mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
hs : MeasurableSet s
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
t : Set E
ht : MeasurableSet t
h't : μ t ≠ 0
h''t : μ t ≠ ∞
I : ∀ (u v : Set E), μ u ≠ 0 → μ u ≠ ∞ → MeasurableSet v → μ u / μ u - μ (vᶜ ∩ u) / μ u = μ (v ∩ u) / μ u
L : Tendsto (fun r ↦ μ (sᶜ ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
L' : Tendsto (fun r ↦ μ (sᶜ ∩ ({x} + r • t)) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0)
L'' : Tendsto (fun r ↦ μ ({x} + r • t) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 1)
this : Tendsto (fun a ↦ μ ({x} + a • t) / μ ({x} + a • t) - μ (sᶜ ∩ ({x} + a • t)) / μ ({x} + a • t)) (𝓝[>] 0) (𝓝 1)
⊢ (fun a ↦ μ ({x} + a • t) / μ ({x} + a • t) - μ (sᶜ ∩ ({x} + a • t)) / μ ({x} + a • t)) =ᶠ[𝓝[>] 0] fun r ↦
    μ (s ∩ ({x} + r • t)) / μ ({x} + r • t)

Tactic:
filter_upwards [self_mem_nhdsWithin]

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
hs : MeasurableSet s
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
t : Set E
ht : MeasurableSet t
h't : μ t ≠ 0
h''t : μ t ≠ ∞
I : ∀ (u v : Set E), μ u ≠ 0 → μ u ≠ ∞ → MeasurableSet v → μ u / μ u - μ (vᶜ ∩ u) / μ u = μ (v ∩ u) / μ u
L : Tendsto (fun r ↦ μ (sᶜ ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
L' : Tendsto (fun r ↦ μ (sᶜ ∩ ({x} + r • t)) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0)
L'' : Tendsto (fun r ↦ μ ({x} + r • t) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 1)
this : Tendsto (fun a ↦ μ ({x} + a • t) / μ ({x} + a • t) - μ (sᶜ ∩ ({x} + a • t)) / μ ({x} + a • t)) (𝓝[>] 0) (𝓝 1)
⊢ ∀ a ∈ Ioi 0,
    μ ({x} + a • t) / μ ({x} + a • t) - μ (sᶜ ∩ ({x} + a • t)) / μ ({x} + a • t) =
      μ (s ∩ ({x} + a • t)) / μ ({x} + a • t)
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: F : Type u_5
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace ℝ F
F' : Type u_6
inst✝¹ : NormedAddCommGroup F'
inst✝ : InnerProductSpace ℝ F'
v : OrthonormalBasis (Fin 2) ℝ F
f : F ≃ₗᵢ[ℝ] F'
⊢ orthonormalBasisOneI.repr.trans (v.repr.symm.trans f) = (orthonormalBasisOneI.repr.trans v.repr.symm).trans f

Tactic:
rw [LinearIsometryEquiv.trans_assoc]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: case hfh
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
t : Set E
ht : MeasurableSet t
h't : μ t ≠ 0
h''t : μ t ≠ ∞
⊢ ∀ᶠ (b : ℝ) in 𝓝[>] 0, μ (toMeasurable μ s ∩ closedBall x b) / μ (closedBall x b) ≤ 1

Tactic:
filter_upwards [self_mem_nhdsWithin]

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
t : Set E
ht : MeasurableSet t
h't : μ t ≠ 0
h''t : μ t ≠ ∞
⊢ ∀ a ∈ Ioi 0, μ (toMeasurable μ s ∩ closedBall x a) / μ (closedBall x a) ≤ 1
Module: Mathlib.Analysis.BoxIntegral.Partition.Measure

Goal state before:
pp: ι : Type u_1
I : Box ι
inst✝ : Countable ι
⊢ MeasurableSet ↑I

Tactic:
rw [coe_eq_pi]

Goal state after:
ι : Type u_1
I : Box ι
inst✝ : Countable ι
⊢ MeasurableSet (univ.pi fun i ↦ Ioc (I.lower i) (I.upper i))
Module: Mathlib.Analysis.BoxIntegral.Partition.Measure

Goal state before:
pp: ι : Type u_1
I : Box ι
inst✝ : Fintype ι
⊢ ↑I =ᵐ[volume] Box.Icc I

Tactic:
rw [coe_eq_pi]

Goal state after:
ι : Type u_1
I : Box ι
inst✝ : Fintype ι
⊢ (univ.pi fun i ↦ Ioc (I.lower i) (I.upper i)) =ᵐ[volume] Box.Icc I
Module: Mathlib.Analysis.Oscillation

Goal state before:
pp: case right
E : Type u
F : Type v
inst✝¹ : PseudoEMetricSpace F
inst✝ : PseudoEMetricSpace E
K : Set E
f : E → F
D : Set E
ε : ℝ≥0∞
comp : IsCompact K
hK : ∀ x ∈ K, oscillationWithin f D x < ε
S : ℝ → Set E := ⋯
S_open : ∀ r > 0, IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (S r)
S_cover : K ⊆ ⋃ r, ⋃ (_ : r > 0), S r
S_antitone : ∀ (r₁ r₂ : ℝ), r₁ ≤ r₂ → S r₂ ⊆ S r₁
δ : ℝ
δ0 : δ > 0
hδ : K ⊆ S δ
⊢ ∀ x ∈ K, ediam (f '' (eball x (ENNReal.ofReal δ) ∩ D)) ≤ ε

Tactic:
intro x xK

Goal state after:
case right
E : Type u
F : Type v
inst✝¹ : PseudoEMetricSpace F
inst✝ : PseudoEMetricSpace E
K : Set E
f : E → F
D : Set E
ε : ℝ≥0∞
comp : IsCompact K
hK : ∀ x ∈ K, oscillationWithin f D x < ε
S : ℝ → Set E := fun r ↦ {x | ∃ a > r, ediam (f '' (eball x (ENNReal.ofReal a) ∩ D)) ≤ ε}
S_open : ∀ r > 0, IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (S r)
S_cover : K ⊆ ⋃ r, ⋃ (_ : r > 0), S r
S_antitone : ∀ (r₁ r₂ : ℝ), r₁ ≤ r₂ → S r₂ ⊆ S r₁
δ : ℝ
δ0 : δ > 0
hδ : K ⊆ S δ
x : E
xK : x ∈ K
⊢ ediam (f '' (eball x (ENNReal.ofReal δ) ∩ D)) ≤ ε
Module: Mathlib.Analysis.BoxIntegral.Integrability

Goal state before:
pp: case hbc.refine_2
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
hl : l.bRiemann = false
s : Set (ι → ℝ)
hs : MeasurableSet s
I : Box ι
y : E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
ε : ℝ≥0
ε0 : 0 < ε
A : μ (s ∩ Box.Icc I) ≠ ∞
B : μ (s ∩ ↑I) ≠ ∞
F : Set (ι → ℝ)
hFs : F ⊆ s ∩ Box.Icc I
hFc : IsClosed F
hμF : μ ((s ∩ Box.Icc I) \ F) < ↑ε
U : Set (ι → ℝ)
hsU : s ∩ Box.Icc I ⊆ U
hUo : IsOpen U
hUt : μ U < ∞
hμU : μ (U \ (s ∩ Box.Icc I)) < ↑ε
rs : (ι → ℝ) → ↑(Set.Ioi 0)
hrsU : ∀ x ∈ s ∩ Box.Icc I, closedBall x ↑(rs x) ⊆ U
rs' : (ι → ℝ) → ↑(Set.Ioi 0)
hrs'F : ∀ x ∈ Box.Icc I \ s, closedBall x ↑(rs' x) ⊆ Fᶜ
r : (ι → ℝ) → ↑(Set.Ioi 0) := s.piecewise rs rs'
c : ℝ≥0
π : TaggedPrepartition I
hπ : l.MemBaseSet I c ((fun x ↦ r) c) π
hπp : π.IsPartition
t : Set (ι → ℝ) := (π.filter fun x ↦ π.tag x ∈ s).iUnion
htU : t ⊆ U ∩ ↑I
hμt : μ t ≠ ∞
x : ι → ℝ
hxt : x ∉ t
hxs : x ∈ s
hxI : x ∈ ↑I
hxF : x ∈ F
J : Box ι
hJπ : J ∈ π.toPrepartition
hxJ : x ∈ J
⊢ π.tag J ∈ s

Tactic:
contrapose hxF

Goal state after:
case hbc.refine_2
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
hl : l.bRiemann = false
s : Set (ι → ℝ)
hs : MeasurableSet s
I : Box ι
y : E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
ε : ℝ≥0
ε0 : 0 < ε
A : μ (s ∩ Box.Icc I) ≠ ∞
B : μ (s ∩ ↑I) ≠ ∞
F : Set (ι → ℝ)
hFs : F ⊆ s ∩ Box.Icc I
hFc : IsClosed F
hμF : μ ((s ∩ Box.Icc I) \ F) < ↑ε
U : Set (ι → ℝ)
hsU : s ∩ Box.Icc I ⊆ U
hUo : IsOpen U
hUt : μ U < ∞
hμU : μ (U \ (s ∩ Box.Icc I)) < ↑ε
rs : (ι → ℝ) → ↑(Set.Ioi 0)
hrsU : ∀ x ∈ s ∩ Box.Icc I, closedBall x ↑(rs x) ⊆ U
rs' : (ι → ℝ) → ↑(Set.Ioi 0)
hrs'F : ∀ x ∈ Box.Icc I \ s, closedBall x ↑(rs' x) ⊆ Fᶜ
r : (ι → ℝ) → ↑(Set.Ioi 0) := s.piecewise rs rs'
c : ℝ≥0
π : TaggedPrepartition I
hπ : l.MemBaseSet I c ((fun x ↦ r) c) π
hπp : π.IsPartition
t : Set (ι → ℝ) := (π.filter fun x ↦ π.tag x ∈ s).iUnion
htU : t ⊆ U ∩ ↑I
hμt : μ t ≠ ∞
x : ι → ℝ
hxt : x ∉ t
hxs : x ∈ s
hxI : x ∈ ↑I
J : Box ι
hJπ : J ∈ π.toPrepartition
hxJ : x ∈ J
hxF : π.tag J ∉ s
⊢ x ∉ F
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: ι : Type u_1
s : Set (ι → ℝ)
F : (ι → ℝ) → ℝ
inst✝ : Fintype ι
hF : Continuous F
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
B : Box ι
hB : hasIntegralVertices B
hs₀ : s ⊆ ↑B
ε : ℝ
hε : ε > 0
h₁ : ∃ C, ∀ x ∈ Box.Icc B, ‖s.indicator F x‖ ≤ C
h₂ : ∀ᵐ (x : ι → ℝ), ContinuousAt (s.indicator F) x
r : NNReal → (ι → ℝ) → ↑(Set.Ioi 0)
hr₁ : ∀ (c : NNReal), IntegrationParams.Riemann.RCond (r c)
hr₂ :
  ∀ (c : NNReal) (π : TaggedPrepartition B),
    IntegrationParams.Riemann.MemBaseSet B c (r c) π →
      π.IsPartition →
        dist (integralSum (s.indicator F) volume.toBoxAdditive.toSMul π)
            (∫ (x : ι → ℝ) in ↑B, s.indicator F x ∂volume) ≤
          ε / 2
n : ℕ
hn : n ≥ ⌈(↑(r 0 0))⁻¹⌉₊
this : NeZero n
⊢ dist (integralSum (s.indicator F) volume.toBoxAdditive.toSMul (prepartition n B))
      (∫ (x : ι → ℝ) in s, F x ∂volume.restrict ↑B) ≤
    ε / 2

Tactic:
← integral_indicator hs₂

Goal state after:
ι : Type u_1
s : Set (ι → ℝ)
F : (ι → ℝ) → ℝ
inst✝ : Fintype ι
hF : Continuous F
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
B : Box ι
hB : hasIntegralVertices B
hs₀ : s ⊆ ↑B
ε : ℝ
hε : ε > 0
h₁ : ∃ C, ∀ x ∈ Box.Icc B, ‖s.indicator F x‖ ≤ C
h₂ : ∀ᵐ (x : ι → ℝ), ContinuousAt (s.indicator F) x
r : NNReal → (ι → ℝ) → ↑(Set.Ioi 0)
hr₁ : ∀ (c : NNReal), IntegrationParams.Riemann.RCond (r c)
hr₂ :
  ∀ (c : NNReal) (π : TaggedPrepartition B),
    IntegrationParams.Riemann.MemBaseSet B c (r c) π →
      π.IsPartition →
        dist (integralSum (s.indicator F) volume.toBoxAdditive.toSMul π)
            (∫ (x : ι → ℝ) in ↑B, s.indicator F x ∂volume) ≤
          ε / 2
n : ℕ
hn : n ≥ ⌈(↑(r 0 0))⁻¹⌉₊
this : NeZero n
⊢ dist (integralSum (s.indicator F) volume.toBoxAdditive.toSMul (prepartition n B))
      (∫ (x : ι → ℝ) in ↑B, s.indicator F x ∂volume) ≤
    ε / 2
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
ι : Type u_2
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
e f : OrthonormalBasis ι ℝ E
h : e.toBasis.orientation ≠ f.toBasis.orientation
⊢ e.toBasis.det ⇑f = -1

Tactic:
by
  contrapose! h
  simp [e.toBasis.orientation_eq_iff_det_pos,
    (e.det_to_matrix_orthonormalBasis_real f).resolve_right h]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: case zero
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
_i : Fact (finrank ℝ E = 0)
o : Orientation ℝ E (Fin 0)
b : OrthonormalBasis (Fin 0) ℝ E
hb : b.toBasis.orientation = o
this : o = positiveOrientation
⊢ AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1 = b.toBasis.det

Tactic:
Basis.det_isEmpty

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: case zero
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
_i : Fact (finrank ℝ E = 0)
o : Orientation ℝ E (Fin 0)
b : OrthonormalBasis (Fin 0) ℝ E
hb : b.toBasis.orientation ≠ o
this : positiveOrientation ≠ o
⊢ -AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1 = -b.toBasis.det

Tactic:
Basis.det_isEmpty

Goal state after:
Module: Mathlib.LinearAlgebra.Orientation

Goal state before:
pp: case neg
R : Type u_1
inst✝⁷ : CommRing R
inst✝⁶ : LinearOrder R
inst✝⁵ : IsStrictOrderedRing R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
ι : Type u_3
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : Nonempty ι
e : Basis ι R M
x : Orientation R M ι
h✝ : ¬e.orientation = x
⊢ (e.unitsSMul (Function.update 1 (Classical.arbitrary ι) (-1))).det = e.det ∨
    (e.unitsSMul (Function.update 1 (Classical.arbitrary ι) (-1))).det = -e.det

Tactic:
right
    simp only [e.det_unitsSMul, Finset.mem_univ, Finset.prod_update_of_mem,
      Pi.one_apply, Finset.prod_const_one, mul_one, inv_neg, inv_one, Units.val_neg, Units.val_one]
    ext
    simp

Goal state after:
Module: Mathlib.Analysis.SumOverResidueClass

Goal state before:
pp: f : ℕ → ℝ
hf : Antitone f
n : ℕ
hn : f n < 0
hs : Summable f
this : ∀ ε > 0, ∃ N, ∀ n ≥ N, |f n| < ε
N : ℕ
hN : True
⊢ |f n| ≤ |f (max n N)|

Tactic:
clear hN

Goal state after:
f : ℕ → ℝ
hf : Antitone f
n : ℕ
hn : f n < 0
hs : Summable f
this : ∀ ε > 0, ∃ N, ∀ n ≥ N, |f n| < ε
N : ℕ
⊢ |f n| ≤ |f (max n N)|
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: p : ℕ
h : 1 < p
n : ℕ
⊢ 1 / (-1) ^ p * (1 / ↑n ^ p) = 1 / ((-1) ^ p * ↑n ^ p)

Tactic:
mul_one_div,

Goal state after:
p : ℕ
h : 1 < p
n : ℕ
⊢ 1 / (-1) ^ p / ↑n ^ p = 1 / ((-1) ^ p * ↑n ^ p)
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: case e_a
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
L : Submodule ℤ E
inst✝⁶ : DiscreteTopology ↥L
inst✝⁵ : IsZLattice ℝ L
ι : Type u_2
inst✝⁴ : Fintype ι
b : Basis ι ℤ ↥L
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : Nonempty ι
X : Set E
hX : ∀ ⦃x : E⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
F : E → ℝ
h₁ : ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ card ι * F x
h₂ : Bornology.IsBounded {x | x ∈ X ∧ F x ≤ 1}
h₃ : MeasurableSet {x | x ∈ X ∧ F x ≤ 1}
h₄ : volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1})) = 0
c : ℝ
hc : 0 < c
aux₁ : ↑(card ι) ≠ 0
aux₂ : 0 < c ^ (↑(card ι))⁻¹
aux₃ : (c ^ (↑(card ι))⁻¹)⁻¹ ≠ 0
aux₄ : c ^ (-(↑(card ι))⁻¹) ≠ 0
hc₁ : 0 ≤ c
hc₂ : 0 ≠ c
x✝ : E
⊢ x✝ ∈ {x | x ∈ X ∧ F x ≤ c} ∧ x✝ ∈ ↑L ↔
    (Basis.ofZLatticeBasis ℝ L b).equivFun x✝ ∈
        ⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ (c ^ (↑(card ι))⁻¹) ^ ↑(card ι)} ∧
      (Basis.ofZLatticeBasis ℝ L b).equivFun x✝ ∈ ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))

Tactic:
← Real.rpow_mul hc₁,

Goal state after:
case e_a
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
L : Submodule ℤ E
inst✝⁶ : DiscreteTopology ↥L
inst✝⁵ : IsZLattice ℝ L
ι : Type u_2
inst✝⁴ : Fintype ι
b : Basis ι ℤ ↥L
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : Nonempty ι
X : Set E
hX : ∀ ⦃x : E⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
F : E → ℝ
h₁ : ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ card ι * F x
h₂ : Bornology.IsBounded {x | x ∈ X ∧ F x ≤ 1}
h₃ : MeasurableSet {x | x ∈ X ∧ F x ≤ 1}
h₄ : volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1})) = 0
c : ℝ
hc : 0 < c
aux₁ : ↑(card ι) ≠ 0
aux₂ : 0 < c ^ (↑(card ι))⁻¹
aux₃ : (c ^ (↑(card ι))⁻¹)⁻¹ ≠ 0
aux₄ : c ^ (-(↑(card ι))⁻¹) ≠ 0
hc₁ : 0 ≤ c
hc₂ : 0 ≠ c
x✝ : E
⊢ x✝ ∈ {x | x ∈ X ∧ F x ≤ c} ∧ x✝ ∈ ↑L ↔
    (Basis.ofZLatticeBasis ℝ L b).equivFun x✝ ∈
        ⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ c ^ ((↑(card ι))⁻¹ * ↑(card ι))} ∧
      (Basis.ofZLatticeBasis ℝ L b).equivFun x✝ ∈ ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))
Module: Mathlib.Algebra.MvPolynomial.Division

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝¹ : CommSemiring R
inst✝ : Nontrivial R
i j : σ
⊢ i = j ∧ 1 ≠ 0 ↔ i = j

Tactic:
ne_eq,

Goal state after:
σ : Type u_1
R : Type u_2
inst✝¹ : CommSemiring R
inst✝ : Nontrivial R
i j : σ
⊢ i = j ∧ ¬1 = 0 ↔ i = j
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: case cons
α : Type u_1
R : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Semiring R
f✝ : α → R
a : α
s : Finset α
has : a ∉ s
ih :
  ∀ (hc : (↑s).Pairwise (Commute on f✝)) (n : ℕ),
    (∑ i ∈ s, f✝ i) ^ n = ∑ k ∈ s.piAntidiag n, ↑(multinomial s k) * s.noncommProd (fun i ↦ f✝ i ^ k i) ⋯
hc : (↑(cons a s has)).Pairwise (Commute on f✝)
n : ℕ
p : ℕ × ℕ
hp : p ∈ antidiagonal n
f : α → ℕ
hf : s.sum f = p.2 ∧ ∀ (i : α), f i ≠ 0 → i ∈ s
⊢ ↑((f a + p.1 + s.sum f).choose (f a + p.1)) * ↑(multinomial s (f + fun i ↦ if i = a then p.1 else 0)) *
      (f✝ a ^ (f a + p.1) * s.noncommProd (fun i ↦ f✝ i ^ (f i + if i = a then p.1 else 0)) ⋯) =
    ↑(n.choose p.1) * ↑(multinomial s f) * (f✝ a ^ p.1 * s.noncommProd (fun i ↦ f✝ i ^ f i) ⋯)

Tactic:
not_imp_comm.1 (hf.2 _) has,

Goal state after:
case cons
α : Type u_1
R : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Semiring R
f✝ : α → R
a : α
s : Finset α
has : a ∉ s
ih :
  ∀ (hc : (↑s).Pairwise (Commute on f✝)) (n : ℕ),
    (∑ i ∈ s, f✝ i) ^ n = ∑ k ∈ s.piAntidiag n, ↑(multinomial s k) * s.noncommProd (fun i ↦ f✝ i ^ k i) ⋯
hc : (↑(cons a s has)).Pairwise (Commute on f✝)
n : ℕ
p : ℕ × ℕ
hp : p ∈ antidiagonal n
f : α → ℕ
hf : s.sum f = p.2 ∧ ∀ (i : α), f i ≠ 0 → i ∈ s
⊢ ↑((0 + p.1 + s.sum f).choose (0 + p.1)) * ↑(multinomial s (f + fun i ↦ if i = a then p.1 else 0)) *
      (f✝ a ^ (0 + p.1) * s.noncommProd (fun i ↦ f✝ i ^ (f i + if i = a then p.1 else 0)) ⋯) =
    ↑(n.choose p.1) * ↑(multinomial s f) * (f✝ a ^ p.1 * s.noncommProd (fun i ↦ f✝ i ^ f i) ⋯)
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: case empty.zero
α : Type u_1
R : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Semiring R
x : α → R
hc : (↑∅).Pairwise (Commute on x)
⊢ 0 ^ 0 = ∑ k, ↑(↑↑k).countPerms * (Multiset.map x ↑↑k).noncommProd ⋯

Tactic:
rw [_root_.pow_zero, Fintype.sum_subsingleton]

Goal state after:
case empty.zero
α : Type u_1
R : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Semiring R
x : α → R
hc : (↑∅).Pairwise (Commute on x)
⊢ 1 = ↑(↑↑?empty.zero.a).countPerms * (Multiset.map x ↑↑?empty.zero.a).noncommProd ⋯

case empty.zero.a
α : Type u_1
R : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Semiring R
x : α → R
hc : (↑∅).Pairwise (Commute on x)
⊢ ↥(∅.sym 0)
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: case cons
m : Multiset ℕ
l✝ l'✝ : List ℕ
x : ℕ
l l' : List ℕ
hl : l.Perm l'
hl' : l.multinomial = l'.multinomial
⊢ (x :: l).multinomial = (x :: l').multinomial

Tactic:
| @cons x l l' hl hl' =>

Goal state after:
Module: Mathlib.Algebra.MvPolynomial.NoZeroDivisors

Goal state before:
pp: case neg
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
p : MvPolynomial σ R
j : σ
c : R
hc : c ∈ R⁰
hp : ¬p = 0
hp' : ((optionEquivLeft R { b // b ≠ j }) ((rename ⇑(optionSubtypeNe j).symm) p)).leadingCoeff ≠ 0
⊢ ((optionEquivLeft R { b // b ≠ j }) ((renameEquiv R (optionSubtypeNe j).symm) (C c))).leadingCoeff *
      ((optionEquivLeft R { b // b ≠ j }) ((renameEquiv R (optionSubtypeNe j).symm) p)).leadingCoeff ≠
    0

Tactic:
ne_eq,

Goal state after:
case neg
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
p : MvPolynomial σ R
j : σ
c : R
hc : c ∈ R⁰
hp : ¬p = 0
hp' : ((optionEquivLeft R { b // b ≠ j }) ((rename ⇑(optionSubtypeNe j).symm) p)).leadingCoeff ≠ 0
⊢ ¬((optionEquivLeft R { b // ¬b = j }) ((renameEquiv R (optionSubtypeNe j).symm) (C c))).leadingCoeff *
        ((optionEquivLeft R { b // ¬b = j }) ((renameEquiv R (optionSubtypeNe j).symm) p)).leadingCoeff =
      0
Module: Mathlib.RingTheory.MvPolynomial.MonomialOrder

Goal state before:
pp: case neg
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommSemiring R
f : MvPolynomial σ R
hf : m.degree f = 0
d : σ →₀ ℕ
hd : ¬d = 0
⊢ 0 ≤ m.toSyn d ∧ 0 ≠ m.toSyn d

Tactic:
ne_eq,

Goal state after:
case neg
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommSemiring R
f : MvPolynomial σ R
hf : m.degree f = 0
d : σ →₀ ℕ
hd : ¬d = 0
⊢ 0 ≤ m.toSyn d ∧ ¬0 = m.toSyn d
Module: Mathlib.RingTheory.MvPolynomial.MonomialOrder

Goal state before:
pp: case neg
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommSemiring R
f g : MvPolynomial σ R
a b : σ →₀ ℕ
ha : m.toSyn (m.degree f) ≤ m.toSyn a
hb : m.toSyn (m.degree g) ≤ m.toSyn b
c d : σ →₀ ℕ
h : (c, d) ≠ (a, b)
hcd : c + d = a + b
hf : m.toSyn c ≤ m.toSyn (m.degree f)
hf' : m.toSyn d ≤ m.toSyn (m.degree g)
⊢ m.toSyn d ≤ m.toSyn b

Tactic:
exact le_trans hf' hb

Goal state after:
Module: Mathlib.Data.Nat.Squarefree

Goal state before:
pp: n k : ℕ
pk : Prime k
dk : k ∣ n
dkk : ¬k * k ∣ n
o : Option ℕ
H : (n / k).MinSqFacProp o
p : ℕ
pp : Prime p
dp : p * p ∣ n
e : k = p
⊢ False

Tactic:
subst e

Goal state after:
n k : ℕ
pk : Prime k
dk : k ∣ n
dkk : ¬k * k ∣ n
o : Option ℕ
H : (n / k).MinSqFacProp o
pp : Prime k
dp : k * k ∣ n
⊢ False
Module: Mathlib.Data.Nat.Squarefree

Goal state before:
pp: case none
n k : ℕ
pk : Prime k
dk : k ∣ n
dkk : ¬k * k ∣ n
this : ∀ (p : ℕ), Prime p → p * p ∣ n → k * (p * p) ∣ n
H : (n / k).MinSqFacProp none
⊢ n.MinSqFacProp none

pp: case some
n k : ℕ
pk : Prime k
dk : k ∣ n
dkk : ¬k * k ∣ n
this : ∀ (p : ℕ), Prime p → p * p ∣ n → k * (p * p) ∣ n
d : ℕ
H : (n / k).MinSqFacProp (some d)
⊢ n.MinSqFacProp (some d)

Tactic:
· rw [MinSqFacProp, squarefree_iff_prime_squarefree] at H ⊢
    exact fun p pp dp => H p pp ((dvd_div_iff_mul_dvd dk).2 (this _ pp dp))

Goal state after:
case some
n k : ℕ
pk : Prime k
dk : k ∣ n
dkk : ¬k * k ∣ n
this : ∀ (p : ℕ), Prime p → p * p ∣ n → k * (p * p) ∣ n
d : ℕ
H : (n / k).MinSqFacProp (some d)
⊢ n.MinSqFacProp (some d)
Module: Mathlib.Algebra.MvPolynomial.SchwartzZippel

Goal state before:
pp: case a
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DecidableEq R
n : ℕ
p : MvPolynomial (Fin (n + 1)) R
hp : p ≠ 0
S : Fin (n + 1) → Finset R
p' : Polynomial (MvPolynomial (Fin n) R) := (finSuccEquiv R n) p
hp' : p' = (finSuccEquiv R n) p
k : ℕ := p'.natDegree
hk : k = p'.natDegree
pₖ : MvPolynomial (Fin n) R := p'.leadingCoeff
hpₖ : pₖ = p'.leadingCoeff
hp'₀ : p' ≠ 0
hpₖ₀ : pₖ ≠ 0
⊢ ∑ xₜ ∈ piFinset fun i ↦ tail S i with (eval xₜ) pₖ ≠ 0, #({x₀ ∈ S 0 | (eval (Fin.cons x₀ xₜ)) p = 0}) ≤
    ∑ xₜ ∈ piFinset fun i ↦ tail S i with (eval xₜ) pₖ ≠ 0, degreeOf 0 p

Tactic:
gcongr with xₜ hxₜ

Goal state after:
case h
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DecidableEq R
n : ℕ
p : MvPolynomial (Fin (n + 1)) R
hp : p ≠ 0
S : Fin (n + 1) → Finset R
p' : Polynomial (MvPolynomial (Fin n) R) := (finSuccEquiv R n) p
hp' : p' = (finSuccEquiv R n) p
k : ℕ := p'.natDegree
hk : k = p'.natDegree
pₖ : MvPolynomial (Fin n) R := p'.leadingCoeff
hpₖ : pₖ = p'.leadingCoeff
hp'₀ : p' ≠ 0
hpₖ₀ : pₖ ≠ 0
xₜ : Fin n → R
hxₜ : xₜ ∈ {xₜ ∈ piFinset fun i ↦ tail S i | (eval xₜ) pₖ ≠ 0}
⊢ #({x₀ ∈ S 0 | (eval (Fin.cons x₀ xₜ)) p = 0}) ≤ degreeOf 0 p
Module: Mathlib.RingTheory.MvPolynomial.MonomialOrder

Goal state before:
pp: σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommSemiring R
f g : MvPolynomial σ R
⊢ m.toWithBotSyn (m.withBotDegree f) ≤ m.toWithBotSyn (m.withBotDegree g) ↔
    m.toSyn (m.degree f) ≤ m.toSyn (m.degree g) ∧ (g = 0 → f = 0)

Tactic:
by
  classical
  by_cases! +distrib h : f ≠ 0 ∧ g ≠ 0
  · simp [m.withBotDegree_eq, h, m.toWithBotSyn_apply]
  rcases h with h | _
  · simp [h]
  · aesop

Goal state after:
Module: Mathlib.Data.Nat.Factorization.PrimePow

Goal state before:
pp: n : ℕ
h : IsPrimePow n
⊢ ∃ p, Nat.Prime p ∧ n / p ^ n.factorization p = 1

Tactic:
by
  rcases eq_or_ne n 0 with (rfl | hn0); · cases not_isPrimePow_zero h
  rcases isPrimePow_iff_factorization_eq_single.mp h with ⟨p, k, hk0, h1⟩
  rcases em' p.Prime with (pp | pp)
  · refine absurd ?_ hk0.ne'
    simp [← Nat.factorization_eq_zero_of_not_prime n pp, h1]
  refine ⟨p, pp, ?_⟩
  refine Nat.eq_of_factorization_eq (Nat.ordCompl_pos p hn0).ne' (by simp) fun q => ?_
  rw [Nat.factorization_ordCompl n p, h1]
  simp

Goal state after:
Module: Mathlib.NumberTheory.ArithmeticFunction.Defs

Goal state before:
pp: case insert
R : Type u_1
ι : Type u_2
inst✝ : CommMonoidWithZero R
g : ι → ℕ
f : ArithmeticFunction R
hf : f.IsMultiplicative
a✝ : ι
s✝ : Finset ι
has : a✝ ∉ s✝
ih : (↑s✝).Pairwise (Coprime on g) → f (∏ i ∈ s✝, g i) = ∏ i ∈ s✝, f (g i)
hs : (↑(insert a✝ s✝)).Pairwise (Coprime on g)
⊢ f (∏ i ∈ insert a✝ s✝, g i) = ∏ i ∈ insert a✝ s✝, f (g i)

Tactic:
rw [coe_insert, Set.pairwise_insert_of_symm] at hs
      rw [prod_insert has, prod_insert has, hf.map_mul_of_coprime, ih hs.1]
      exact Coprime.prod_right fun i hi => hs.2 _ hi (hi.ne_of_notMem has).symm

Goal state after: