mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : TopologicalSpace X
μ : Measure ℝ
inst✝¹ : NullSingletonClass μ
inst✝ : IsLocallyFiniteMeasure μ
f : X → ℝ → E
a₀ : ℝ
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
q : X
b₀ ε : ℝ
εpos : ε > 0
a : ℝ
a_lt : a < a₀ ∧ a < b₀
b : ℝ
lt_b : a₀ < b ∧ b₀ < b
this✝ : IsCompact ({q} ×ˢ Icc a b)
M : ℝ
hM : M ∈ upperBounds ((fun x ↦ ‖Function.uncurry f x‖) '' {q} ×ˢ Icc a b)
δ : ℝ
δpos : 0 < δ
hδ : δ < 1
h'δ : Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
h''δ : (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
v : Set X
v_mem : v ∈ 𝓝[univ] q
this : v ×ˢ Ioo (b₀ - δ) (b₀ + δ) ∈ 𝓝 (q, b₀)
p : X
s : ℝ
hp : p ∈ v
hs : s ∈ Ioo (b₀ - δ) (b₀ + δ)
hv : ∀ p ∈ v, ∀ x ∈ Icc a b, ‖f p x - f q x‖ < δ
J : ∀ (r : X) (u v : ℝ), IntervalIntegrable (f r) μ u v
⊢ ∫ (t : ℝ) in Icc (b₀ - δ) (b₀ + δ), M + 1 ∂μ + ∫ (_t : ℝ) in Icc a b, δ ∂μ =
    (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b)

Tactic:
by simp [mul_comm]

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

Goal state before:
pp: R : ℝ
z w : ℂ
hw : w ∈ ball z R
⊢ ∀ (θ : ℝ), circleMap z R θ - w ≠ 0

Tactic:
sub_ne_zero

Goal state after:
R : ℝ
z w : ℂ
hw : w ∈ ball z R
⊢ ∀ (θ : ℝ), circleMap z R θ ≠ w
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic

Goal state before:
pp: case negSucc
E : Type u_1
inst✝¹ : NormedAddCommGroup E
f : ℝ → E
T : ℝ
inst✝ : NormedSpace ℝ E
hf : Periodic f T
t : ℝ
h_int : ∀ (t₁ t₂ : ℝ), IntervalIntegrable f volume t₁ t₂
this : ∀ (m : ℕ), ∫ (x : ℝ) in 0..m • T, f x = m • ∫ (x : ℝ) in 0..T, f x
n : ℕ
h₀ : Int.negSucc n • T + (n + 1) • T = 0
⊢ ∫ (x : ℝ) in 0..(n + 1) • T, f x = (n + 1) • ∫ (x : ℝ) in (n + 1) • T..T + (n + 1) • T, f x

Tactic:
this (n + 1),

Goal state after:
case negSucc
E : Type u_1
inst✝¹ : NormedAddCommGroup E
f : ℝ → E
T : ℝ
inst✝ : NormedSpace ℝ E
hf : Periodic f T
t : ℝ
h_int : ∀ (t₁ t₂ : ℝ), IntervalIntegrable f volume t₁ t₂
this : ∀ (m : ℕ), ∫ (x : ℝ) in 0..m • T, f x = m • ∫ (x : ℝ) in 0..T, f x
n : ℕ
h₀ : Int.negSucc n • T + (n + 1) • T = 0
⊢ (n + 1) • ∫ (x : ℝ) in 0..T, f x = (n + 1) • ∫ (x : ℝ) in (n + 1) • T..T + (n + 1) • T, f x
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: case mp
c : ℂ
R : ℝ
n : ℤ
hR : R ≠ 0
hn : n < 0
θ : ℝ
hθ : θ ∈ [[0, 2 * π]]
f : ℝ → ℂ := fun θ' ↦ circleMap c R θ' - circleMap c R θ
this : ∀ᶠ (θ' : ℝ) in 𝓝[≠] θ, f θ' ∈ ball 0 1 \ {0}
θ' : ℝ
hθ' : f θ' ∈ ball 0 1 \ {0}
⊢ ‖(circleMap c R θ' - circleMap c R θ)⁻¹‖ ≤ |R|⁻¹ * ‖(circleMap 0 R θ' * I) • (circleMap c R θ' - circleMap c R θ) ^ n‖

Tactic:
set x := ‖f θ'‖

Goal state after:
case mp
c : ℂ
R : ℝ
n : ℤ
hR : R ≠ 0
hn : n < 0
θ : ℝ
hθ : θ ∈ [[0, 2 * π]]
f : ℝ → ℂ := fun θ' ↦ circleMap c R θ' - circleMap c R θ
this : ∀ᶠ (θ' : ℝ) in 𝓝[≠] θ, f θ' ∈ ball 0 1 \ {0}
θ' : ℝ
hθ' : f θ' ∈ ball 0 1 \ {0}
x : ℝ := ‖f θ'‖
⊢ ‖(circleMap c R θ' - circleMap c R θ)⁻¹‖ ≤ |R|⁻¹ * ‖(circleMap 0 R θ' * I) • (circleMap c R θ' - circleMap c R θ) ^ n‖
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: case hF
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
ι : Type u_2
f : ι → ℂ → E
g : ℂ → E
c : ℂ
R : ℝ
l : Filter ι
inst✝ : l.IsCountablyGenerated
hR : 0 ≤ R
hf : ∀ᶠ (i : ι) in l, ContinuousOn (f i) (sphere c R)
h : TendstoUniformlyOn f g l (sphere c R)
⊢ ∀ᶠ (i : ι) in l, ContinuousOn (fun x ↦ deriv (circleMap c R) x • (fun z ↦ f i z) (circleMap c R x)) [[0, 2 * π]]

pp: case h_lim
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
ι : Type u_2
f : ι → ℂ → E
g : ℂ → E
c : ℂ
R : ℝ
l : Filter ι
inst✝ : l.IsCountablyGenerated
hR : 0 ≤ R
hf : ∀ᶠ (i : ι) in l, ContinuousOn (f i) (sphere c R)
h : TendstoUniformlyOn f g l (sphere c R)
⊢ TendstoUniformlyOn (fun n x ↦ deriv (circleMap c R) x • (fun z ↦ f n z) (circleMap c R x))
    (fun x ↦ deriv (circleMap c R) x • (fun z ↦ g z) (circleMap c R x)) l [[0, 2 * π]]

Tactic:
· refine hf.mono fun i hi ↦ .smul ?_ (hi.comp ?_ ?_)
    · rw [funext (deriv_circleMap _ _)]
      fun_prop
    · fun_prop
    · simp [hR, MapsTo]

Goal state after:
case h_lim
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
ι : Type u_2
f : ι → ℂ → E
g : ℂ → E
c : ℂ
R : ℝ
l : Filter ι
inst✝ : l.IsCountablyGenerated
hR : 0 ≤ R
hf : ∀ᶠ (i : ι) in l, ContinuousOn (f i) (sphere c R)
h : TendstoUniformlyOn f g l (sphere c R)
⊢ TendstoUniformlyOn (fun n x ↦ deriv (circleMap c R) x • (fun z ↦ f n z) (circleMap c R x))
    (fun x ↦ deriv (circleMap c R) x • (fun z ↦ g z) (circleMap c R x)) l [[0, 2 * π]]
Module: Mathlib.Analysis.BoxIntegral.DivergenceTheorem

Goal state before:
pp: case refine_3
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
f : (Fin (n + 1) → ℝ) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x
Hd : ∀ x ∈ Box.Icc I \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x
i : Fin (n + 1)
Hc : ContinuousOn f (Box.Icc I)
fI : ℝ → Box (Fin n) → E := fun y J ↦ integral J GP (fun x ↦ f (i.insertNth y x)) BoxAdditiveMap.volume
fb : ↑(Set.Icc (I.lower i) (I.upper i)) → Fin n →ᵇᵃ[↑(I.face i)] E := fun x ↦ ⋯.toBoxAdditive
F : Fin (n + 1) →ᵇᵃ[↑I] E := BoxAdditiveMap.upperSubLower I i fI fb ⋯
⊢ ∀ (c : ℝ≥0),
    ∀ x ∈ Box.Icc I ∩ s,
      ∀ ε > 0,
        ∃ δ > 0,
          ∀ J ≤ I,
            Box.Icc J ⊆ closedBall x δ →
              x ∈ Box.Icc J →
                (GP.bDistortion = true → J.distortion ≤ c) →
                  dist ((BoxAdditiveMap.volume J) ((f' x) (Pi.single i 1))) (F J) ≤ ε

Tactic:
intro c

Goal state after:
case refine_3
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
f : (Fin (n + 1) → ℝ) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x
Hd : ∀ x ∈ Box.Icc I \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x
i : Fin (n + 1)
Hc : ContinuousOn f (Box.Icc I)
fI : ℝ → Box (Fin n) → E := fun y J ↦ integral J GP (fun x ↦ f (i.insertNth y x)) BoxAdditiveMap.volume
fb : ↑(Set.Icc (I.lower i) (I.upper i)) → Fin n →ᵇᵃ[↑(I.face i)] E := fun x ↦ ⋯.toBoxAdditive
F : Fin (n + 1) →ᵇᵃ[↑I] E := BoxAdditiveMap.upperSubLower I i fI fb ⋯
c : ℝ≥0
⊢ ∀ x ∈ Box.Icc I ∩ s,
    ∀ ε > 0,
      ∃ δ > 0,
        ∀ J ≤ I,
          Box.Icc J ⊆ closedBall x δ →
            x ∈ Box.Icc J →
              (GP.bDistortion = true → J.distortion ≤ c) →
                dist ((BoxAdditiveMap.volume J) ((f' x) (Pi.single i 1))) (F J) ≤ ε
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: case inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ≥0
n : ℕ
hR : R ^ n = 0
⊢ ((2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, ‖f (circleMap c (↑R) θ)‖) * ↑(R⁻¹ ^ n) * 0 ≤
    (2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, ‖f (circleMap c (↑R) θ)‖

Tactic:
mul_zero

Goal state after:
case inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ≥0
n : ℕ
hR : R ^ n = 0
⊢ 0 ≤ (2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, ‖f (circleMap c (↑R) θ)‖
Module: Mathlib.MeasureTheory.Integral.Prod

Goal state before:
pp: case pos.h_ind
α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝³ : NormedAddCommGroup E
inst✝² : SFinite ν
inst✝¹ : NormedSpace ℝ E
inst✝ : SFinite μ
hE : CompleteSpace E
⊢ ∀ (c : E) ⦃s : Set (α × β)⦄,
    MeasurableSet s →
      (μ.prod ν) s < ∞ →
        ∫ (z : α × β), s.indicator (fun x ↦ c) z ∂μ.prod ν = ∫ (x : α), ∫ (y : β), s.indicator (fun x ↦ c) (x, y) ∂ν ∂μ

Tactic:
intro c

Goal state after:
case pos.h_ind
α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝³ : NormedAddCommGroup E
inst✝² : SFinite ν
inst✝¹ : NormedSpace ℝ E
inst✝ : SFinite μ
hE : CompleteSpace E
c : E
⊢ ∀ ⦃s : Set (α × β)⦄,
    MeasurableSet s →
      (μ.prod ν) s < ∞ →
        ∫ (z : α × β), s.indicator (fun x ↦ c) z ∂μ.prod ν = ∫ (x : α), ∫ (y : β), s.indicator (fun x ↦ c) (x, y) ∂ν ∂μ
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: c w : ℂ
R : ℝ
hw : w ∈ ball c R
hR : 0 < R
H : ∀ (n : ℕ), n ≠ 0 → ∮ (z : ℂ) in C(c, R), (z - c) ^ (-↑n - 1) = 0
this : ∮ (z : ℂ) in C(c, R), ((w - c) / (z - c)) ^ 0 * (z - c)⁻¹ = 2 * ↑π * I
n : ℕ
hn : n ≠ 0
⊢ (w - c) ^ n * 0 = 0

pp: c w : ℂ
R : ℝ
hw : w ∈ ball c R
hR : 0 < R
H : ∀ (n : ℕ), n ≠ 0 → ∮ (z : ℂ) in C(c, R), (z - c) ^ (-↑n - 1) = 0
this : ∮ (z : ℂ) in C(c, R), ((w - c) / (z - c)) ^ 0 * (z - c)⁻¹ = 2 * ↑π * I
n : ℕ
hn : n ≠ 0
⊢ ∀ z ∈ sphere c R, (z - c)⁻¹ ^ n * (z - c)⁻¹ = (z - c) ^ (-↑n - 1)

Tactic:
mul_zero

Goal state after:
c w : ℂ
R : ℝ
hw : w ∈ ball c R
hR : 0 < R
H : ∀ (n : ℕ), n ≠ 0 → ∮ (z : ℂ) in C(c, R), (z - c) ^ (-↑n - 1) = 0
this : ∮ (z : ℂ) in C(c, R), ((w - c) / (z - c)) ^ 0 * (z - c)⁻¹ = 2 * ↑π * I
n : ℕ
hn : n ≠ 0
⊢ 0 = 0

c w : ℂ
R : ℝ
hw : w ∈ ball c R
hR : 0 < R
H : ∀ (n : ℕ), n ≠ 0 → ∮ (z : ℂ) in C(c, R), (z - c) ^ (-↑n - 1) = 0
this : ∮ (z : ℂ) in C(c, R), ((w - c) / (z - c)) ^ 0 * (z - c)⁻¹ = 2 * ↑π * I
n : ℕ
hn : n ≠ 0
⊢ ∀ z ∈ sphere c R, (z - c)⁻¹ ^ n * (z - c)⁻¹ = (z - c) ^ (-↑n - 1)
Module: Mathlib.Analysis.BoxIntegral.DivergenceTheorem

Goal state before:
pp: case refine_4
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
f : (Fin (n + 1) → ℝ) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x
Hd : ∀ x ∈ Box.Icc I \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x
i : Fin (n + 1)
Hc : ContinuousOn f (Box.Icc I)
fI : ℝ → Box (Fin n) → E := fun y J ↦ integral J GP (fun x ↦ f (i.insertNth y x)) BoxAdditiveMap.volume
fb : ↑(Set.Icc (I.lower i) (I.upper i)) → Fin n →ᵇᵃ[↑(I.face i)] E := fun x ↦ ⋯.toBoxAdditive
F : Fin (n + 1) →ᵇᵃ[↑I] E := BoxAdditiveMap.upperSubLower I i fI fb ⋯
⊢ ∀ (c : ℝ≥0),
    ∀ x ∈ Box.Icc I \ s,
      ∀ ε > 0,
        ∃ δ > 0,
          ∀ J ≤ I,
            Box.Icc J ⊆ closedBall x δ →
              x ∈ Box.Icc J →
                (GP.bDistortion = true → J.distortion ≤ c) →
                  dist ((BoxAdditiveMap.volume J) ((f' x) (Pi.single i 1))) (F J) ≤
                    ε * (volume.toBoxAdditive.restrict ↑I ⋯) J

Tactic:
intro c

Goal state after:
case refine_4
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
f : (Fin (n + 1) → ℝ) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x
Hd : ∀ x ∈ Box.Icc I \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x
i : Fin (n + 1)
Hc : ContinuousOn f (Box.Icc I)
fI : ℝ → Box (Fin n) → E := fun y J ↦ integral J GP (fun x ↦ f (i.insertNth y x)) BoxAdditiveMap.volume
fb : ↑(Set.Icc (I.lower i) (I.upper i)) → Fin n →ᵇᵃ[↑(I.face i)] E := fun x ↦ ⋯.toBoxAdditive
F : Fin (n + 1) →ᵇᵃ[↑I] E := BoxAdditiveMap.upperSubLower I i fI fb ⋯
c : ℝ≥0
⊢ ∀ x ∈ Box.Icc I \ s,
    ∀ ε > 0,
      ∃ δ > 0,
        ∀ J ≤ I,
          Box.Icc J ⊆ closedBall x δ →
            x ∈ Box.Icc J →
              (GP.bDistortion = true → J.distortion ≤ c) →
                dist ((BoxAdditiveMap.volume J) ((f' x) (Pi.single i 1))) (F J) ≤
                  ε * (volume.toBoxAdditive.restrict ↑I ⋯) J
Module: Mathlib.Analysis.Complex.Liouville

Goal state before:
pp: F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : ℂ → F
hf : Differentiable ℂ f
hb : Bornology.IsBounded (range f)
⊢ ∀ (c : ℂ), deriv f c = 0

Tactic:
intro c

Goal state after:
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : ℂ → F
hf : Differentiable ℂ f
hb : Bornology.IsBounded (range f)
c : ℂ
⊢ deriv f c = 0
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.Defs

Goal state before:
pp: σ : Type u_5
R : Type u_6
inst✝² : CommSemiring R
inst✝¹ : Fintype σ
inst✝ : DecidableEq σ
⊢ hsymm σ R 1 = ∑ i, X i

Tactic:
symm
  apply Fintype.sum_equiv oneEquiv
  simp only [oneEquiv_apply, Multiset.map_singleton, Multiset.prod_singleton, implies_true]

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 w : ℂ
f : ℂ → E
s : Set ℂ
hs : s.Countable
R : ℝ≥0
hw : w ∈ ball c ↑R
hc : ContinuousOn f (closedBall c ↑R)
hd : ∀ x ∈ ball c ↑R \ s, DifferentiableAt ℂ f x
hR : 0 < ↑R
this : w ∈ closure (ball c ↑R \ s)
⊢ (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - w)⁻¹ • f z = f w

Tactic:
have A : ContinuousAt (fun w => (2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - w)⁻¹ • f z) w := by
      have := hasFPowerSeriesOn_cauchy_integral
        ((hc.mono sphere_subset_closedBall).circleIntegrable R.coe_nonneg) hR
      refine this.continuousOn.continuousAt (Metric.isOpen_eball.mem_nhds ?_)
      rwa [Metric.eball_coe]

Goal state after:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c w : ℂ
f : ℂ → E
s : Set ℂ
hs : s.Countable
R : ℝ≥0
hw : w ∈ ball c ↑R
hc : ContinuousOn f (closedBall c ↑R)
hd : ∀ x ∈ ball c ↑R \ s, DifferentiableAt ℂ f x
hR : 0 < ↑R
this : w ∈ closure (ball c ↑R \ s)
A : ContinuousAt (fun w ↦ (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - w)⁻¹ • f z) w
⊢ (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - w)⁻¹ • f z = f w
Module: Mathlib.RingTheory.Polynomial.Vieta

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
s : Multiset R
k : ℕ
x : Multiset R
hx : x ∈ powersetCard k s
⊢ (prod ∘ map Neg.neg) x = (map (Function.const R (-1)) x).prod * (map (fun x ↦ x) x).prod

Tactic:
rw [← prod_map_mul, map_congr rfl, Function.comp_apply]

Goal state after:
R : Type u_1
inst✝ : CommRing R
s : Multiset R
k : ℕ
x : Multiset R
hx : x ∈ powersetCard k s
⊢ ∀ x_1 ∈ x, Function.const R (-1) x_1 * x_1 = -x_1
Module: Mathlib.MeasureTheory.Integral.DivergenceTheorem

Goal state before:
pp: E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f f' : ℝ → E
a b : ℝ
hle : a ≤ b
s : Set ℝ
hs : s.Countable
Hc : ContinuousOn f (Set.Icc a b)
Hd : ∀ x ∈ Set.Ioo a b \ s, HasDerivAt f (f' x) x
Hi : IntervalIntegrable f' volume a b
e : ℝ ≃L[ℝ] Fin 1 → ℝ := (ContinuousLinearEquiv.funUnique (Fin 1) ℝ ℝ).symm
F' : ℝ → ℝ →L[ℝ] E := fun x ↦ smulRight 1 (f' x)
⊢ ∫ (x : ℝ) in a..b, f' x = f b - f a

Tactic:
calc
    ∫ x in a..b, f' x = ∫ x in Icc a b, f' x := by
      rw [intervalIntegral.integral_of_le hle, setIntegral_congr_set Ioc_ae_eq_Icc]
    _ = ∑ i : Fin 1,
          ((∫ x in Icc (e a ∘ i.succAbove) (e b ∘ i.succAbove),
              f (e.symm <| i.insertNth (e b i) x)) -
            ∫ x in Icc (e a ∘ i.succAbove) (e b ∘ i.succAbove),
              f (e.symm <| i.insertNth (e a i) x)) := by
      simp only [← interior_Icc] at Hd
      refine
        integral_divergence_of_hasFDerivAt_off_countable_of_equiv e ?_ ?_ (fun _ => f)
          (fun _ => F') s hs a b hle (fun _ => Hc) (fun x hx _ => Hd x hx) _ ?_ ?_
      · exact fun x y => (OrderIso.funUnique (Fin 1) ℝ).symm.le_iff_le
      · exact (volume_preserving_funUnique (Fin 1) ℝ).symm _
      · simp [F', e]
      · rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hle] at Hi
        exact Hi.congr_set_ae Ioc_ae_eq_Icc.symm
    _ = f b - f a := by
      simp [e, Subsingleton.elim (const (Fin 0) _) isEmptyElim, volume_pi,
        Measure.pi_of_empty fun _ : Fin 0 ↦ _]

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

Goal state before:
pp: case inr
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f f' : ℝ → E
a b : ℝ
s : Set ℝ
hs : s.Countable
Hi : IntervalIntegrable f' volume a b
hab : b ≤ a
Hc : ContinuousOn f (Set.Icc b a)
Hd : ∀ x ∈ Set.Ioo b a \ s, HasDerivAt f (f' x) x
⊢ -∫ (x : ℝ) in b..a, f' x = f b - f a

Tactic:
neg_eq_iff_eq_neg,

Goal state after:
case inr
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f f' : ℝ → E
a b : ℝ
s : Set ℝ
hs : s.Countable
Hi : IntervalIntegrable f' volume a b
hab : b ≤ a
Hc : ContinuousOn f (Set.Icc b a)
Hd : ∀ x ∈ Set.Ioo b a \ s, HasDerivAt f (f' x) x
⊢ ∫ (x : ℝ) in b..a, f' x = -(f b - f a)
Module: Mathlib.Analysis.Complex.Polynomial.Basic

Goal state before:
pp: p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
a : Finset ℂ := (p.rootSet ℂ).toFinset
b : Finset ℂ := Finset.image (⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)) (p.rootSet ℝ).toFinset
c : Finset ℂ :=
  Finset.image (fun a ↦ ↑a) ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support
ha : ∀ (z : ℂ), z ∈ a ↔ (aeval z) p = 0
hb : ∀ (z : ℂ), z ∈ b ↔ (aeval z) p = 0 ∧ z.im = 0
w : ↑(p.rootSet ℂ)
⊢ ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))) w = w ↔ (↑w).im = 0

Tactic:
Subtype.ext_iff,

Goal state after:
p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
a : Finset ℂ := (p.rootSet ℂ).toFinset
b : Finset ℂ := Finset.image (⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)) (p.rootSet ℝ).toFinset
c : Finset ℂ :=
  Finset.image (fun a ↦ ↑a) ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support
ha : ∀ (z : ℂ), z ∈ a ↔ (aeval z) p = 0
hb : ∀ (z : ℂ), z ∈ b ↔ (aeval z) p = 0 ∧ z.im = 0
w : ↑(p.rootSet ℂ)
⊢ ↑(((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))) w) = ↑w ↔ (↑w).im = 0
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Embeddings

Goal state before:
pp: K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
A : Type u_2
inst✝² : NormedField A
inst✝¹ : IsAlgClosed A
inst✝ : NormedAlgebra ℚ A
x : K
hx₀ : x ≠ 0
hxi : IsIntegral ℤ x
hx : ∀ (φ : K →+* A), ‖φ x‖ ≤ 1
a b : ℕ
habne : a ≠ b
h : x ^ a = x ^ b
⊢ ∃ n, ∃ (_ : 0 < n), x ^ n = 1

Tactic:
wlog hlt : b < a

Goal state after:
case inr
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
A : Type u_2
inst✝² : NormedField A
inst✝¹ : IsAlgClosed A
inst✝ : NormedAlgebra ℚ A
x : K
hx₀ : x ≠ 0
hxi : IsIntegral ℤ x
hx : ∀ (φ : K →+* A), ‖φ x‖ ≤ 1
a b : ℕ
habne : a ≠ b
h : x ^ a = x ^ b
this :
  ∀ (K : Type u_1) [inst : Field K] [NumberField K] (A : Type u_2) [inst_2 : NormedField A] [IsAlgClosed A]
    [NormedAlgebra ℚ A] {x : K},
    x ≠ 0 →
      IsIntegral ℤ x →
        (∀ (φ : K →+* A), ‖φ x‖ ≤ 1) → ∀ (a b : ℕ), a ≠ b → x ^ a = x ^ b → b < a → ∃ n, ∃ (_ : 0 < n), x ^ n = 1
hlt : ¬b < a
⊢ ∃ n, ∃ (_ : 0 < n), x ^ n = 1

K✝ : Type u_1
inst✝⁶ : Field K✝
A✝ : Type u_2
inst✝⁵ : NormedField A✝
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
A : Type u_2
inst✝² : NormedField A
inst✝¹ : IsAlgClosed A
inst✝ : NormedAlgebra ℚ A
x : K
hx₀ : x ≠ 0
hxi : IsIntegral ℤ x
hx : ∀ (φ : K →+* A), ‖φ x‖ ≤ 1
a b : ℕ
habne : a ≠ b
h : x ^ a = x ^ b
hlt : b < a
⊢ ∃ n, ∃ (_ : 0 < n), x ^ n = 1
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Embeddings

Goal state before:
pp: K : Type u_1
inst✝³ : Field K
k : Type u_2
inst✝² : Field k
inst✝¹ : Algebra k K
inst✝ : Algebra.IsAlgebraic k K
φ : k →+* ℂ
⊢ (lift K φ).comp (algebraMap k K) = φ

Tactic:
unfold lift
  let := φ.toAlgebra
  rw [AlgHom.toRingHom_eq_coe, AlgHom.comp_algebraMap_of_tower, RingHom.algebraMap_toAlgebra']

Goal state after:
Module: Mathlib.Analysis.Complex.Polynomial.Basic

Goal state before:
pp: p : ℝ[X]
z : ℂ
h0 : (aeval z) p = 0
hz : z.im ≠ 0
⊢ X ^ 2 - (C z + C ((starRingEnd ℂ) z)) * X + C (z * (starRingEnd ℂ) z) = (X - C ((starRingEnd ℂ) z)) * (X - C z)

Tactic:
map_mul

Goal state after:
p : ℝ[X]
z : ℂ
h0 : (aeval z) p = 0
hz : z.im ≠ 0
⊢ X ^ 2 - (C z + C ((starRingEnd ℂ) z)) * X + C z * C ((starRingEnd ℂ) z) = (X - C ((starRingEnd ℂ) z)) * (X - C z)
Module: Mathlib.Data.Real.Embedding

Goal state before:
pp: case a
M : Type u_1
inst✝⁶ : AddCommGroup M
inst✝⁵ : LinearOrder M
inst✝⁴ : IsOrderedAddMonoid M
inst✝³ : One M
inst✝² : ZeroLEOneClass M
inst✝¹ : NeZero 1
inst✝ : Archimedean M
⊢ 0 ≤ embedRealFun 0

Tactic:
· rw [le_csSup_iff (ratLt'_bddAbove (0 : M)) (ratLt'_nonempty 0)]
    intro x
    rw [mem_upperBounds]
    suffices (∀ (y : ℚ), y.num • (1 : M) < 0 → y ≤ x) → 0 ≤ x by simpa using this
    intro h
    have h' (y : ℚ) (hy : y < 0) : y ≤ x := by
      exact h _ <| (smul_neg_iff_of_neg_left (by simpa using hy)).mpr zero_lt_one
    contrapose! h'
    obtain ⟨y, hxy, hy⟩ := exists_rat_btwn h'
    exact ⟨y, by simpa using hy, hxy⟩

Goal state after:
Module: Mathlib.RingTheory.Valuation.RankOne

Goal state before:
pp: case pos
R : Type u_3
inst✝¹ : Ring R
inst✝ : ValuativeRel R
h : MulArchimedean (ValueGroupWithZero R)
H : IsNontrivial R
⊢ IsRankLeOne R

pp: case neg
R : Type u_3
inst✝¹ : Ring R
inst✝ : ValuativeRel R
h : MulArchimedean (ValueGroupWithZero R)
H : ¬IsNontrivial R
⊢ IsRankLeOne R

Tactic:
· rw [isNontrivial_iff_isNontrivial (valuation R)] at H
      have h' : MulArchimedean (ValueGroup₀ (.ofClass (valuation R))) :=
        MulArchimedean.comap embedding.toMonoidHom embedding_strictMono
      rw [← (valuation R).nonempty_rankOne_iff_mulArchimedean] at h'
      obtain ⟨f⟩ := h'
      exact isRankLeOne_of_rankOne

Goal state after:
case neg
R : Type u_3
inst✝¹ : Ring R
inst✝ : ValuativeRel R
h : MulArchimedean (ValueGroupWithZero R)
H : ¬IsNontrivial R
⊢ IsRankLeOne R
Module: Mathlib.Topology.MetricSpace.Ultra.Basic

Goal state before:
pp: case inr.inl
X : Type u_1
inst✝¹ : PseudoMetricSpace X
inst✝ : IsUltrametricDist X
x : X
r : ℝ
hr : r ≠ 0
h : 0 < r
y : X
hy : y ∈ closedBall x r
hd : closedBall x r = closedBall y r
⊢ ∃ ε, 0 < ε ∧ ball y ε ⊆ closedBall x r

Tactic:
use r
      simp [h, hd, ball_subset_closedBall]

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

Goal state before:
pp: case inr.inr
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
p q : σ →₀ ℕ
hp : φ.lexOrder = ↑(toLex p)
hq : ψ.lexOrder = ↑(toLex q)
u v : σ →₀ ℕ
h' : (u, v) ≠ (p, q)
h : u + v = p + q
h'' : v < q
⊢ (coeff (u, v).1) φ * 0 = 0

pp: case inr.inr
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
p q : σ →₀ ℕ
hp : φ.lexOrder = ↑(toLex p)
hq : ψ.lexOrder = ↑(toLex q)
u v : σ →₀ ℕ
h' : (u, v) ≠ (p, q)
h : u + v = p + q
h'' : v < q
⊢ ↑(toLex v) < ψ.lexOrder

Tactic:
mul_zero

Goal state after:
case inr.inr
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
p q : σ →₀ ℕ
hp : φ.lexOrder = ↑(toLex p)
hq : ψ.lexOrder = ↑(toLex q)
u v : σ →₀ ℕ
h' : (u, v) ≠ (p, q)
h : u + v = p + q
h'' : v < q
⊢ 0 = 0

case inr.inr
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
p q : σ →₀ ℕ
hp : φ.lexOrder = ↑(toLex p)
hq : ψ.lexOrder = ↑(toLex q)
u v : σ →₀ ℕ
h' : (u, v) ≠ (p, q)
h : u + v = p + q
h'' : v < q
⊢ ↑(toLex v) < ψ.lexOrder
Module: Mathlib.RingTheory.MvPowerSeries.LexOrder

Goal state before:
pp: case inr
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
d : σ →₀ ℕ
hd : ↑(toLex d) < φ.lexOrder + ψ.lexOrder
u v : σ →₀ ℕ
h : u + v = d
hv : ↑(toLex v) < ψ.lexOrder
⊢ (coeff u) φ * 0 = 0

Tactic:
mul_zero

Goal state after:
case inr
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
d : σ →₀ ℕ
hd : ↑(toLex d) < φ.lexOrder + ψ.lexOrder
u v : σ →₀ ℕ
h : u + v = d
hv : ↑(toLex v) < ψ.lexOrder
⊢ 0 = 0
Module: Mathlib.Topology.Algebra.Valued.NormedValued

Goal state before:
pp: L : Type u_1
inst✝¹ : Field L
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
val : Valued L Γ₀
hv : v.RankOne
x y z : L
⊢ dist x z ≤ max (dist x y) (dist y z)

Tactic:
refine (Valuation.norm_add_le _ (x - y) (y - z)).trans_eq' ?_
    simp only [sub_add_sub_cancel]
    rfl

Goal state after: