
Module: Mathlib.MeasureTheory.Group.Prod

Goal state before:
pp: G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
inst✝⁴ : MeasurableInv G
μ' ν' : Measure G
inst✝³ : SigmaFinite μ'
inst✝² : SigmaFinite ν'
inst✝¹ : μ'.IsMulLeftInvariant
inst✝ : ν'.IsMulLeftInvariant
s t : Set G
h2s : ν' s ≠ 0
h3s : ν' s ≠ ∞
this : ∀ (s : Set G), ν' s ≠ 0 → ν' s ≠ ∞ → MeasurableSet s → μ' s * ν' t = ν' s * μ' t
hs : ¬MeasurableSet s
s' : Set G
hm : MeasurableSet s'
hμ : μ' s' = μ' s
hν : ν' s' = ν' s
⊢ ν' s' ≠ 0

Tactic:
hν

Goal state after:
G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
inst✝⁴ : MeasurableInv G
μ' ν' : Measure G
inst✝³ : SigmaFinite μ'
inst✝² : SigmaFinite ν'
inst✝¹ : μ'.IsMulLeftInvariant
inst✝ : ν'.IsMulLeftInvariant
s t : Set G
h2s : ν' s ≠ 0
h3s : ν' s ≠ ∞
this : ∀ (s : Set G), ν' s ≠ 0 → ν' s ≠ ∞ → MeasurableSet s → μ' s * ν' t = ν' s * μ' t
hs : ¬MeasurableSet s
s' : Set G
hm : MeasurableSet s'
hμ : μ' s' = μ' s
hν : ν' s' = ν' s
⊢ ν' s ≠ 0
Module: Mathlib.MeasureTheory.Group.Prod

Goal state before:
pp: G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
inst✝⁴ : MeasurableInv G
μ' ν' : Measure G
inst✝³ : SigmaFinite μ'
inst✝² : SigmaFinite ν'
inst✝¹ : μ'.IsMulLeftInvariant
inst✝ : ν'.IsMulLeftInvariant
s t : Set G
h2s : ν' s ≠ 0
h3s : ν' s ≠ ∞
this : ∀ (s : Set G), ν' s ≠ 0 → ν' s ≠ ∞ → MeasurableSet s → μ' s * ν' t = ν' s * μ' t
hs : ¬MeasurableSet s
s' : Set G
hm : MeasurableSet s'
hμ : μ' s' = μ' s
hν : ν' s' = ν' s
⊢ ν' s' ≠ ∞

Tactic:
hν

Goal state after:
G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
inst✝⁴ : MeasurableInv G
μ' ν' : Measure G
inst✝³ : SigmaFinite μ'
inst✝² : SigmaFinite ν'
inst✝¹ : μ'.IsMulLeftInvariant
inst✝ : ν'.IsMulLeftInvariant
s t : Set G
h2s : ν' s ≠ 0
h3s : ν' s ≠ ∞
this : ∀ (s : Set G), ν' s ≠ 0 → ν' s ≠ ∞ → MeasurableSet s → μ' s * ν' t = ν' s * μ' t
hs : ¬MeasurableSet s
s' : Set G
hm : MeasurableSet s'
hμ : μ' s' = μ' s
hν : ν' s' = ν' s
⊢ ν' s ≠ ∞
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
β : Type u_2
inst✝¹ : MeasurableSpace β
f : β → α → ℝ≥0∞
inst✝ : SFinite μ
hf : Measurable (Function.uncurry f)
⊢ Measurable fun b ↦ μ.withDensity (f b)

Tactic:
rw [Measure.measurable_measure]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
β : Type u_2
inst✝¹ : MeasurableSpace β
f : β → α → ℝ≥0∞
inst✝ : SFinite μ
hf : Measurable (Function.uncurry f)
⊢ ∀ (s : Set α), MeasurableSet s → Measurable fun b ↦ (μ.withDensity (f b)) s
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x y : ℝ
hy : 0 ≤ y
⊢ √x ≤ √y ↔ x ≤ y

Tactic:
by
  rw [Real.sqrt, Real.sqrt, NNReal.coe_le_coe, NNReal.sqrt_le_sqrt, toNNReal_le_toNNReal_iff hy]

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

Goal state before:
pp: a : ℕ
⊢ ↑a.sqrt ≤ √↑a

Tactic:
Real.le_sqrt (Nat.cast_nonneg _) (Nat.cast_nonneg _)

Goal state after:
a : ℕ
⊢ ↑a.sqrt ^ 2 ≤ ↑a
Module: Mathlib.Analysis.Complex.Norm

Goal state before:
pp: f : CauSeq ℂ fun x ↦ ‖x‖
ε : ℝ
ε0 : ε > 0
i : ℕ
hi : ∀ j ≥ i, (fun x ↦ ‖x‖) (↑f j - ↑f i) < ε
j : ℕ
hj : j ≥ i
⊢ (fun x ↦ ‖x‖) ((fun n ↦ (starRingEnd ℂ) (↑f n)) j - (fun n ↦ (starRingEnd ℂ) (↑f n)) i) < ε

Tactic:
simp_rw [← map_sub, norm_conj]; exact hi j hj

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Hom

Goal state before:
pp: V : Type u_1
V₁ : Type u_2
V₂ : Type u_3
V₃ : Type u_4
inst✝³ : SeminormedAddCommGroup V
inst✝² : SeminormedAddCommGroup V₁
inst✝¹ : SeminormedAddCommGroup V₂
inst✝ : SeminormedAddCommGroup V₃
f✝ g✝ f g : NormedAddGroupHom V₁ V₂
v : V₁
⊢ ‖f v‖ + ‖g v‖ ≤ ‖f‖ * ‖v‖ + ‖g‖ * ‖v‖

Tactic:
gcongr <;> apply le_opNorm

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

Goal state before:
pp: ι : Type u_5
R : Type u_7
R₂ : Type u_8
M : Type u_9
M₂ : Type u_10
inst✝⁹ : Semiring R
inst✝⁸ : Semiring R₂
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : Module R₂ M₂
inst✝³ : TopologicalSpace M
inst✝² : TopologicalSpace M₂
σ : R →+* R₂
σ' : R₂ →+* R
inst✝¹ : RingHomInvPair σ σ'
inst✝ : RingHomInvPair σ' σ
L : SummationFilter ι
f : ι → M
e : M ≃SL[σ] M₂
y : M₂
h : HasSum (fun b ↦ e (f b)) y L
⊢ HasSum f (e.symm y) L

Tactic:
by simpa only [e.symm.coe_coe, e.symm_apply_apply] using h.mapL (e.symm : M₂ →SL[σ'] M)

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

Goal state before:
pp: case neg
ι : Type u_5
R : Type u_7
R₂ : Type u_8
M : Type u_9
M₂ : Type u_10
inst✝¹¹ : Semiring R
inst✝¹⁰ : Semiring R₂
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
inst✝⁷ : AddCommMonoid M₂
inst✝⁶ : Module R₂ M₂
inst✝⁵ : TopologicalSpace M
inst✝⁴ : TopologicalSpace M₂
σ : R →+* R₂
σ' : R₂ →+* R
inst✝³ : RingHomInvPair σ σ'
inst✝² : RingHomInvPair σ' σ
L : SummationFilter ι
inst✝¹ : T2Space M
inst✝ : T2Space M₂
f : ι → M
e : M ≃SL[σ] M₂
y : M₂
hf : Summable f L
hL : ¬L.NeBot
⊢ ∑'[L] (z : ι), e (f z) = y ↔ ∑'[L] (z : ι), f z = e.symm y

Tactic:
· simp only [tsum_bot hL, eq_symm_apply]
      constructor <;> rintro rfl
      exacts [e.map_finsum f, (e.map_finsum f).symm]

Goal state after:
Module: Mathlib.LinearAlgebra.Complex.Module

Goal state before:
pp: case refine_1
A : Type u_1
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : Module ℂ A
inst✝² : SMulCommClass ℂ A A
inst✝¹ : IsScalarTower ℂ A A
inst✝ : StarModule ℂ A
x : A
x✝ : star x * x = 1 ∧ x * star x = 1
h : star x * x = 1
h' : x * star x = 1
this : IsStarNormal x
⊢ IsStarNormal x ∧ ↑(ℜ x) ^ 2 + ↑(ℑ x) ^ 2 = 1

Tactic:
exact ⟨this, by simp [sq, ← star_mul_self_eq_realPart_sq_add_imaginaryPart_sq x, h]⟩

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

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
z : K
⊢ ‖z‖ ≤ re z ↔ z = ↑‖z‖

Tactic:
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
  · have h' : ‖z‖ = re z := (le_antisymm (re_le_norm z) h).symm
    rw [h', re_eq_self_of_le h]
  · rw [h]
    simp

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

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
y z : K
r : ℝ
hx : 0 ≤ re ↑r ∧ im ↑r = 0
hyz : ↑r * y < ↑r * z
⊢ y < z

Tactic:
simp only [RCLike.lt_iff_re_im (K := K), mul_re, ofReal_re, ofReal_im, zero_mul, sub_zero,
      mul_im, add_zero, mul_eq_mul_left_iff] at hyz ⊢

Goal state after:
K : Type u_1
inst✝ : RCLike K
y z : K
r : ℝ
hx : 0 ≤ re ↑r ∧ im ↑r = 0
hyz : r * re y < r * re z ∧ (im y = im z ∨ r = 0)
⊢ re y < re z ∧ im y = im z
Module: Mathlib.Analysis.RCLike.Basic

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

Tactic:
← mul_sub,

Goal state after:
K : Type u_1
inst✝ : RCLike K
a : { x // 0 < x }
b c : K
a' : ℝ
h : 0 ≤ ↑a' * (c - b)
ha1 : a' > 0
ha2 : ↑a' = ↑a
⊢ 0 ≤ c - b
Module: Mathlib.Analysis.Normed.Operator.LinearIsometry

Goal state before:
pp: R : Type u_1
R₂ : Type u_2
E : Type u_5
E₂ : Type u_6
inst✝⁷ : Semiring R
inst✝⁶ : Semiring R₂
σ₁₂ : R →+* R₂
σ₂₁ : R₂ →+* R
inst✝⁵ : RingHomInvPair σ₁₂ σ₂₁
inst✝⁴ : RingHomInvPair σ₂₁ σ₁₂
inst✝³ : SeminormedAddCommGroup E
inst✝² : SeminormedAddCommGroup E₂
inst✝¹ : Module R E
inst✝ : Module R₂ E₂
e : E ≃ₛₗᵢ[σ₁₂] E₂
⊢ range ⇑e = univ

Tactic:
rw [← coe_toIsometryEquiv]
  exact IsometryEquiv.range_eq_univ _

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

Goal state before:
pp: case cons
α : Type u_1
E' : Type u_6
inst✝ : SeminormedAddCommGroup E'
l : Filter α
ι : Type u_18
A : ι → α → E'
s✝ : Finset ι
B : ι → α → ℝ
i : ι
s : Finset ι
his : i ∉ s
h : (∀ i ∈ s, A i =o[l] B i) → (fun H ↦ ∑ i ∈ s, A i H) =o[l] fun H ↦ ∑ i ∈ s, ‖B i H‖
hAB : ∀ i_1 ∈ Finset.cons i s his, A i_1 =o[l] B i_1
⊢ (fun H ↦ ∑ i ∈ Finset.cons i s his, A i H) =o[l] fun H ↦ ∑ i ∈ Finset.cons i s his, ‖B i H‖

Tactic:
simp_rw [Finset.sum_cons]
  calc (fun H => A i H + ∑ j ∈ s, A j H)
      =o[l] fun H => ‖B i H‖ + ‖∑ j ∈ s, ‖B j H‖‖ :=
          (hAB i (by simp)).add_add (h (fun j hj => hAB j (by simp [hj])))
    _ =ᶠ[l] fun H => ‖B i H‖ + ∑ j ∈ s, ‖B j H‖ := by
        refine Eventually.of_forall fun H ↦ congr_arg (‖B i H‖ + ·) ?_
        exact Real.norm_of_nonneg (Finset.sum_nonneg fun _ _ => norm_nonneg _)

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

Goal state before:
pp: α : Type u_1
E' : Type u_6
F' : Type u_7
R : Type u_13
𝕜' : Type u_16
inst✝⁷ : SeminormedAddCommGroup E'
inst✝⁶ : SeminormedAddCommGroup F'
inst✝⁵ : SeminormedRing R
inst✝⁴ : NormedDivisionRing 𝕜'
f' : α → E'
g' : α → F'
l : Filter α
inst✝³ : Module R E'
inst✝² : IsBoundedSMul R E'
inst✝¹ : Module 𝕜' F'
inst✝ : NormSMulClass 𝕜' F'
k₁ : α → R
k₂ : α → 𝕜'
h₁ : k₁ =o[l] k₂
h₂ : f' =O[l] g'
⊢ (fun x ↦ k₁ x • f' x) =o[l] fun x ↦ k₂ x • g' x

Tactic:
by
  simp only [IsLittleO_def] at *
  intro c cpos
  rcases h₂.exists_pos with ⟨c', c'pos, hc'⟩
  exact ((h₁ (div_pos cpos c'pos)).smul hc').congr_const (div_mul_cancel₀ _ (ne_of_gt c'pos))

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

Goal state before:
pp: α : Type u_1
F : Type u_4
E'' : Type u_9
inst✝¹ : Norm F
inst✝ : NormedAddCommGroup E''
g : α → F
l : Filter α
c : E''
hc : c ≠ 0
⊢ (fun _x ↦ c) =o[l] g ↔ Tendsto (fun x ↦ ‖g x‖) l atTop

Tactic:
simp only [← isLittleO_one_left_iff ℝ]

Goal state after:
α : Type u_1
F : Type u_4
E'' : Type u_9
inst✝¹ : Norm F
inst✝ : NormedAddCommGroup E''
g : α → F
l : Filter α
c : E''
hc : c ≠ 0
⊢ (fun _x ↦ c) =o[l] g ↔ (fun _x ↦ 1) =o[l] g
Module: Mathlib.Analysis.Complex.Exponential

Goal state before:
pp: α : Type u_1
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
n j : ℕ
hn : 0 < n
⊢ ∑ m ∈ range j with n ≤ m, 1 / ↑m.factorial = ∑ m ∈ range (j - n), 1 / ↑(m + n).factorial

Tactic:
refine sum_nbij' (· - n) (· + n) ?_ ?_ ?_ ?_ ?_ <;>
          simp +contextual [lt_tsub_iff_right, tsub_add_cancel_of_le]

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

Goal state before:
pp: 𝕜 : Type u_15
inst✝ : NormedDivisionRing 𝕜
m p : ℕ
hp0 : 0 < p
h : m < m + p
this : (fun x ↦ x ^ m * x ^ p) =o[𝓝 0] fun x ↦ x ^ m * 1 ^ p
⊢ (fun x ↦ x ^ (m + p)) =o[𝓝 0] fun x ↦ x ^ m

Tactic:
simpa only [pow_add, one_pow, mul_one]

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

Goal state before:
pp: x : ℂ
hx : ‖x‖ ≤ 1
⊢ ‖-x * I‖ ^ 5 * (↑(Nat.succ 5) * (↑(Nat.factorial 5) * ↑5)⁻¹) / 2 +
      ‖x * I‖ ^ 5 * (↑(Nat.succ 5) * (↑(Nat.factorial 5) * ↑5)⁻¹) / 2 =
    ‖x‖ ^ 5 / 100

Tactic:
by norm_num [mul_one_div]

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

Goal state before:
pp: x : ℝ
⊢ ‖cexp (I * ↑x)‖₊ = 1

Tactic:
rw [← nnnorm_norm, norm_exp_I_mul_ofReal, ← NNReal.coe_eq_one]

Goal state after:
x : ℝ
⊢ ↑‖1‖₊ = 1
Module: Mathlib.Analysis.Complex.Exponential

Goal state before:
pp: n : ℕ
a₁ b₁ : ℝ
m : ℕ
en : n + 1 = m
h : |rexp 1 - expNear m 1 ((a₁ - 1) * ↑m)| ≤ |1| ^ m / ↑m.factorial * (b₁ * ↑m)
⊢ |1 + (↑m)⁻¹ * ((a₁ - 1) * ↑m) - a₁| ≤ b₁ - (↑m)⁻¹ * (b₁ * ↑m)

Tactic:
field_simp [show (m : ℝ) ≠ 0 by norm_cast; lia]

Goal state after:
n : ℕ
a₁ b₁ : ℝ
m : ℕ
en : n + 1 = m
h : |rexp 1 - expNear m 1 ((a₁ - 1) * ↑m)| ≤ |1| ^ m / ↑m.factorial * (b₁ * ↑m)
⊢ |1 + (a₁ - 1) - a₁| ≤ b₁ * (1 - 1)
Module: Mathlib.Algebra.Order.Field.Power

Goal state before:
pp: α : Type u_1
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
a b : α
n : ℤ
hb : b ≠ 0
m : ℕ
⊢ a = -b ∧ Odd m.succ ↔ a = -b ∧ Odd (-↑m.succ)

Tactic:
simp [parity_simps]

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

Goal state before:
pp: α : Type u_1
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : Archimedean α
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, |f n| ≤ a
hnm : ∀ n ≥ m, f n.succ ≤ f n
ε : α
ε0 : ε > 0
k : ℕ
hk : a ≤ k • ε
h : ∃ l, ∀ n ≥ m, a - l • ε < f n
l : ℕ := Nat.find h
hl : ∀ n ≥ m, f n > a - l • ε
hl0 : l ≠ 0
i : ℕ
hi : i ≥ m ∧ f i ≤ a - (Nat.find h).pred • ε
j : ℕ
hj : j ≥ i
hfij : f j ≤ f i
| a - l.pred • ε = a - l • ε + ε

Tactic:
rhs

Goal state after:
α : Type u_1
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : Archimedean α
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, |f n| ≤ a
hnm : ∀ n ≥ m, f n.succ ≤ f n
ε : α
ε0 : ε > 0
k : ℕ
hk : a ≤ k • ε
h : ∃ l, ∀ n ≥ m, a - l • ε < f n
l : ℕ := Nat.find h
hl : ∀ n ≥ m, f n > a - l • ε
hl0 : l ≠ 0
i : ℕ
hi : i ≥ m ∧ f i ≤ a - (Nat.find h).pred • ε
j : ℕ
hj : j ≥ i
hfij : f j ≤ f i
| a - l • ε + ε
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic

Goal state before:
pp: ⊢ sin π = 0

Tactic:
by
  rw [← mul_div_cancel_left₀ π two_ne_zero, two_mul, add_div, sin_add, cos_pi_div_two]; simp

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Pointwise

Goal state before:
pp: E : Type u_1
inst✝ : SeminormedCommGroup E
δ : ℝ
x y : E
⊢ {x} * (sphere y δ)⁻¹ = sphere (x * y⁻¹) δ

Tactic:
inv_sphere,

Goal state after:
E : Type u_1
inst✝ : SeminormedCommGroup E
δ : ℝ
x y : E
⊢ {x} * sphere y⁻¹ δ = sphere (x * y⁻¹) δ
Module: Mathlib.Analysis.Normed.Module.RCLike.Real

Goal state before:
pp: case inr.inr
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x y : E
hr✝ : dist y x ≠ 0
hr : 0 < dist y x
hy : y ∈ interior (closedBall x (dist y x))
⊢ y ∈ ball x (dist y x)

Tactic:
set f : ℝ → E := fun c : ℝ => c • (y - x) + x

Goal state after:
case inr.inr
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x y : E
hr✝ : dist y x ≠ 0
hr : 0 < dist y x
hy : y ∈ interior (closedBall x (dist y x))
f : ℝ → E := fun c ↦ c • (y - x) + x
⊢ y ∈ ball x (dist y x)
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: case e'_6
R : Type u_4
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
r : R
hr : ‖r‖ < 1
k : ℕ
ih : HasSum (fun n ↦ ↑((n + k).choose k) * r ^ n) ((1 - r)⁻¹ʳ ^ (k + 1))
I1 : Summable fun n ↦ ‖↑((n + k).choose k) * r ^ n‖
⊢ (1 - r)⁻¹ʳ ^ (k + 1 + 1) = (∑' (n : ℕ), ↑((n + k).choose k) * r ^ n) * ∑' (n : ℕ), r ^ n

Tactic:
· rw [ih.tsum_eq, (hasSum_geom_series_inverse r hr).tsum_eq, pow_succ]

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Ball.Pointwise

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace ℝ E
inst✝ : Nontrivial E
c : 𝕜
x : E
r : ℝ
hr : 0 ≤ r
⊢ c • sphere x r = sphere (c • x) (‖c‖ * r)

Tactic:
rcases eq_or_ne c 0 with (rfl | hc)
  · simp [zero_smul_set, Set.singleton_zero, hr]
  · exact smul_sphere' hc x r

Goal state after:
Module: Mathlib.Topology.Path

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
x y z : X
γ₁ : Path x y
γ₂ : Path y z
t : ℝ
ht : 1 / 2 ≤ t
⊢ (γ₁.trans γ₂).symm.extend (1 - t) = γ₂.extend (2 * t - 1)

Tactic:
trans_symm,

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
x y z : X
γ₁ : Path x y
γ₂ : Path y z
t : ℝ
ht : 1 / 2 ≤ t
⊢ (γ₂.symm.trans γ₁.symm).extend (1 - t) = γ₂.extend (2 * t - 1)
Module: Mathlib.Topology.Path

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
x y z : X
γ₁ : Path x y
γ₂ : Path y z
t : ℝ
ht : 1 / 2 ≤ t
⊢ (γ₂.symm.trans γ₁.symm).extend (1 - t) = γ₂.extend (2 * t - 1)

Tactic:
extend_trans_of_le_half _ _ (by linarith),

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
x y z : X
γ₁ : Path x y
γ₂ : Path y z
t : ℝ
ht : 1 / 2 ≤ t
⊢ γ₂.symm.extend (2 * (1 - t)) = γ₂.extend (2 * t - 1)
Module: Mathlib.Analysis.Normed.Group.AddCircle

Goal state before:
pp: case inr.inr.inl.inl
p x ε : ℝ
s : Set ℝ
hε : ε < |p| / 2
z : ℤ
hs : s ⊆ Icc (x - |p| / 2) (x + |p| / 2)
hz : z ≠ 0
y : ℝ
hy₀ : y ∈ s
hy₁ : x + ↑z * p - ε ≤ y
hy₂ : y ≤ x + ↑z * p + ε
hy₃ : x - |p| / 2 ≤ y
hy₄ : y ≤ x + |p| / 2
hp : 0 < p
hz' : ↑z ≤ -1
⊢ False

Tactic:
have : ↑z * p ≤ -p := by nlinarith

Goal state after:
case inr.inr.inl.inl
p x ε : ℝ
s : Set ℝ
hε : ε < |p| / 2
z : ℤ
hs : s ⊆ Icc (x - |p| / 2) (x + |p| / 2)
hz : z ≠ 0
y : ℝ
hy₀ : y ∈ s
hy₁ : x + ↑z * p - ε ≤ y
hy₂ : y ≤ x + ↑z * p + ε
hy₃ : x - |p| / 2 ≤ y
hy₄ : y ≤ x + |p| / 2
hp : 0 < p
hz' : ↑z ≤ -1
this : ↑z * p ≤ -p
⊢ False