mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.LocallyConvex.Separation

Goal state before:
pp: E : Type u_2
inst✝⁵ : TopologicalSpace E
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
s t : Set E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : LocallyConvexSpace ℝ E
hs₁ : Convex ℝ s
hs₂ : IsCompact s
ht₁ : Convex ℝ t
ht₂ : IsClosed t
disj : Disjoint s t
⊢ ∃ f u v, (∀ a ∈ s, f a < u) ∧ u < v ∧ ∀ b ∈ t, v < f b

Tactic:
obtain rfl | hs := s.eq_empty_or_nonempty
  · exact ⟨0, -2, -1, by simp⟩
  obtain rfl | _ht := t.eq_empty_or_nonempty
  · exact ⟨0, 1, 2, by simp⟩
  obtain ⟨U, V, hU, hV, hU₁, hV₁, sU, tV, disj'⟩ := disj.exists_open_convexes hs₁ hs₂ ht₁ ht₂
  obtain ⟨f, u, hf₁, hf₂⟩ := geometric_hahn_banach_open_open hU₁ hU hV₁ hV disj'
  obtain ⟨x, hx₁, hx₂⟩ := hs₂.exists_isMaxOn hs f.continuous.continuousOn
  have : f x < u := hf₁ x (sU hx₁)
  exact
    ⟨f, (f x + u) / 2, u,
      fun a ha => by have := hx₂ ha; dsimp at this; linarith,
      by linarith,
      fun b hb => hf₂ b (tV hb)⟩

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

Goal state before:
pp: case h
𝕜 : Type u_1
E : Type u_2
inst✝⁸ : TopologicalSpace E
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
s t : Set E
inst✝⁵ : RCLike 𝕜
inst✝⁴ : Module 𝕜 E
inst✝³ : IsScalarTower ℝ 𝕜 E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
hs₁ : Convex ℝ s
hs₂ : IsCompact s
ht₁ : Convex ℝ t
ht₂ : IsClosed t
disj : Disjoint s t
this : ContinuousSMul ℝ E
g : StrongDual ℝ E
u v : ℝ
h1 : (∀ a ∈ s, g a < u) ∧ u < v ∧ ∀ b ∈ t, v < g b
⊢ ∃ u v, (∀ a ∈ s, re ((StrongDual.extendRCLikeₗ g) a) < u) ∧ u < v ∧ ∀ b ∈ t, v < re ((StrongDual.extendRCLikeₗ g) b)

Tactic:
simpa [g.extendRCLikeₗ_apply, exists_and_left] using ⟨u, h1.1, v, h1.2⟩

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
x : ℝ
hx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}
⊢ x ∈ ⋂ e, ⋃ n, ⋂ p, ⋂ (_ : p ≥ n), ⋂ q, ⋂ (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)

Tactic:
mem_iInter

Goal state after:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
x : ℝ
hx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}
⊢ ∀ (i : ℕ), x ∈ ⋃ n, ⋂ p, ⋂ (_ : p ≥ n), ⋂ q, ⋂ (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ i)
Module: Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

Goal state before:
pp: X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
f_meas : Measurable f
g : X → E
hE : CompleteSpace E
hg : Integrable g (μ.withDensity fun x ↦ ↑(f x))
C1 : Continuous fun u ↦ ∫ (x : X), ↑↑u x ∂μ.withDensity fun x ↦ ↑(f x)
this : Continuous ((fun u ↦ ∫ (x : X), ↑↑u x ∂μ) ∘ ⇑(withDensitySMulLI μ f_meas))
u : ↥(Lp E 1 (μ.withDensity fun x ↦ ↑(f x)))
⊢ ∫ (x : X), f x • ↑↑u x ∂μ = ∫ (x : X), ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯) x ∂μ

Tactic:
exact integral_congr_ae (memL1_smul_of_L1_withDensity f_meas u).coeFn_toLp.symm

Goal state after:
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
ι : Type u_3
inst✝ : Countable ι
F : ι → ℝ → E
bound : ι → ℝ → ℝ
hF_meas : ∀ (n : ι), AEStronglyMeasurable (F n) (μ.restrict (Ι a b))
h_bound : ∀ (n : ι), ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), ‖F n x‖ ≤ bound n x
bound_summable : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), Summable fun n ↦ bound n x
bound_integrable : IntegrableOn (fun t ↦ ∑' (n : ι), bound n t) (Ι a b) μ
h_lim : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), HasSum (fun n ↦ F n x) (f x)
⊢ HasSum (fun n ↦ (if a ≤ b then 1 else -1) • ∫ (t : ℝ) in Ι a b, F n t ∂μ)
    ((if a ≤ b then 1 else -1) • ∫ (t : ℝ) in Ι a b, f t ∂μ)

Tactic:
exact
    (hasSum_integral_of_dominated_convergence bound hF_meas h_bound bound_summable bound_integrable
          h_lim).const_smul
      _

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_5
F : Type u_6
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
a b : ℝ
μ : Measure ℝ
f : ℝ → E
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : NormedSpace ℝ F
inst✝¹ : CompleteSpace F
inst✝ : CompleteSpace E
L : E →L[𝕜] F
hf : IntervalIntegrable f μ a b
⊢ ∫ (x : ℝ) in a..b, L (f x) ∂μ = L (∫ (x : ℝ) in a..b, f x ∂μ)

Tactic:
by
  simp_rw [intervalIntegral, L.integral_comp_comm hf.1, L.integral_comp_comm hf.2, L.map_sub]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c d : ℝ
f : ℝ → E
μ : Measure ℝ
hab : IntervalIntegrable f μ a b
hcd : IntervalIntegrable f μ c d
hac : IntervalIntegrable f μ a c
⊢ ∫ (x : ℝ) in a..c, f x ∂μ - ∫ (x : ℝ) in b..d, f x ∂μ = ∫ (x : ℝ) in d..b, f x ∂μ - ∫ (x : ℝ) in c..a, f x ∂μ

Tactic:
integral_symm b d,

Goal state after:
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c d : ℝ
f : ℝ → E
μ : Measure ℝ
hab : IntervalIntegrable f μ a b
hcd : IntervalIntegrable f μ c d
hac : IntervalIntegrable f μ a c
⊢ ∫ (x : ℝ) in a..c, f x ∂μ - ∫ (x : ℝ) in b..d, f x ∂μ = -∫ (x : ℝ) in b..d, f x ∂μ - ∫ (x : ℝ) in c..a, f x ∂μ
Module: Mathlib.Order.Filter.AtTopBot.Floor

Goal state before:
pp: case calc_1
K : Type u_1
inst✝³ : Ring K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorSemiring K
a c : K
d n : ℕ
h : ⌈|a|⌉₊ * ⌈|c|⌉₊ ^ n < (n - d)!
⊢ |a| * |c| ^ n ≤ ↑⌈|a|⌉₊ * ↑⌈|c|⌉₊ ^ n

Tactic:
gcongr <;> try first | positivity | apply Nat.le_ceil

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.Quotient

Goal state before:
pp: G : Type u_1
inst✝¹⁵ : Group G
inst✝¹⁴ : MeasurableSpace G
inst✝¹³ : TopologicalSpace G
inst✝¹² : IsTopologicalGroup G
inst✝¹¹ : BorelSpace G
inst✝¹⁰ : PolishSpace G
Γ : Subgroup G
inst✝⁹ : Γ.Normal
inst✝⁸ : T2Space (G ⧸ Γ)
inst✝⁷ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝⁶ : ν.IsMulLeftInvariant
inst✝⁵ : Countable ↥Γ
inst✝⁴ : ν.IsMulRightInvariant
inst✝³ : SigmaFinite ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : SigmaFinite μ
inst✝ : IsFiniteMeasure μ
hasFun : HasFundamentalDomain (↥Γ.op) G ν
h : covolume (↥Γ.op) G ν = μ univ
s : Set G
fund_dom_s : IsFundamentalDomain (↥Γ.op) s ν
finiteCovol : μ univ < ∞
⊢ QuotientMeasureEqMeasurePreimage ν μ

Tactic:
rw [fund_dom_s.covolume_eq_volume] at h

Goal state after:
G : Type u_1
inst✝¹⁵ : Group G
inst✝¹⁴ : MeasurableSpace G
inst✝¹³ : TopologicalSpace G
inst✝¹² : IsTopologicalGroup G
inst✝¹¹ : BorelSpace G
inst✝¹⁰ : PolishSpace G
Γ : Subgroup G
inst✝⁹ : Γ.Normal
inst✝⁸ : T2Space (G ⧸ Γ)
inst✝⁷ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝⁶ : ν.IsMulLeftInvariant
inst✝⁵ : Countable ↥Γ
inst✝⁴ : ν.IsMulRightInvariant
inst✝³ : SigmaFinite ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : SigmaFinite μ
inst✝ : IsFiniteMeasure μ
hasFun : HasFundamentalDomain (↥Γ.op) G ν
s : Set G
h : ν s = μ univ
fund_dom_s : IsFundamentalDomain (↥Γ.op) s ν
finiteCovol : μ univ < ∞
⊢ QuotientMeasureEqMeasurePreimage ν μ
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure ℝ
f : ℝ → E
inst✝ : NullSingletonClass μ
a₀ : ℝ
hf : IntegrableOn f (Ici a₀) μ
⊢ ContinuousOn (fun b ↦ ∫ (x : ℝ) in Ici b, f x ∂μ) (Ici a₀)

Tactic:
simp_rw [integral_Ici_eq_integral_Ioi]

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure ℝ
f : ℝ → E
inst✝ : NullSingletonClass μ
a₀ : ℝ
hf : IntegrableOn f (Ici a₀) μ
⊢ ContinuousOn (fun b ↦ ∫ (x : ℝ) in Ioi b, f x ∂μ) (Ici a₀)
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure ℝ
f : ℝ → E
inst✝ : NullSingletonClass μ
a₀ : ℝ
hf : IntegrableOn f (Ici a₀) μ
⊢ ContinuousOn (fun b ↦ ∫ (x : ℝ) in Ici b, f x ∂μ) (Ici a₀)

Tactic:
simp_rw [integral_Ici_eq_integral_Ioi]
  exact (hf.mono_set Ioi_subset_Ici_self).continuousOn_Ici_primitive_Ioi

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic

Goal state before:
pp: T : ℝ
g : ℝ → ℝ
hg : Periodic g T
h₀ : 0 < ∫ (x : ℝ) in 0..T, g x
hT : 0 < T
h_int : IntervalIntegrable g volume 0 T
⊢ Tendsto (fun n ↦ sSup ((fun t ↦ ∫ (x : ℝ) in 0..t, g x) '' Icc 0 T) + ⌊n / T⌋ • ∫ (x : ℝ) in 0..T, g x) atBot atBot

Tactic:
apply atBot.tendsto_atBot_add_const_left (sSup <| (fun t => ∫ x in 0..t, g x) '' Icc 0 T)

Goal state after:
T : ℝ
g : ℝ → ℝ
hg : Periodic g T
h₀ : 0 < ∫ (x : ℝ) in 0..T, g x
hT : 0 < T
h_int : IntervalIntegrable g volume 0 T
⊢ Tendsto (fun x ↦ ⌊x / T⌋ • ∫ (x : ℝ) in 0..T, g x) atBot atBot
Module: Mathlib.MeasureTheory.Integral.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : StronglyMeasurable f
this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖
this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q))
h2f : ∫⁻ (x : α), ∫⁻ (y : β), ‖f (x, y)‖ₑ ∂ν ∂μ < ∞
⊢ Measurable fun x ↦ ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν

Tactic:
exact h1f.enorm.lintegral_prod_right'

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

Goal state before:
pp: E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → E
y : E
s : Set ℂ
hs : s.Countable
hc : ContinuousOn f (closedBall c R \ {c})
hd : ∀ z ∈ (ball c R \ {c}) \ s, DifferentiableAt ℂ f z
hy : Tendsto f (𝓝[≠] c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
δ : ℝ
δ0 : δ > 0
hδ : ∀ z ∈ closedBall c δ \ {c}, dist (f z) y < ε / (2 * π)
r : ℝ
hr0 : 0 < r
hrδ : r ≤ δ
hrR : r ≤ R
hsub : closedBall c R \ ball c r ⊆ closedBall c R \ {c}
hsub' : ball c R \ closedBall c r ⊆ ball c R \ {c}
hzne : ∀ z ∈ sphere c r, z ≠ c
hc' : ContinuousOn (fun z ↦ (z - c)⁻¹) (sphere c r)
⊢ ‖(∮ (z : ℂ) in C(c, r), (z - c)⁻¹ • f z) - ∮ (z : ℂ) in C(c, r), (z - c)⁻¹ • y‖ =
    ‖∮ (z : ℂ) in C(c, r), (z - c)⁻¹ • f z - (z - c)⁻¹ • y‖

Tactic:
rw [circleIntegral.integral_sub] <;> refine (hc'.smul ?_).circleIntegrable hr0.le

Goal state after:
case hf
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → E
y : E
s : Set ℂ
hs : s.Countable
hc : ContinuousOn f (closedBall c R \ {c})
hd : ∀ z ∈ (ball c R \ {c}) \ s, DifferentiableAt ℂ f z
hy : Tendsto f (𝓝[≠] c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
δ : ℝ
δ0 : δ > 0
hδ : ∀ z ∈ closedBall c δ \ {c}, dist (f z) y < ε / (2 * π)
r : ℝ
hr0 : 0 < r
hrδ : r ≤ δ
hrR : r ≤ R
hsub : closedBall c R \ ball c r ⊆ closedBall c R \ {c}
hsub' : ball c R \ closedBall c r ⊆ ball c R \ {c}
hzne : ∀ z ∈ sphere c r, z ≠ c
hc' : ContinuousOn (fun z ↦ (z - c)⁻¹) (sphere c r)
⊢ ContinuousOn f (sphere c r)

case hg
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → E
y : E
s : Set ℂ
hs : s.Countable
hc : ContinuousOn f (closedBall c R \ {c})
hd : ∀ z ∈ (ball c R \ {c}) \ s, DifferentiableAt ℂ f z
hy : Tendsto f (𝓝[≠] c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
δ : ℝ
δ0 : δ > 0
hδ : ∀ z ∈ closedBall c δ \ {c}, dist (f z) y < ε / (2 * π)
r : ℝ
hr0 : 0 < r
hrδ : r ≤ δ
hrR : r ≤ R
hsub : closedBall c R \ ball c r ⊆ closedBall c R \ {c}
hsub' : ball c R \ closedBall c r ⊆ ball c R \ {c}
hzne : ∀ z ∈ sphere c r, z ≠ c
hc' : ContinuousOn (fun z ↦ (z - c)⁻¹) (sphere c r)
⊢ ContinuousOn (fun z ↦ y) (sphere c r)
Module: Mathlib.Analysis.Complex.Liouville

Goal state before:
pp: F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
c : ℂ
R C : ℝ
f : ℂ → F
hR : 0 < R
hf : DiffContOnCl ℂ f (ball c R)
hC : ∀ z ∈ sphere c R, ‖f z‖ ≤ C
⊢ ‖deriv f c‖ ≤ C / R

Tactic:
by
  simpa using norm_iteratedDeriv_le_of_forall_mem_sphere_norm_le 1 hR hf hC

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

Goal state before:
pp: E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : Nontrivial E
f : E → F
hf : Differentiable ℂ f
c : F
hb : Tendsto f (cocompact E) (𝓝 c)
s : Set E
hs : s ∈ cocompact E
hs_bdd : Bornology.IsBounded (f '' s)
t : Set E
ht : IsCompact t
hts : tᶜ ⊆ s
⊢ Bornology.IsBounded (range f)

Tactic:
apply ht.image hf.continuous |>.isBounded.union hs_bdd |>.subset

Goal state after:
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : Nontrivial E
f : E → F
hf : Differentiable ℂ f
c : F
hb : Tendsto f (cocompact E) (𝓝 c)
s : Set E
hs : s ∈ cocompact E
hs_bdd : Bornology.IsBounded (f '' s)
t : Set E
ht : IsCompact t
hts : tᶜ ⊆ s
⊢ range f ⊆ f '' t ∪ f '' s
Module: Mathlib.Analysis.Complex.Liouville

Goal state before:
pp: E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : Nontrivial E
f : E → F
hf : Differentiable ℂ f
c : F
hb : Tendsto f (cocompact E) (𝓝 c)
s : Set E
hs : s ∈ cocompact E
hs_bdd : Bornology.IsBounded (f '' s)
t : Set E
ht : IsCompact t
hts : tᶜ ⊆ s
⊢ range f ⊆ f '' t ∪ f '' s

Tactic:
simpa [Set.image_union, Set.image_univ] using! Set.image_mono <| calc
      Set.univ = t ∪ tᶜ := t.union_compl_self.symm
      _        ⊆ t ∪ s  := by gcongr

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

Goal state before:
pp: case refine_2
R : Type u_1
S : Type u_2
k : Type u_3
α : Type u_4
inst✝⁵ : Semiring R
inst✝⁴ : Ring S
inst✝³ : Field k
inst✝² : LinearOrder k
inst✝¹ : IsStrictOrderedRing k
f : R →+* S
abv : S → k
inst✝ : IsAbsoluteValue abv
l : Filter α
z : α → S
hz : Tendsto (abv ∘ z) l atTop
⊢ ∀ {p : R[X]},
    0 < p.degree →
      (f p.leadingCoeff ≠ 0 → Tendsto (fun x ↦ abv (eval₂ f (z x) p)) l atTop) →
        f (p * X).leadingCoeff ≠ 0 → Tendsto (fun x ↦ abv (eval₂ f (z x) (p * X))) l atTop

pp: case refine_3
R : Type u_1
S : Type u_2
k : Type u_3
α : Type u_4
inst✝⁵ : Semiring R
inst✝⁴ : Ring S
inst✝³ : Field k
inst✝² : LinearOrder k
inst✝¹ : IsStrictOrderedRing k
f : R →+* S
abv : S → k
inst✝ : IsAbsoluteValue abv
l : Filter α
z : α → S
hz : Tendsto (abv ∘ z) l atTop
⊢ ∀ {p : R[X]} {a : R},
    0 < p.degree →
      (f p.leadingCoeff ≠ 0 → Tendsto (fun x ↦ abv (eval₂ f (z x) p)) l atTop) →
        f (p + C a).leadingCoeff ≠ 0 → Tendsto (fun x ↦ abv (eval₂ f (z x) (p + C a))) l atTop

Tactic:
· intro _ _ ihp hf
    rw [leadingCoeff_mul_X] at hf
    simpa [abv_mul abv] using (ihp hf).atTop_mul_atTop₀ hz

Goal state after:
case refine_3
R : Type u_1
S : Type u_2
k : Type u_3
α : Type u_4
inst✝⁵ : Semiring R
inst✝⁴ : Ring S
inst✝³ : Field k
inst✝² : LinearOrder k
inst✝¹ : IsStrictOrderedRing k
f : R →+* S
abv : S → k
inst✝ : IsAbsoluteValue abv
l : Filter α
z : α → S
hz : Tendsto (abv ∘ z) l atTop
⊢ ∀ {p : R[X]} {a : R},
    0 < p.degree →
      (f p.leadingCoeff ≠ 0 → Tendsto (fun x ↦ abv (eval₂ f (z x) p)) l atTop) →
        f (p + C a).leadingCoeff ≠ 0 → Tendsto (fun x ↦ abv (eval₂ f (z x) (p + C a))) l atTop
Module: Mathlib.Topology.MetricSpace.Ultra.Basic

Goal state before:
pp: case inr
X : Type u_1
inst✝¹ : PseudoMetricSpace X
inst✝ : IsUltrametricDist X
x y z : X
h✝ : dist x y ≠ dist y z
h : dist y z < dist x y
⊢ max (dist x y) (dist y z) ≤ dist x z

Tactic:
rw [max_eq_left h.le, dist_comm x y, dist_comm x z]

Goal state after:
case inr
X : Type u_1
inst✝¹ : PseudoMetricSpace X
inst✝ : IsUltrametricDist X
x y z : X
h✝ : dist x y ≠ dist y z
h : dist y z < dist x y
⊢ dist y x ≤ dist z x
Module: Mathlib.Analysis.Normed.Group.Ultra

Goal state before:
pp: S' : Type u_2
inst✝ : SeminormedGroup S'
h : ∀ (x y : S'), ‖x * y‖ ≤ max ‖x‖ ‖y‖
x y z : S'
⊢ dist x z ≤ max (dist x y) (dist y z)

Tactic:
by
    simpa [dist_eq_norm_inv_mul] using h (x⁻¹ * y) (y⁻¹ * z)

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors

Goal state before:
pp: case cons
σ : Type u_1
R : Type u_3
inst✝² : CommSemiring R
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
ι : Type u_4
w : σ → ℕ
f : ι → MvPowerSeries σ R
a : ι
s : Finset ι
ha : a ∉ s
ih : weightedOrder w (∏ i ∈ s, f i) = ∑ i ∈ s, weightedOrder w (f i)
⊢ weightedOrder w (f a) + weightedOrder w (∏ x ∈ s, f x) = weightedOrder w (f a) + ∑ x ∈ s, weightedOrder w (f x)

Tactic:
ih

Goal state after:
case cons
σ : Type u_1
R : Type u_3
inst✝² : CommSemiring R
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
ι : Type u_4
w : σ → ℕ
f : ι → MvPowerSeries σ R
a : ι
s : Finset ι
ha : a ∉ s
ih : weightedOrder w (∏ i ∈ s, f i) = ∑ i ∈ s, weightedOrder w (f i)
⊢ weightedOrder w (f a) + ∑ i ∈ s, weightedOrder w (f i) = weightedOrder w (f a) + ∑ x ∈ s, weightedOrder w (f x)
Module: Mathlib.RingTheory.PowerSeries.Inverse

Goal state before:
pp: case right
k : Type u_2
inst✝ : Field k
I : Ideal k⟦X⟧
⊢ ∀ (x : k⟦X⟧), Ideal.span {X} ≤ I → x ∉ Ideal.span {X} → x ∈ I → 1 ∈ I

Tactic:
f

Goal state after:
case right
k : Type u_2
inst✝ : Field k
I : Ideal k⟦X⟧
f : k⟦X⟧
⊢ Ideal.span {X} ≤ I → f ∉ Ideal.span {X} → f ∈ I → 1 ∈ I
Module: Mathlib.AlgebraicGeometry.Geometrically.Connected

Goal state before:
pp: ⊢ @GeometricallyConnected = geometrically fun x ↦ ConnectedSpace ↥x

Tactic:
by
  ext; exact geometricallyConnected_iff _

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes

Goal state before:
pp: case trans
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Limits.HasPullbacks C
K : Precoverage C
inst✝ : K.IsStableUnderBaseChange
S : C
F : Cᵒᵖ ⥤ Type u_2
𝒰 : K.ZeroHypercover S
𝒱 : K.ZeroHypercover S
f : Hom K 𝒰 𝒱
H₁ : Presieve.IsSheafFor F (Presieve.ofArrows 𝒰.X 𝒰.f)
H₂ : ∀ {X : C} (f : X ⟶ S), Presieve.IsSeparatedFor F (Presieve.ofArrows (pullback₂ f 𝒰).X (pullback₂ f 𝒰).f)
Y : C
⊢ ∀ ⦃f : Y ⟶ S⦄,
    (Sieve.generate (Presieve.ofArrows 𝒱.X 𝒱.f)).arrows f →
      Presieve.IsSeparatedFor F (Sieve.pullback f (Sieve.generate (Presieve.ofArrows 𝒰.X 𝒰.f))).arrows

Tactic:
f

Goal state after:
case trans
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Limits.HasPullbacks C
K : Precoverage C
inst✝ : K.IsStableUnderBaseChange
S : C
F : Cᵒᵖ ⥤ Type u_2
𝒰 : K.ZeroHypercover S
𝒱 : K.ZeroHypercover S
f✝ : Hom K 𝒰 𝒱
H₁ : Presieve.IsSheafFor F (Presieve.ofArrows 𝒰.X 𝒰.f)
H₂ : ∀ {X : C} (f : X ⟶ S), Presieve.IsSeparatedFor F (Presieve.ofArrows (pullback₂ f 𝒰).X (pullback₂ f 𝒰).f)
Y : C
f : Y ⟶ S
⊢ (Sieve.generate (Presieve.ofArrows 𝒱.X 𝒱.f)).arrows f →
    Presieve.IsSeparatedFor F (Sieve.pullback f (Sieve.generate (Presieve.ofArrows 𝒰.X 𝒰.f))).arrows
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic

Goal state before:
pp: F : Type u
inst✝ : Field F
P : Fin 3 → F
hPz : P z ≠ 0
⊢ P y * ![P x / P z, P y / P z, 1] z = ![P x / P z, P y / P z, 1] y * P z

Tactic:
by linear_combination (norm := (matrix_simp; ring1)) -P y * div_self hPz

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Projective R
⊢ W'.polynomialX = C W'.a₁ * Y * Z - (C 3 * X ^ 2 + C (2 * W'.a₂) * X * Z + C W'.a₄ * Z ^ 2)

Tactic:
rw [polynomialX, polynomial]
  pderiv_simp
  ring1

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point

Goal state before:
pp: F : Type u
inst✝¹ : Field F
W : Projective F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Equation P
hQ : W.Equation Q
hPz : P z ≠ 0
hQz : Q z ≠ 0
hx : P x * Q z ≠ Q x * P z
⊢ W.add P Q =
    W.addZ P Q •
      ![W.toAffine.addX (P x / P z) (Q x / Q z) (W.toAffine.slope (P x / P z) (Q x / Q z) (P y / P z) (Q y / Q z)),
        W.toAffine.addY (P x / P z) (Q x / Q z) (P y / P z)
          (W.toAffine.slope (P x / P z) (Q x / Q z) (P y / P z) (Q y / Q z)),
        1]

Tactic:
add_of_not_equiv <| not_equiv_of_X_ne hx,

Goal state after:
F : Type u
inst✝¹ : Field F
W : Projective F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Equation P
hQ : W.Equation Q
hPz : P z ≠ 0
hQz : Q z ≠ 0
hx : P x * Q z ≠ Q x * P z
⊢ W.addXYZ P Q =
    W.addZ P Q •
      ![W.toAffine.addX (P x / P z) (Q x / Q z) (W.toAffine.slope (P x / P z) (Q x / Q z) (P y / P z) (Q y / Q z)),
        W.toAffine.addY (P x / P z) (Q x / Q z) (P y / P z)
          (W.toAffine.slope (P x / P z) (Q x / Q z) (P y / P z) (Q y / Q z)),
        1]
Module: Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
J : GrothendieckTopology C
X : C
E : J.OneHypercover X
F : Cᵒᵖ ⥤ Type u_2
hF : Presieve.IsSheaf J F
S : Sieve X
h₁ : ∀ (i : E.I₀), Presieve.IsSheafFor F (Sieve.pullback (E.f i) S).arrows
h₂ : ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), Presieve.IsSeparatedFor F (Sieve.pullback (E.p₁ k ≫ E.f i) S).arrows
Y : C
⊢ ∀ (f : Y ⟶ X), Presieve.IsSeparatedFor F (Sieve.pullback f E.sieve₀).arrows

Tactic:
f

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
J : GrothendieckTopology C
X : C
E : J.OneHypercover X
F : Cᵒᵖ ⥤ Type u_2
hF : Presieve.IsSheaf J F
S : Sieve X
h₁ : ∀ (i : E.I₀), Presieve.IsSheafFor F (Sieve.pullback (E.f i) S).arrows
h₂ : ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), Presieve.IsSeparatedFor F (Sieve.pullback (E.p₁ k ≫ E.f i) S).arrows
Y : C
f : Y ⟶ X
⊢ Presieve.IsSeparatedFor F (Sieve.pullback f E.sieve₀).arrows
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point

Goal state before:
pp: case mk.mk
F : Type u
inst✝ : Field F
W : Projective F
P Q : PointClass F
a✝¹ : Fin 3 → F
hP : W.NonsingularLift (Quot.mk (⇑(MulAction.orbitRel Fˣ (Fin 3 → F))) a✝¹)
a✝ : Fin 3 → F
hQ : W.NonsingularLift (Quot.mk (⇑(MulAction.orbitRel Fˣ (Fin 3 → F))) a✝)
⊢ W.NonsingularLift
    (W.addMap (Quot.mk (⇑(MulAction.orbitRel Fˣ (Fin 3 → F))) a✝¹) (Quot.mk (⇑(MulAction.orbitRel Fˣ (Fin 3 → F))) a✝))

Tactic:
exact nonsingular_add hP hQ

Goal state after:
Module: Mathlib.AlgebraicGeometry.Sites.BigZariski

Goal state before:
pp: F : Schemeᵒᵖ ⥤ Type v
ι : Type u_1
inst✝¹ : Small.{u, u_1} ι
inst✝ : Small.{v, u_1} ι
hF : Presieve.IsSheaf Scheme.zariskiTopology F
⊢ PreservesLimitsOfShape (Discrete ι) F

Tactic:
apply (config := { allowSynthFailures := true }) preservesLimitsOfShape_of_discrete
  intro X
  have (i : ι) : Mono (Cofan.inj (Sigma.cocone (Discrete.functor <| unop ∘ X)) i) :=
    inferInstanceAs <| Mono (Sigma.ι _ _)
  refine Presieve.preservesProduct_of_isSheafFor F ?_ initialIsInitial
      (Sigma.cocone (Discrete.functor <| unop ∘ X)) (coproductIsCoproduct' _) ?_ ?_
  · apply hF.isSheafFor
    convert! (⊥_ Scheme).bot_mem_grothendieckTopology
    rw [eq_bot_iff]
    rintro Y f ⟨g, _, _, ⟨i⟩, _⟩
    exact i.elim
  · intro i j
    exact CoproductDisjoint.isPullback_of_isInitial
      (coproductIsCoproduct' <| Discrete.functor <| unop ∘ X) initialIsInitial
  · exact hF.isSheafFor _ (sigmaOpenCover _).mem_grothendieckTopology

Goal state after:
Module: Mathlib.Topology.Category.TopCat.GrothendieckTopology

Goal state before:
pp: case refine_2
X Y : TopCat
𝒰 : precoverage.ZeroHypercover Y
heq : ∀ (i : 𝒰.I₀) (y : ↑(𝒰.X i)), (ConcreteCategory.hom (𝒰.f i)) y = ↑⟨(ConcreteCategory.hom (𝒰.f i)) y, ⋯⟩
x : (i : 𝒰.I₀) → (yoneda.obj X).obj (Opposite.op (𝒰.X i))
hx : Presieve.Arrows.Compatible (yoneda.obj X) 𝒰.f x
i : 𝒰.I₀
x✝ : ↑(Opposite.unop (Opposite.op (𝒰.X i)))
⊢ (ContinuousMap.liftCover (fun i ↦ Set.range ⇑(ConcreteCategory.hom (𝒰.f i)))
        (fun i ↦ { toFun := ⇑(Hom.hom (x i)) ∘ ⇑⋯.toHomeomorph.symm, continuous_toFun := ⋯ }) ⋯ ⋯)
      ↑⟨(ConcreteCategory.hom (𝒰.f i)) x✝, ⋯⟩ =
    (ConcreteCategory.hom (x i)) x✝

Tactic:
ContinuousMap.liftCover_coe

Goal state after:
case refine_2
X Y : TopCat
𝒰 : precoverage.ZeroHypercover Y
heq : ∀ (i : 𝒰.I₀) (y : ↑(𝒰.X i)), (ConcreteCategory.hom (𝒰.f i)) y = ↑⟨(ConcreteCategory.hom (𝒰.f i)) y, ⋯⟩
x : (i : 𝒰.I₀) → (yoneda.obj X).obj (Opposite.op (𝒰.X i))
hx : Presieve.Arrows.Compatible (yoneda.obj X) 𝒰.f x
i : 𝒰.I₀
x✝ : ↑(Opposite.unop (Opposite.op (𝒰.X i)))
⊢ { toFun := ⇑(Hom.hom (x i)) ∘ ⇑⋯.toHomeomorph.symm, continuous_toFun := ⋯ } ⟨(ConcreteCategory.hom (𝒰.f i)) x✝, ⋯⟩ =
    (ConcreteCategory.hom (x i)) x✝