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

Goal state before:
pp: X₁ : Type u₁
inst✝³ : TopologicalSpace X₁
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : NormalSpace X
e : X₁ → X
he : IsClosedEmbedding e
Y : Type v
inst✝ : TopologicalSpace Y
f : C(X₁, Y)
t : Set Y
hf : ∀ (x : X₁), f x ∈ t
ht : TietzeExtension ↑t
g : C(X, ↑t)
hg : g.comp { toFun := e, continuous_toFun := ⋯ } = { toFun := Set.codRestrict (⇑f) t hf, continuous_toFun := ⋯ }
⊢ ({ toFun := Subtype.val, continuous_toFun := ⋯ }.comp g).comp { toFun := e, continuous_toFun := ⋯ } = f

Tactic:
ext x; congrm(($(hg) x : Y))

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

Goal state before:
pp: c : ℂ
r R M : ℝ
f : ℂ → ℂ
r_pos : 0 < |r|
r_lt_R : |r| < |R|
hM : 1 ≤ M
h₁f : AnalyticOnNhd ℂ f (closedBall c |R|)
h₂f : f c ≠ 0
f_bound : ∀ z ∈ sphere c |R|, ‖f z‖ ≤ M
hrR : 1 < |R / r|
jensen :
  circleAverage (fun x ↦ Real.log ‖f x‖) c R =
    ∑ᶠ (u : ℂ), ↑((divisor f (closedBall c |R|)) u) * Real.log (R * ‖c - u‖⁻¹) + Real.log ‖f c‖
integral_bound : circleAverage (fun x ↦ Real.log ‖f x‖) c R ≤ Real.log M
u : ℂ
h1 : u ∈ closedBall c |R|
h2 : u ∈ closedBall c |r|
h3 : u ≠ c
⊢ ‖c - u‖ ≠ 0

Tactic:
simpa [sub_eq_zero] using h3.symm

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Projective

Goal state before:
pp: n : Type u_1
R : Type u_2
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq n
inst✝³ : CommRing R
inst✝² : Nonempty n
F : Type u_3
inst✝¹ : Field F
inst✝ : IsAlgClosed F
r : Fˣ
x : F
hx : (X ^ Fintype.card n - C ↑r).IsRoot x
hx' : x ≠ 0
⊢ ∃ k, k ^ Fintype.card n = r

Tactic:
exact ⟨⟨x, x⁻¹, mul_inv_cancel₀ hx', inv_mul_cancel₀ hx'⟩, by
    simpa [Units.ext_iff, sub_eq_zero] using hx⟩

Goal state after:
Module: Mathlib.Analysis.Complex.UpperHalfPlane.FixedPoints

Goal state before:
pp: g : GL (Fin 2) ℝ
z : ℍ
h : 0 < (↑g).det
⊢ g • z = z ↔ ↑(↑g 1 0) * (↑z * ↑z) + (↑(↑g 1 1) - ↑(↑g 0 0)) * ↑z + -↑(↑g 0 1) = 0

Tactic:
simp [gl_smul_eq_iff_num_eq, σ, h, num, denom]

Goal state after:
g : GL (Fin 2) ℝ
z : ℍ
h : 0 < (↑g).det
⊢ ↑(↑g 0 0) * ↑z + ↑(↑g 0 1) = ↑z * (↑(↑g 1 0) * ↑z + ↑(↑g 1 1)) ↔
    ↑(↑g 1 0) * (↑z * ↑z) + (↑(↑g 1 1) - ↑(↑g 0 0)) * ↑z + -↑(↑g 0 1) = 0
Module: Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction

Goal state before:
pp: g h : GL (Fin 2) ℝ
z : ℍ
u : ℂ
hu✝ : (σ g) ((σ h) ↑z) = u
hu : u.im ≠ 0
hu' : (num h u / denom h u).im ≠ 0
| num g (num h u / denom h u) * denom (g * h) u

Tactic:
rw [num]

Goal state after:
g h : GL (Fin 2) ℝ
z : ℍ
u : ℂ
hu✝ : (σ g) ((σ h) ↑z) = u
hu : u.im ≠ 0
hu' : (num h u / denom h u).im ≠ 0
| (↑(↑g 0 0) * (num h u / denom h u) + ↑(↑g 0 1)) * denom (g * h) u
Module: Mathlib.Analysis.Complex.UpperHalfPlane.FixedPoints

Goal state before:
pp: g : GL (Fin 2) ℝ
⊢ (∀ (z : ℍ), g • z = z) ↔ g ∈ Subgroup.center (GL (Fin 2) ℝ)

Tactic:
by
  constructor
  · intro hg
    by_contra! hgc
    rcases g.det_ne_zero.lt_or_gt with hlt | hgt
    · obtain ⟨ha, hb⟩ := (gl_smul_I_eq_I_iff_of_neg hlt).mp (hg _)
      rw [eq_neg_iff_add_eq_zero, ← Matrix.trace_fin_two] at ha
      rcases eq_or_ne (g 1 0) 0 with hc | hc
      · specialize hg ⟨1 + .I, by simp⟩
        rw [gl_smul_eq_self_iff_re_eq ha hc] at hg
        simp_all
      · have : 0 < 1 + √(-g.val.det) / |g 1 0| := by simp [add_pos, *]
        specialize hg ⟨⟨-g 1 1 / g 1 0, 1 + √(-g.val.det) / |g 1 0|⟩, this⟩
        simp [gl_smul_eq_self_iff_dist_eq hlt ha hc, Complex.dist_eq_re_im, Real.sqrt_sq this.le]
          at hg
    · have := isElliptic_of_exists_smul_eq_self hgt hgc ⟨.I, hg _⟩
      contrapose! hg
      simp [gl_smul_eq_self_iff_eq_fixedPt hgt this, exists_ne]
  · aesop (add simp GeneralLinearGroup.center_eq_range_scalar)

Goal state after:
Module: Mathlib.Topology.TietzeExtension

Goal state before:
pp: case refine_1
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : NormalSpace Y
f : X →ᵇ ℝ
a b : ℝ
e : X → Y
hf : ∀ (x : X), f x ∈ Icc a b
hle : a ≤ b
he : IsClosedEmbedding e
g : Y →ᵇ ℝ
hgf : ‖g‖ = ‖f - const X ((a + b) / 2)‖
hge : ⇑g ∘ e = ⇑(f - const X ((a + b) / 2))
y : Y
⊢ (const Y ((a + b) / 2) + g) y ∈ Icc a b

pp: case refine_2
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : NormalSpace Y
f : X →ᵇ ℝ
a b : ℝ
e : X → Y
hf : ∀ (x : X), f x ∈ Icc a b
hle : a ≤ b
he : IsClosedEmbedding e
g : Y →ᵇ ℝ
hgf : ‖g‖ = ‖f - const X ((a + b) / 2)‖
hge : ⇑g ∘ e = ⇑(f - const X ((a + b) / 2))
⊢ ⇑(const Y ((a + b) / 2) + g) ∘ e = ⇑f

Tactic:
· suffices ‖f - const X ((a + b) / 2)‖ ≤ (b - a) / 2 by
      simpa [Real.Icc_eq_closedBall, add_mem_closedBall_iff_norm] using
        (norm_coe_le_norm g y).trans (hgf.trans_le this)
    refine (norm_le <| div_nonneg (sub_nonneg.2 hle) zero_le_two).2 fun x => ?_
    simpa only [Real.Icc_eq_closedBall] using! hf x

Goal state after:
case refine_2
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : NormalSpace Y
f : X →ᵇ ℝ
a b : ℝ
e : X → Y
hf : ∀ (x : X), f x ∈ Icc a b
hle : a ≤ b
he : IsClosedEmbedding e
g : Y →ᵇ ℝ
hgf : ‖g‖ = ‖f - const X ((a + b) / 2)‖
hge : ⇑g ∘ e = ⇑(f - const X ((a + b) / 2))
⊢ ⇑(const Y ((a + b) / 2) + g) ∘ e = ⇑f
Module: Mathlib.Topology.TietzeExtension

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : NormalSpace Y
inst✝ : Nonempty X
f : X →ᵇ ℝ
e : X → Y
he : IsClosedEmbedding e
inhabited_h : Inhabited X
⊢ ∃ g, (∀ (y : Y), ∃ x₁ x₂, g y ∈ Icc (f x₁) (f x₂)) ∧ ⇑g ∘ e = ⇑f

Tactic:
obtain ⟨a, ha⟩ : ∃ a, IsGLB (range f) a := ⟨_, isGLB_ciInf f.isBounded_range.bddBelow⟩

Goal state after:
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : NormalSpace Y
inst✝ : Nonempty X
f : X →ᵇ ℝ
e : X → Y
he : IsClosedEmbedding e
inhabited_h : Inhabited X
a : ℝ
ha : IsGLB (range ⇑f) a
⊢ ∃ g, (∀ (y : Y), ∃ x₁ x₂, g y ∈ Icc (f x₁) (f x₂)) ∧ ⇑g ∘ e = ⇑f
Module: Mathlib.Analysis.Complex.UpperHalfPlane.FunctionsBoundedAtInfty

Goal state before:
pp: t : ℝ
⊢ 0 < t → t ∈ Set.range im

Tactic:
ht

Goal state after:
t : ℝ
ht : 0 < t
⊢ t ∈ Set.range im
Module: Mathlib.RingTheory.Norm.Transitivity

Goal state before:
pp: S : Type u_2
m : Type u_5
inst✝² : CommRing S
M : Matrix m m S
inst✝¹ : DecidableEq m
k : m
inst✝ : Fintype m
i j : m
lt : ¬(i = k → j = k)
⊢ (M * auxMat M k) i j = 0

Tactic:
Classical.not_imp

Goal state after:
S : Type u_2
m : Type u_5
inst✝² : CommRing S
M : Matrix m m S
inst✝¹ : DecidableEq m
k : m
inst✝ : Fintype m
i j : m
lt : i = k ∧ ¬j = k
⊢ (M * auxMat M k) i j = 0
Module: Mathlib.Analysis.Complex.UpperHalfPlane.Measure

Goal state before:
pp: s : Set ℍ
this :
  MeasurePreserving UpperHalfPlane.coe (Measure.comap UpperHalfPlane.coe volume)
    (volume.restrict (Set.range UpperHalfPlane.coe))
⊢ ∫⁻ (a : ℍ) in s, ↑((1 / NNReal.mk a.im ⋯) ^ 2) ∂Measure.comap UpperHalfPlane.coe volume =
    ∫⁻ (a : ℍ) in s, ↑((1 / ‖(↑a).im‖₊) ^ 2) ∂Measure.comap UpperHalfPlane.coe volume

pp: s : Set ℍ
this :
  MeasurePreserving UpperHalfPlane.coe (Measure.comap UpperHalfPlane.coe volume)
    (volume.restrict (Set.range UpperHalfPlane.coe))
⊢ MeasurableSet (Set.range UpperHalfPlane.coe)

Tactic:
· simp [Real.nnnorm_of_nonneg (im_pos _).le]

Goal state after:
s : Set ℍ
this :
  MeasurePreserving UpperHalfPlane.coe (Measure.comap UpperHalfPlane.coe volume)
    (volume.restrict (Set.range UpperHalfPlane.coe))
⊢ MeasurableSet (Set.range UpperHalfPlane.coe)
Module: Mathlib.Analysis.Complex.UpperHalfPlane.Manifold

Goal state before:
pp: f : ℍ → ℂ
hf : DifferentiableOn ℂ (f ∘ ↑ofComplex) {z | 0 < z.im}
τ : ℍ
hτ : ∃ᶠ (z : ℍ) in 𝓝[≠] τ, f z = 0
this : AnalyticOnNhd ℂ (f ∘ ↑ofComplex) {z | 0 < z.im}
⊢ f = 0

Tactic:
ext w

Goal state after:
f : ℍ → ℂ
hf : DifferentiableOn ℂ (f ∘ ↑ofComplex) {z | 0 < z.im}
τ : ℍ
hτ : ∃ᶠ (z : ℍ) in 𝓝[≠] τ, f z = 0
this : AnalyticOnNhd ℂ (f ∘ ↑ofComplex) {z | 0 < z.im}
w : ℍ
⊢ f w = 0 w
Module: Mathlib.Analysis.Complex.UpperHalfPlane.Metric

Goal state before:
pp: z w : ℍ
⊢ cosh (dist z w / 2) = dist (↑z) ((starRingEnd ℂ) ↑w) / (2 * √(z.im * w.im))

Tactic:
← sq_eq_sq₀,

Goal state after:
z w : ℍ
⊢ cosh (dist z w / 2) ^ 2 = (dist (↑z) ((starRingEnd ℂ) ↑w) / (2 * √(z.im * w.im))) ^ 2

case ha
z w : ℍ
⊢ 0 ≤ cosh (dist z w / 2)

case hb
z w : ℍ
⊢ 0 ≤ dist (↑z) ((starRingEnd ℂ) ↑w) / (2 * √(z.im * w.im))
Module: Mathlib.Analysis.Complex.UpperHalfPlane.Metric

Goal state before:
pp: z w : ℍ
r : ℝ
hr : 0 ≤ r
⊢ dist ↑z ↑w / (2 * √(z.im * w.im)) = sinh (r / 2) ↔ dist ↑z ↑w ^ 2 / (4 * z.im * w.im) = sinh (r / 2) ^ 2

Tactic:
← sq_eq_sq₀,

Goal state after:
z w : ℍ
r : ℝ
hr : 0 ≤ r
⊢ (dist ↑z ↑w / (2 * √(z.im * w.im))) ^ 2 = sinh (r / 2) ^ 2 ↔ dist ↑z ↑w ^ 2 / (4 * z.im * w.im) = sinh (r / 2) ^ 2

case ha
z w : ℍ
r : ℝ
hr : 0 ≤ r
⊢ 0 ≤ dist ↑z ↑w / (2 * √(z.im * w.im))

case hb
z w : ℍ
r : ℝ
hr : 0 ≤ r
⊢ 0 ≤ sinh (r / 2)
Module: Mathlib.Topology.Algebra.Group.Matrix

Goal state before:
pp: n : Type u_1
R : Type u_2
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : CommRing R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalRing R
⊢ Continuous[Units.instTopologicalSpaceUnits, Units.instTopologicalSpaceUnits] ⇑det

Tactic:
by
  simp_rw [Units.continuous_iff, ← map_inv]
  constructor <;> fun_prop

Goal state after:
Module: Mathlib.Topology.Algebra.Group.Matrix

Goal state before:
pp: n : Type u_1
R : Type u_2
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : CommRing R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalRing R
⊢ Continuous[instTopologicalSpace, Units.instTopologicalSpaceUnits] ⇑toGL

Tactic:
by
  simp_rw [Units.continuous_iff, ← map_inv]
  constructor <;> fun_prop

Goal state after:
Module: Mathlib.Analysis.Complex.UpperHalfPlane.Metric

Goal state before:
pp: z : ℍ
r : ℝ
⊢ UpperHalfPlane.coe '' closedBall z r = closedBall (↑(z.center r)) (z.im * Real.sinh r)

Tactic:
ext w

Goal state after:
z : ℍ
r : ℝ
w : ℂ
⊢ w ∈ UpperHalfPlane.coe '' closedBall z r ↔ w ∈ closedBall (↑(z.center r)) (z.im * Real.sinh r)
Module: Mathlib.Analysis.Complex.UpperHalfPlane.Metric

Goal state before:
pp: z : ℍ
r : ℝ
⊢ UpperHalfPlane.coe '' ball z r = ball (↑(z.center r)) (z.im * Real.sinh r)

Tactic:
ext w

Goal state after:
z : ℍ
r : ℝ
w : ℂ
⊢ w ∈ UpperHalfPlane.coe '' ball z r ↔ w ∈ ball (↑(z.center r)) (z.im * Real.sinh r)
Module: Mathlib.Analysis.Complex.UpperHalfPlane.Metric

Goal state before:
pp: z : ℍ
r : ℝ
⊢ UpperHalfPlane.coe '' sphere z r = sphere (↑(z.center r)) (z.im * Real.sinh r)

Tactic:
ext w

Goal state after:
z : ℍ
r : ℝ
w : ℂ
⊢ w ∈ UpperHalfPlane.coe '' sphere z r ↔ w ∈ sphere (↑(z.center r)) (z.im * Real.sinh r)
Module: Mathlib.Topology.Compactification.StoneCech

Goal state before:
pp: case e'_1.e'_3
α : Type u
inst✝ : TopologicalSpace α
x : α
g : Ultrafilter α
gx : ↑g ≤ 𝓝 x
this✝ : ↑(Ultrafilter.map pure g) ≤ 𝓝 g
this : map preStoneCechUnit ↑g ≤ 𝓝 (Quot.mk (fun F G ↦ ∃ x, ↑F ≤ 𝓝 x ∧ ↑G ≤ 𝓝 x) g)
e_1✝ : PreStoneCech α = Quot fun F G ↦ ∃ x, ↑F ≤ 𝓝 x ∧ ↑G ≤ 𝓝 x
⊢ preStoneCechUnit x = Quot.mk (fun F G ↦ ∃ x, ↑F ≤ 𝓝 x ∧ ↑G ≤ 𝓝 x) g

Tactic:
exact Quot.sound ⟨x, pure_le_nhds x, gx⟩

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
Y : Type u_3
inst✝ : TopologicalSpace Y
f : OnePoint X → Y
x : X
⊢ ContinuousAt f ↑x ↔ ContinuousAt (f ∘ some) x

Tactic:
by
  rw [ContinuousAt, nhds_coe_eq, tendsto_map'_iff, ContinuousAt]; rfl

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

Goal state before:
pp: case neg
E : Type u_1
inst✝ : NormedAddCommGroup E
f g : ℂ → E
a : WithTop E
a₀ : E
h : ¬a = ⊤
⊢ ℝ → ℝ

Tactic:
exact circleAverage (log⁺ ‖f · - a.untop₀‖⁻¹) 0

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

Goal state before:
pp: case e_a
E✝ : Type u_1
inst✝² : NormedAddCommGroup E✝
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : ProperSpace E
D₁ D₂ : locallyFinsupp E ℤ
r : ℝ
this : ∀ {A B C D : ℝ}, A + B + (C + D) = A + C + (B + D)
⊢ ∑ᶠ (z : E), (↑(((toClosedBall r) D₁) z) + ↑(((toClosedBall r) D₂) z)) * log (r * ‖z‖⁻¹) =
    ∑ᶠ (z : E), ↑(((toClosedBall r) D₁) z) * log (r * ‖z‖⁻¹) + ∑ᶠ (z : E), ↑(((toClosedBall r) D₂) z) * log (r * ‖z‖⁻¹)

Tactic:
have h₁s : ((D₁.toClosedBall r).support ∪ (D₂.toClosedBall r).support).Finite := by
        apply Set.finite_union.2
        constructor
        <;> apply finiteSupport _ (isCompact_closedBall 0 |r|)

Goal state after:
case e_a
E✝ : Type u_1
inst✝² : NormedAddCommGroup E✝
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : ProperSpace E
D₁ D₂ : locallyFinsupp E ℤ
r : ℝ
this : ∀ {A B C D : ℝ}, A + B + (C + D) = A + C + (B + D)
h₁s : (((toClosedBall r) D₁).support ∪ ((toClosedBall r) D₂).support).Finite
⊢ ∑ᶠ (z : E), (↑(((toClosedBall r) D₁) z) + ↑(((toClosedBall r) D₂) z)) * log (r * ‖z‖⁻¹) =
    ∑ᶠ (z : E), ↑(((toClosedBall r) D₁) z) * log (r * ‖z‖⁻¹) + ∑ᶠ (z : E), ↑(((toClosedBall r) D₂) z) * log (r * ‖z‖⁻¹)
Module: Mathlib.Analysis.Complex.ValueDistribution.Cartan

Goal state before:
pp: f : ℂ → ℂ
R : ℝ
h₁f : Meromorphic f
h₂f : 0 < meromorphicOrderAt f 0
hR : R ≠ 0
⊢ characteristic f ⊤ R = circleAverage (fun x ↦ logCounting f (↑x) R) 0 1

Tactic:
rw [characteristic_top_eq_circleAverage_add_circleAverage h₁f hR]
  simp [circleAverage_log_norm_meromorphicTrailingCoeffAt_of_meromorphicOrderAt_pos h₂f]

Goal state after:
Module: Mathlib.Analysis.ConstantSpeed

Goal state before:
pp: E : Type u_2
inst✝ : PseudoEMetricSpace E
f : ℝ → E
s : Set ℝ
l : ℝ≥0
t : Set ℝ
hfs : ∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → x ≤ y → eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (↑l * (y - x))
hft : ∀ ⦃x : ℝ⦄, x ∈ t → ∀ ⦃y : ℝ⦄, y ∈ t → x ≤ y → eVariationOn f (t ∩ Icc x y) = ENNReal.ofReal (↑l * (y - x))
x : ℝ
hs : IsGreatest s x
ht : IsLeast t x
z : ℝ
zs : z ∈ s
y : ℝ
ys : y ∈ s
zy : z ≤ y
⊢ (s ∪ t) ∩ Icc z y = s ∩ Icc z y

Tactic:
ext w

Goal state after:
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : ℝ → E
s : Set ℝ
l : ℝ≥0
t : Set ℝ
hfs : ∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → x ≤ y → eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (↑l * (y - x))
hft : ∀ ⦃x : ℝ⦄, x ∈ t → ∀ ⦃y : ℝ⦄, y ∈ t → x ≤ y → eVariationOn f (t ∩ Icc x y) = ENNReal.ofReal (↑l * (y - x))
x : ℝ
hs : IsGreatest s x
ht : IsLeast t x
z : ℝ
zs : z ∈ s
y : ℝ
ys : y ∈ s
zy : z ≤ y
w : ℝ
⊢ w ∈ (s ∪ t) ∩ Icc z y ↔ w ∈ s ∩ Icc z y
Module: Mathlib.Topology.Semicontinuity.Lindelof

Goal state before:
pp: X : Type u_1
E : Type u_2
inst✝⁶ : TopologicalSpace X
inst✝⁵ : HereditarilyLindelofSpace X
inst✝⁴ : LinearOrder E
inst✝³ : TopologicalSpace E
inst✝² : OrderClosedTopology E
inst✝¹ : DenselyOrdered E
inst✝ : SeparableSpace E
s : X → E
𝓕 : Set (X → E)
h𝓕_cont : ∀ f ∈ 𝓕, UpperSemicontinuous f
h𝓕 : ∀ (a : X), IsGLB (Function.eval a '' 𝓕) (s a)
D : Set E
D_count : D.Countable
D_dense : Dense D
U : (X → E) → E → Set X := fun f d ↦ {x | f x < d}
U_open : ∀ {f : X → E}, f ∈ 𝓕 → ∀ (d : E), IsOpen (U f d)
d : E
x : X
⊢ x ∈ {x | s x < d} ↔ x ∈ ⋃ f, U (↑f) d

Tactic:
simp [U, isGLB_lt_iff (h𝓕 x)]

Goal state after:
Module: Mathlib.Analysis.ConstantSpeed

Goal state before:
pp: E : Type u_2
inst✝ : PseudoEMetricSpace E
f : ℝ → E
s : Set ℝ
l : ℝ≥0
t : Set ℝ
hfs : ∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → x ≤ y → eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (↑l * (y - x))
hft : ∀ ⦃x : ℝ⦄, x ∈ t → ∀ ⦃y : ℝ⦄, y ∈ t → x ≤ y → eVariationOn f (t ∩ Icc x y) = ENNReal.ofReal (↑l * (y - x))
x : ℝ
hs : IsGreatest s x
ht : IsLeast t x
z : ℝ
zs : z ∈ s
y : ℝ
yt : y ∈ t
zy : z ≤ y
⊢ (s ∪ t) ∩ Icc z y = s ∩ Icc z x ∪ t ∩ Icc x y

Tactic:
ext w

Goal state after:
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : ℝ → E
s : Set ℝ
l : ℝ≥0
t : Set ℝ
hfs : ∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → x ≤ y → eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (↑l * (y - x))
hft : ∀ ⦃x : ℝ⦄, x ∈ t → ∀ ⦃y : ℝ⦄, y ∈ t → x ≤ y → eVariationOn f (t ∩ Icc x y) = ENNReal.ofReal (↑l * (y - x))
x : ℝ
hs : IsGreatest s x
ht : IsLeast t x
z : ℝ
zs : z ∈ s
y : ℝ
yt : y ∈ t
zy : z ≤ y
w : ℝ
⊢ w ∈ (s ∪ t) ∩ Icc z y ↔ w ∈ s ∩ Icc z x ∪ t ∩ Icc x y
Module: Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Asymptotic

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : ProperSpace E
D : locallyFinsupp E ℤ
h : 0 ≤ D
hO : logCounting D =O[atTop] log
C : ℝ
hC : ∀ᶠ (x : ℝ) in atTop, ‖logCounting D x‖ ≤ C * ‖log x‖
N : ℕ
hCN : max C 0 < ↑N
hCN' : C < ↑N
⊢ (support D).Finite

Tactic:
by_contra! hInf

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : ProperSpace E
D : locallyFinsupp E ℤ
h : 0 ≤ D
hO : logCounting D =O[atTop] log
C : ℝ
hC : ∀ᶠ (x : ℝ) in atTop, ‖logCounting D x‖ ≤ C * ‖log x‖
N : ℕ
hCN : max C 0 < ↑N
hCN' : C < ↑N
hInf : (support D).Infinite
⊢ False
Module: Mathlib.Analysis.ConstantSpeed

Goal state before:
pp: E : Type u_2
inst✝ : PseudoEMetricSpace E
f : ℝ → E
s : Set ℝ
l : ℝ≥0
t : Set ℝ
hfs : ∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → x ≤ y → eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (↑l * (y - x))
hft : ∀ ⦃x : ℝ⦄, x ∈ t → ∀ ⦃y : ℝ⦄, y ∈ t → x ≤ y → eVariationOn f (t ∩ Icc x y) = ENNReal.ofReal (↑l * (y - x))
x : ℝ
hs : IsGreatest s x
ht : IsLeast t x
z : ℝ
zt : z ∈ t
y : ℝ
yt : y ∈ t
zy : z ≤ y
⊢ (s ∪ t) ∩ Icc z y = t ∩ Icc z y

Tactic:
ext w

Goal state after:
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : ℝ → E
s : Set ℝ
l : ℝ≥0
t : Set ℝ
hfs : ∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → x ≤ y → eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (↑l * (y - x))
hft : ∀ ⦃x : ℝ⦄, x ∈ t → ∀ ⦃y : ℝ⦄, y ∈ t → x ≤ y → eVariationOn f (t ∩ Icc x y) = ENNReal.ofReal (↑l * (y - x))
x : ℝ
hs : IsGreatest s x
ht : IsLeast t x
z : ℝ
zt : z ∈ t
y : ℝ
yt : y ∈ t
zy : z ≤ y
w : ℝ
⊢ w ∈ (s ∪ t) ∩ Icc z y ↔ w ∈ t ∩ Icc z y
Module: Mathlib.Analysis.Convex.BetweenList

Goal state before:
pp: case refine_1
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
l : List P
hl : l ≠ []
x✝ : ∃ l', l'.SortedLE ∧ map (⇑(lineMap (l.head hl) (l.getLast hl))) l' = l
l' : List R
hl's : l'.SortedLE
hl'l : map (⇑(lineMap (l.head hl) (l.getLast hl))) l' = l
⊢ List.Wbtw R (map (⇑(lineMap (l.head hl) (l.getLast hl))) l')

Tactic:
exact Wbtw.map hl's.wbtw _

Goal state after:
Module: Mathlib.Analysis.Convex.Between

Goal state before:
pp: R : Type u_1
V : Type u_2
inst✝⁴ : Ring R
inst✝³ : PartialOrder R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : IsOrderedRing R
x y : V
⊢ affineSegment R x y = segment R x y

Tactic:
rw [segment_eq_image_lineMap, affineSegment]

Goal state after:
Module: Mathlib.Analysis.Convex.BetweenList

Goal state before:
pp: case refine_1
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
p₁ p₂ : P
l' : List R
hl's : l'.SortedLE
x✝ : ∃ p₁_1 p₂_1 l'_1, l'_1.SortedLE ∧ map (⇑(lineMap p₁_1 p₂_1)) l'_1 = map (⇑(lineMap p₁ p₂)) l'
⊢ List.Wbtw R (map (⇑(lineMap p₁ p₂)) l')

Tactic:
exact Wbtw.map hl's.wbtw _

Goal state after:
Module: Mathlib.Analysis.Convex.BetweenList

Goal state before:
pp: case refine_2.refine_1
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
l : List P
hl : l ≠ []
hp : Pairwise (fun x1 x2 ↦ x1 ≠ x2) l
l' : List R
hl's : l'.SortedLE
hl'l : map (⇑(lineMap (l.head hl) (l.getLast hl))) l' = l
⊢ l'.SortedLT

Tactic:
rw [← hl'l] at hp

Goal state after:
case refine_2.refine_1
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
l : List P
hl : l ≠ []
l' : List R
hp : Pairwise (fun x1 x2 ↦ x1 ≠ x2) (map (⇑(lineMap (l.head hl) (l.getLast hl))) l')
hl's : l'.SortedLE
hl'l : map (⇑(lineMap (l.head hl) (l.getLast hl))) l' = l
⊢ l'.SortedLT
Module: Mathlib.Analysis.Convex.Approximation

Goal state before:
pp: case refine_2
𝕜 : Type u_1
E : Type u_2
φ : E → ℝ
inst✝⁹ : RCLike 𝕜
inst✝⁸ : TopologicalSpace E
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : IsScalarTower ℝ 𝕜 E
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : LocallyConvexSpace ℝ E
inst✝ : HereditarilyLindelofSpace E
hφc : LowerSemicontinuous φ
hφcv : ConvexOn ℝ univ φ
𝓕 : Set (E → ℝ) := {f | f ≤ φ ∧ ∃ l c, f = ⇑re ∘ ⇑l + const E c}
⊢ BddAbove 𝓕

Tactic:
exact (bddAbove_def.2 ⟨φ, fun y hy => hy.1⟩)

Goal state after:
Module: Mathlib.Combinatorics.Hall.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : Fintype β
r : α → β → Prop
inst✝ : DecidableRel r
this : DecidableEq β
r' : α → Finset β := fun a ↦ {b | r a b}
A : Finset α
b : β
⊢ b ∈ {b | ∃ a ∈ A, r a b} ↔ b ∈ A.biUnion r'

Tactic:
simp [r']

Goal state after:
Module: Mathlib.Combinatorics.Hall.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : Fintype β
r : α → β → Prop
inst✝ : DecidableRel r
this : DecidableEq β
r' : α → Finset β := fun a ↦ {b | r a b}
h : ∀ (A : Finset α), {b | ∃ a ∈ A, r a b} = A.biUnion r'
⊢ ∀ (f : α → β) (x : α), r x (f x) ↔ f x ∈ r' x

Tactic:
simp [r']

Goal state after:
Module: Mathlib.Analysis.Convex.DoublyStochasticMatrix

Goal state before:
pp: R : Type u_1
n : Type u_2
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : Semifield R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
M : Matrix n n R
hs : 0 ≤ 0
h₁ : ∀ (i j : n), 0 ≤ M i j
a✝ : ∀ (j : n), ∑ i, M i j = 0
i : n
h₂ : ∑ j, M i j = 0
j : n
⊢ M i j = 0 i j

Tactic:
rw [sum_eq_zero_iff_of_nonneg (by simp [h₁ i])] at h₂

Goal state after:
R : Type u_1
n : Type u_2
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : Semifield R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
M : Matrix n n R
hs : 0 ≤ 0
h₁ : ∀ (i j : n), 0 ≤ M i j
a✝ : ∀ (j : n), ∑ i, M i j = 0
i : n
h₂ : ∀ i_1 ∈ univ, M i i_1 = 0
j : n
⊢ M i j = 0 i j
Module: Mathlib.Analysis.Convex.Body

Goal state before:
pp: case a
V : Type u_1
inst✝² : TopologicalSpace V
inst✝¹ : AddCommGroup V
inst✝ : Module ℝ V
K : ConvexBody V
h_symm : ∀ x ∈ K, -x ∈ K
x : V
hx : x ∈ ↑K
⊢ 0 < 1 / 2

pp: case a
V : Type u_1
inst✝² : TopologicalSpace V
inst✝¹ : AddCommGroup V
inst✝ : Module ℝ V
K : ConvexBody V
h_symm : ∀ x ∈ K, -x ∈ K
x : V
hx : x ∈ ↑K
⊢ 0 < 1 / 2

pp: case a
V : Type u_1
inst✝² : TopologicalSpace V
inst✝¹ : AddCommGroup V
inst✝ : Module ℝ V
K : ConvexBody V
h_symm : ∀ x ∈ K, -x ∈ K
x : V
hx : x ∈ ↑K
⊢ 1 / 2 + 1 / 2 = 1

Tactic:
all_goals linarith

Goal state after:
Module: Mathlib.Analysis.Convex.Cone.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
inst✝⁹ : AddCommMonoid E
inst✝⁸ : TopologicalSpace E
inst✝⁷ : Semifield 𝕜
inst✝⁶ : LinearOrder 𝕜
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : OrderTopology 𝕜
inst✝² : DenselyOrdered 𝕜
inst✝¹ : NoMaxOrder 𝕜
inst✝ : ContinuousSMul 𝕜 E
C : ConvexCone 𝕜 E
hSclos : IsClosed ↑C
x : E
hx : x ∈ ↑C
f : 𝕜 → E := fun x_1 ↦ x_1 • x
w✝ : 𝕜
h : w✝ ∈ Ioi 0
⊢ f w✝ ∈ ↑C

Tactic:
exact C.smul_mem h hx

Goal state after:
Module: Mathlib.Analysis.Convex.Caratheodory

Goal state before:
pp: 𝕜 : Type u_1
E : Type u
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
x : E
hx : x ∈ (convexHull 𝕜) s
⊢ ((convexHull 𝕜) ↑(minCardFinsetOfMemConvexHull hx)).Nonempty

Tactic:
exact ⟨x, mem_minCardFinsetOfMemConvexHull hx⟩

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.TensorProduct

Goal state before:
pp: R : Type u_1
inst✝¹⁰ : CommRing R
inst✝⁹ : LinearOrder R
inst✝⁸ : IsStrictOrderedRing R
G : Type u_2
inst✝⁷ : AddCommGroup G
inst✝⁶ : Module R G
H : Type u_3
inst✝⁵ : AddCommGroup H
inst✝⁴ : Module R H
G' : Type u_4
H' : Type u_5
inst✝³ : AddCommGroup G'
inst✝² : Module R G'
inst✝¹ : AddCommGroup H'
inst✝ : Module R H'
f : G →ₗ[R] G'
g : H →ₗ[R] H'
C₁ : PointedCone R G
C₂ : PointedCone R H
⊢ map (TensorProduct.map f g) (C₁.maxTensorProduct C₂) ≤ (map f C₁).maxTensorProduct (map g C₂)

Tactic:
rintro _ ⟨w, hw, rfl⟩
  simp only [SetLike.mem_coe, mem_maxTensorProduct] at hw ⊢
  intro φ hφ ψ hψ
  have h_eq : ((dualDistrib R G' H') (φ ⊗ₜ[R] ψ)).comp (TensorProduct.map f g) =
      ((dualDistrib R G H) ((φ.comp f) ⊗ₜ[R] (ψ.comp g))) :=
    TensorProduct.ext' fun x y ↦ by simp [map_tmul]
  convert! hw (φ.comp f) (fun x hx ↦ hφ ⟨x, hx, rfl⟩) (ψ.comp g) (fun y hy ↦ hψ ⟨y, hy, rfl⟩)
  exact DFunLike.congr_fun h_eq w

Goal state after: