mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: case h
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
E : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : FiniteDimensional 𝕜 E
ι : Type u_7
inst✝ : Fintype ι
card_ι : finrank 𝕜 E = Fintype.card ι
v : ι → E
s : Set ι
hv : Orthonormal 𝕜 (s.domRestrict v)
hsv : Injective (s.domRestrict v)
hX✝ : Orthonormal 𝕜 Subtype.val
Y : Finset E
b₀ : OrthonormalBasis (↥Y) 𝕜 E
hX : range (s.domRestrict v) ⊆ ↑Y
hb₀ : ⇑b₀ = Subtype.val
hιY : Fintype.card ι = Y.card
hvsY : MapsTo v s ↑Y
hsv' : InjOn v s
g : ι ≃ ↥Y
hg : ∀ i ∈ s, ↑(g i) = v i
i : ι
⊢ i ∈ s → (b₀.reindex g.symm) i = v i

Tactic:
hi

Goal state after:
case h
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
E : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : FiniteDimensional 𝕜 E
ι : Type u_7
inst✝ : Fintype ι
card_ι : finrank 𝕜 E = Fintype.card ι
v : ι → E
s : Set ι
hv : Orthonormal 𝕜 (s.domRestrict v)
hsv : Injective (s.domRestrict v)
hX✝ : Orthonormal 𝕜 Subtype.val
Y : Finset E
b₀ : OrthonormalBasis (↥Y) 𝕜 E
hX : range (s.domRestrict v) ⊆ ↑Y
hb₀ : ⇑b₀ = Subtype.val
hιY : Fintype.card ι = Y.card
hvsY : MapsTo v s ↑Y
hsv' : InjOn v s
g : ι ≃ ↥Y
hg : ∀ i ∈ s, ↑(g i) = v i
i : ι
hi : i ∈ s
⊢ (b₀.reindex g.symm) i = v i
Module: Mathlib.Analysis.Oscillation

Goal state before:
pp: 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)
⊢ ∃ δ > 0, ∀ x ∈ K, ediam (f '' (eball x (ENNReal.ofReal δ) ∩ D)) ≤ ε

Tactic:
have S_cover : K ⊆ ⋃ r > 0, S r := by
    intro x hx
    have : oscillationWithin f D x < ε := hK x hx
    simp only [oscillationWithin, Filter.mem_map, iInf_lt_iff] at this
    obtain ⟨n, hn₁, hn₂⟩ := this
    obtain ⟨r, r0, hr⟩ := EMetric.mem_nhdsWithin_iff.1 hn₁
    simp only [gt_iff_lt, mem_iUnion, exists_prop]
    have : ∀ r', (ENNReal.ofReal r') ≤ r →
        ediam (f '' (eball x (ENNReal.ofReal r') ∩ D)) ≤ ε := by
      intro r' hr'
      grw [← hn₂, ← image_subset_iff.2 hr, hr']
    by_cases r_top : r = ⊤
    · exact ⟨1, one_pos, 2, by simp, this 2 (by simp only [r_top, le_top])⟩
    · obtain ⟨r', hr'⟩ := exists_between (toReal_pos (ne_of_gt r0) r_top)
      use r', hr'.1, r.toReal, hr'.2, this r.toReal ofReal_toReal_le

Goal state after:
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
⊢ ∃ δ > 0, ∀ x ∈ K, ediam (f '' (eball x (ENNReal.ofReal δ) ∩ D)) ≤ ε
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
𝕜 : Type u_3
inst✝¹⁰ : RCLike 𝕜
E : Type u_4
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : InnerProductSpace 𝕜 E
F : Type u_5
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : InnerProductSpace ℝ F
F' : Type u_6
inst✝⁵ : NormedAddCommGroup F'
inst✝⁴ : InnerProductSpace ℝ F'
inst✝³ : Fintype ι
V : Type u_7
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace 𝕜 V
inst✝ : FiniteDimensional 𝕜 V
S : Submodule 𝕜 V
L✝ L : ↥S →ₗᵢ[𝕜] V
d : ℕ := finrank 𝕜 ↥Sᗮ
⊢ V →ₗᵢ[𝕜] V

Tactic:
let LS := LinearMap.range L.toLinearMap

Goal state after:
ι : Type u_1
ι' : Type u_2
𝕜 : Type u_3
inst✝¹⁰ : RCLike 𝕜
E : Type u_4
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : InnerProductSpace 𝕜 E
F : Type u_5
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : InnerProductSpace ℝ F
F' : Type u_6
inst✝⁵ : NormedAddCommGroup F'
inst✝⁴ : InnerProductSpace ℝ F'
inst✝³ : Fintype ι
V : Type u_7
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace 𝕜 V
inst✝ : FiniteDimensional 𝕜 V
S : Submodule 𝕜 V
L✝ L : ↥S →ₗᵢ[𝕜] V
d : ℕ := finrank 𝕜 ↥Sᗮ
LS : Submodule 𝕜 V := L.range
⊢ V →ₗᵢ[𝕜] V
Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
n : ι
⊢ gramSchmidt 𝕜 0 n = 0

Tactic:
rw [gramSchmidt_def]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
n : ι
⊢ 0 n - ∑ i < n, (𝕜 ∙ gramSchmidt 𝕜 0 i).starProjection (0 n) = 0
Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho

Goal state before:
pp: case inr
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
a b : ι
h₀ : a ≠ b
this : ∀ (a b : ι), a < b → ⟪gramSchmidt 𝕜 f a, gramSchmidt 𝕜 f b⟫_𝕜 = 0
hb : b < a
⊢ ⟪gramSchmidt 𝕜 f a, gramSchmidt 𝕜 f b⟫_𝕜 = 0

Tactic:
rw [inner_eq_zero_symm]

Goal state after:
case inr
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
a b : ι
h₀ : a ≠ b
this : ∀ (a b : ι), a < b → ⟪gramSchmidt 𝕜 f a, gramSchmidt 𝕜 f b⟫_𝕜 = 0
hb : b < a
⊢ ⟪gramSchmidt 𝕜 f b, gramSchmidt 𝕜 f a⟫_𝕜 = 0
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
inst✝ : Finite ι
x : ι → ℝ
hx : ∀ (i : ι), ↑n * x i ∈ Set.range ⇑(algebraMap ℤ ℝ)
i : ι
a : ℤ
ha : ↑a = ↑n * x i
⊢ ↑⌈↑n * x i⌉ / ↑n = x i

Tactic:
← ha,

Goal state after:
ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
inst✝ : Finite ι
x : ι → ℝ
hx : ∀ (i : ι), ↑n * x i ∈ Set.range ⇑(algebraMap ℤ ℝ)
i : ι
a : ℤ
ha : ↑a = ↑n * x i
⊢ ↑⌈↑a⌉ / ↑n = x i
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
c✝ : ℝ
s : Set (ι → ℝ)
F : (ι → ℝ) → ℝ
inst✝ : Finite ι
c : ℝ
hc : c ≠ 0
x : ι → ℝ
⊢ x ∈ s ∧ x ∈ c⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))) ↔
    (Equiv.smulRight hc) x ∈ c • s ∧ (Equiv.smulRight hc) x ∈ ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))

Tactic:
Equiv.smulRight_apply,

Goal state after:
ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
c✝ : ℝ
s : Set (ι → ℝ)
F : (ι → ℝ) → ℝ
inst✝ : Finite ι
c : ℝ
hc : c ≠ 0
x : ι → ℝ
⊢ x ∈ s ∧ x ∈ c⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))) ↔
    c • x ∈ c • s ∧ c • x ∈ ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))
Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
b✝ b : ι
ih : ∀ y < b, ∀ a < y, ⟪gramSchmidt 𝕜 f a, gramSchmidt 𝕜 f y⟫_𝕜 = 0
a : ι
h₀ : a < b
i : ι
⊢ i ∈ Iio b →
    i ≠ a → ⟪gramSchmidt 𝕜 f i, f b⟫_𝕜 / ↑(‖gramSchmidt 𝕜 f i‖ ^ 2) * ⟪gramSchmidt 𝕜 f a, gramSchmidt 𝕜 f i⟫_𝕜 = 0

Tactic:
hi

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
b✝ b : ι
ih : ∀ y < b, ∀ a < y, ⟪gramSchmidt 𝕜 f a, gramSchmidt 𝕜 f y⟫_𝕜 = 0
a : ι
h₀ : a < b
i : ι
hi : i ∈ Iio b
⊢ i ≠ a → ⟪gramSchmidt 𝕜 f i, f b⟫_𝕜 / ↑(‖gramSchmidt 𝕜 f i‖ ^ 2) * ⟪gramSchmidt 𝕜 f a, gramSchmidt 𝕜 f i⟫_𝕜 = 0
Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho

Goal state before:
pp: case inl
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
b✝ b : ι
ih : ∀ y < b, ∀ a < y, ⟪gramSchmidt 𝕜 f a, gramSchmidt 𝕜 f y⟫_𝕜 = 0
a : ι
h₀ : a < b
i : ι
hi : i ∈ Iio b
hia : i ≠ a
hia₁ : i < a
⊢ ⟪gramSchmidt 𝕜 f a, gramSchmidt 𝕜 f i⟫_𝕜 = 0

Tactic:
rw [inner_eq_zero_symm]

Goal state after:
case inl
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
b✝ b : ι
ih : ∀ y < b, ∀ a < y, ⟪gramSchmidt 𝕜 f a, gramSchmidt 𝕜 f y⟫_𝕜 = 0
a : ι
h₀ : a < b
i : ι
hi : i ∈ Iio b
hia : i ≠ a
hia₁ : i < a
⊢ ⟪gramSchmidt 𝕜 f i, gramSchmidt 𝕜 f a⟫_𝕜 = 0
Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
hf : Pairwise fun x1 x2 ↦ ⟪f x1, f x2⟫_𝕜 = 0
i : ι
⊢ gramSchmidt 𝕜 f i = f i

Tactic:
rw [gramSchmidt_def]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
hf : Pairwise fun x1 x2 ↦ ⟪f x1, f x2⟫_𝕜 = 0
i : ι
⊢ f i - ∑ i_1 < i, (𝕜 ∙ gramSchmidt 𝕜 f i_1).starProjection (f i) = f i
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: case succ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
n : ℕ
_i : Fact (finrank ℝ E = n + 1)
o : Orientation ℝ E (Fin (n + 1))
v : Fin (n + 1) → E
this✝ : FiniteDimensional ℝ E
this : finrank ℝ E = Fintype.card (Fin n.succ)
b : OrthonormalBasis (Fin n.succ) ℝ E := gramSchmidtOrthonormalBasis this v
hb : b.toBasis.det v = ∏ i, ⟪b i, v i⟫
⊢ |∏ i, ⟪b i, v i⟫| ≤ ∏ i, ‖v i‖

Tactic:
Finset.abs_prod

Goal state after:
case succ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
n : ℕ
_i : Fact (finrank ℝ E = n + 1)
o : Orientation ℝ E (Fin (n + 1))
v : Fin (n + 1) → E
this✝ : FiniteDimensional ℝ E
this : finrank ℝ E = Fintype.card (Fin n.succ)
b : OrthonormalBasis (Fin n.succ) ℝ E := gramSchmidtOrthonormalBasis this v
hb : b.toBasis.det v = ∏ i, ⟪b i, v i⟫
⊢ ∏ x, |⟪b x, v x⟫| ≤ ∏ i, ‖v i‖
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: case succ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
n : ℕ
_i : Fact (finrank ℝ E = n + 1)
o : Orientation ℝ E (Fin (n + 1))
v : Fin (n + 1) → E
hv : Pairwise fun i j ↦ ⟪v i, v j⟫ = 0
this : FiniteDimensional ℝ E
hdim : finrank ℝ E = Fintype.card (Fin n.succ)
b : OrthonormalBasis (Fin n.succ) ℝ E := gramSchmidtOrthonormalBasis hdim v
hb : b.toBasis.det v = ∏ i, ⟪b i, v i⟫
⊢ |∏ i, ⟪b i, v i⟫| = ∏ i, ‖v i‖

Tactic:
Finset.abs_prod

Goal state after:
case succ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
n : ℕ
_i : Fact (finrank ℝ E = n + 1)
o : Orientation ℝ E (Fin (n + 1))
v : Fin (n + 1) → E
hv : Pairwise fun i j ↦ ⟪v i, v j⟫ = 0
this : FiniteDimensional ℝ E
hdim : finrank ℝ E = Fintype.card (Fin n.succ)
b : OrthonormalBasis (Fin n.succ) ℝ E := gramSchmidtOrthonormalBasis hdim v
hb : b.toBasis.det v = ∏ i, ⟪b i, v i⟫
⊢ ∏ x, |⟪b x, v x⟫| = ∏ i, ‖v i‖
Module: Mathlib.Analysis.Asymptotics.SpecificAsymptotics

Goal state before:
pp: α : Type u_1
u v : α → ℝ
l : Filter α
hv : 0 ≤ v
h : u ~[l] v
r : ℝ
φ : α → ℝ
hφ : Tendsto φ l (𝓝 1)
huφv : u =ᶠ[l] φ * v
⊢ 1 ≠ 0

Tactic:
norm_num

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

Goal state before:
pp: α : Type u_1
l : Filter α
f g : α → ℝ
hfg : Tendsto (f / g) l (𝓝 1)
g_tendsto : Tendsto g l atTop
hg : ∀ᶠ (x : α) in l, g x ≠ 0
hf : ∀ᶠ (x : α) in l, f x ≠ 0
⊢ 1 ≠ 0

Tactic:
norm_num

Goal state after:
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: M : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : PartialOrder M
inst✝ : IsOrderedAddMonoid M
f : ℕ → M
u : ℕ → ℕ
hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
hu : Monotone u
n : ℕ
ihn : ∑ k ∈ Ico (u 0) (u n), f k ≤ ∑ k ∈ range n, (u (k + 1) - u k) • f (u k)
this : ∑ k ∈ Ico (u n) (u (n + 1)), f k ≤ (u (n + 1) - u n) • f (u n)
⊢ ∑ k ∈ Ico (u 0) (u (n + 1)), f k ≤ ∑ k ∈ range (n + 1), (u (k + 1) - u k) • f (u k)

Tactic:
rw [sum_range_succ, ← sum_Ico_consecutive]
      · exact add_le_add ihn this
      exacts [hu n.zero_le, hu n.le_succ]

Goal state after:
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: f : ℕ → ℝ
h_nonneg : 0 ≤ᶠ[Filter.atTop] f
h_mono : ∀ᶠ (k : ℕ) in Filter.atTop, f (k + 1) ≤ f k
⊢ (Summable fun k ↦ 2 ^ k * f (2 ^ k)) ↔ Summable f

Tactic:
rw [Filter.EventuallyLE, Filter.eventually_atTop] at h_nonneg
  rw [Filter.eventually_atTop] at h_mono
  rcases h_nonneg with ⟨n, hn⟩
  rcases h_mono with ⟨m, hm⟩
  convert! summable_condensed_iff_of_nonneg (f := fun k ↦ f (max k (n + m))) _ _ using 1
  · rw [summable_congr_atTop]
    have h_pow := tendsto_pow_atTop_atTop_of_one_lt (r := 2) (by simp)
    filter_upwards [h_pow.eventually_ge_atTop (n + m)] with _ hk using by simp [max_eq_left hk]
  · rw [summable_congr_atTop]
    filter_upwards [Filter.eventually_ge_atTop (n + m)] with _ hk using by simp [max_eq_left hk]
  · simp_all
  · intro _ _ _ _
    exact antitoneOn_nat_Ici_of_succ_le (k := n + m) (by grind) (by simp) (by simp) (by grind)

Goal state after:
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 ↔
    (Equiv.smulRight aux₄) ((Basis.ofZLatticeBasis ℝ L b).equivFun x✝) ∈
        ⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1} ∧
      (Equiv.smulRight aux₄) ((Basis.ofZLatticeBasis ℝ L b).equivFun x✝) ∈
        (c ^ (↑(card ι))⁻¹)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))

Tactic:
Equiv.smulRight_apply,

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 ↔
    c ^ (-(↑(card ι))⁻¹) • (Basis.ofZLatticeBasis ℝ L b).equivFun x✝ ∈
        ⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1} ∧
      c ^ (-(↑(card ι))⁻¹) • (Basis.ofZLatticeBasis ℝ L b).equivFun x✝ ∈
        (c ^ (↑(card ι))⁻¹)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))
Module: Mathlib.Algebra.Module.ZLattice.Summable

Goal state before:
pp: E✝ : Type u_1
inst✝⁴ : NormedAddCommGroup E✝
L✝ : Submodule ℤ E✝
ι✝ : Type u_2
b✝ : Basis ι✝ ℤ ↥L✝
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : FiniteDimensional ℝ E
L : Submodule ℤ E
inst✝ : DiscreteTopology ↥L
ι : Type u_2
b : Basis ι ℤ ↥L
H : IsZLattice ℝ L
this : Finite ι
b' : Basis ι ℝ E := Basis.ofZLatticeBasis ℝ L b
e : E ≃L[ℝ] ι → ℝ := (b'.repr ≪≫ₗ Finsupp.linearEquivFunOnFinite ℝ ℝ ι).toContinuousLinearEquiv
⊢ ∃ ε, 0 < ε ∧ ∀ (x : ↥L) (i : ι), ε * ↑|(b.repr x) i| ≤ ‖x‖

Tactic:
have := e.continuous.1 (Set.univ.pi fun _ ↦ Set.Ioo (-1) 1)
    (isOpen_set_pi Set.finite_univ fun _ _ ↦ isOpen_Ioo)

Goal state after:
E✝ : Type u_1
inst✝⁴ : NormedAddCommGroup E✝
L✝ : Submodule ℤ E✝
ι✝ : Type u_2
b✝ : Basis ι✝ ℤ ↥L✝
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : FiniteDimensional ℝ E
L : Submodule ℤ E
inst✝ : DiscreteTopology ↥L
ι : Type u_2
b : Basis ι ℤ ↥L
H : IsZLattice ℝ L
this✝ : Finite ι
b' : Basis ι ℝ E := Basis.ofZLatticeBasis ℝ L b
e : E ≃L[ℝ] ι → ℝ := (b'.repr ≪≫ₗ Finsupp.linearEquivFunOnFinite ℝ ℝ ι).toContinuousLinearEquiv
this : IsOpen (⇑e ⁻¹' Set.univ.pi fun x ↦ Set.Ioo (-1) 1)
⊢ ∃ ε, 0 < ε ∧ ∀ (x : ↥L) (i : ι), ε * ↑|(b.repr x) i| ≤ ‖x‖
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: case e'_3
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
L : Submodule ℤ E
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
s : Set E
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
b : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
x✝ : ℕ
⊢ finrank ℝ E = finrank ℤ ↥L

Tactic:
ZLattice.rank ℝ L

Goal state after:
case e'_3
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
L : Submodule ℤ E
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
s : Set E
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
b : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
x✝ : ℕ
⊢ finrank ℝ E = finrank ℝ E
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: case convert_2
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
L : Submodule ℤ E
inst✝² : DiscreteTopology ↥L
inst✝¹ : IsZLattice ℝ L
inst✝ : Nontrivial E
X : Set E
F : E → ℝ
hX : ∀ ⦃x : E⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
h₁ : ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ finrank ℝ E * F x
h₂ : Bornology.IsBounded {x | x ∈ X ∧ F x ≤ 1}
h₃ : MeasurableSet {x | x ∈ X ∧ F x ≤ 1}
h₄ : volume (frontier {x | x ∈ X ∧ F x ≤ 1}) = 0
b : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
⊢ ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ finrank ℤ ↥L * F x

Tactic:
ZLattice.rank ℝ L

Goal state after:
case convert_2
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
L : Submodule ℤ E
inst✝² : DiscreteTopology ↥L
inst✝¹ : IsZLattice ℝ L
inst✝ : Nontrivial E
X : Set E
F : E → ℝ
hX : ∀ ⦃x : E⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
h₁ : ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ finrank ℝ E * F x
h₂ : Bornology.IsBounded {x | x ∈ X ∧ F x ≤ 1}
h₃ : MeasurableSet {x | x ∈ X ∧ F x ≤ 1}
h₄ : volume (frontier {x | x ∈ X ∧ F x ≤ 1}) = 0
b : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
⊢ ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ finrank ℝ E * F x
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: α : Type u_1
s : Finset α
f g : α → ℕ
h : ∀ a ∈ s, f a = g a
⊢ multinomial s f = multinomial s g

Tactic:
simp only [multinomial]; congr 1
  · rw [Finset.sum_congr rfl h]
  · exact Finset.prod_congr rfl fun a ha => by rw [h a ha]

Goal state after:
Module: Mathlib.Data.DFinsupp.WellFounded

Goal state before:
pp: case refine_1
ι : Type u_1
α : ι → Type u_2
inst✝² : (i : ι) → Zero (α i)
inst✝¹ : (i : ι) → Preorder (α i)
inst✝ : ∀ (i : ι), WellFoundedLT (α i)
hbot : ∀ ⦃i : ι⦄ ⦃a : α i⦄, ¬a < 0
β : ι → Type u_2 := fun i ↦ Antisymmetrization (α i) fun x1 x2 ↦ x1 ≤ x2
e : (i : ι) → α i → β i := fun i ↦ toAntisymmetrization fun x1 x2 ↦ x1 ≤ x2
x✝ : (i : ι) → Zero (β i) := fun i ↦ { zero := e i 0 }
⊢ ∀ ⦃i : ι⦄ ⦃a : β i⦄, ¬a < 0

Tactic:
rintro i ⟨a⟩

Goal state after:
case refine_1.mk
ι : Type u_1
α : ι → Type u_2
inst✝² : (i : ι) → Zero (α i)
inst✝¹ : (i : ι) → Preorder (α i)
inst✝ : ∀ (i : ι), WellFoundedLT (α i)
hbot : ∀ ⦃i : ι⦄ ⦃a : α i⦄, ¬a < 0
β : ι → Type u_2 := fun i ↦ Antisymmetrization (α i) fun x1 x2 ↦ x1 ≤ x2
e : (i : ι) → α i → β i := fun i ↦ toAntisymmetrization fun x1 x2 ↦ x1 ≤ x2
x✝ : (i : ι) → Zero (β i) := fun i ↦ { zero := e i 0 }
i : ι
a✝ : β i
a : α i
⊢ ¬Quot.mk (⇑(AntisymmRel.setoid (α i) fun x1 x2 ↦ x1 ≤ x2)) a < 0
Module: Mathlib.Data.Finsupp.MonomialOrder.DegLex

Goal state before:
pp: α : Type u_1
r : α → α → Prop
inst✝ : Std.Trichotomous r
hr : WellFounded (Function.swap r)
s : ℕ → ℕ → Prop
hs : WellFounded s
hs0 : ∀ ⦃n : ℕ⦄, ¬s n 0
⊢ WellFounded (Finsupp.DegLex r s)

Tactic:
have wft := WellFounded.prod_lex hs (Finsupp.Lex.wellFounded' hs0 hs hr)
  rw [← Set.wellFoundedOn_univ] at wft
  unfold Finsupp.DegLex
  rw [← Set.wellFoundedOn_range]
  exact Set.WellFoundedOn.mono wft (le_refl _) (fun _ _ ↦ trivial)

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

Goal state before:
pp: case mpr
R : Type u_1
σ : Type u_2
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
a : R
ha : a ≠ 0
b : R
hab : b ∣ a
⊢ C b ∣ C a

Tactic:
exact map_dvd C hab

Goal state after:
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
h : m.toSyn (m.degree f) < m.toSyn (m.degree g)
⊢ m.degree (f + g) = m.degree g

Tactic:
rw [add_comm]

Goal state after:
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommSemiring R
f g : MvPolynomial σ R
h : m.toSyn (m.degree f) < m.toSyn (m.degree g)
⊢ m.degree (g + f) = m.degree g
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
c : σ →₀ ℕ
hc : m.toSyn (m.degree f + m.degree g) < m.toSyn c
⊢ coeff c (f * g) = 0

Tactic:
rw [coeff_mul]

Goal state after:
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommSemiring R
f g : MvPolynomial σ R
c : σ →₀ ℕ
hc : m.toSyn (m.degree f + m.degree g) < m.toSyn c
⊢ ∑ x ∈ Finset.antidiagonal c, coeff x.1 f * coeff x.2 g = 0
Module: Mathlib.Algebra.Order.Antidiag.FinsuppEquiv

Goal state before:
pp: ι : Type u_1
inst✝ : DecidableEq ι
s : Finset ι
n : ℕ
f : Sym (↥s) n
a : ↥s
⊢ ↑((s.finsuppAntidiagEquiv n).symm f) ↑a = Multiset.count a ↑f

Tactic:
simp [finsuppAntidiagEquiv]

Goal state after:
Module: Mathlib.Algebra.Order.Antidiag.FinsuppEquiv

Goal state before:
pp: ι : Type u_1
inst✝ : DecidableEq ι
s : Finset ι
n : ℕ
f : ↥(s.finsuppAntidiag n)
a : ↥s
⊢ Multiset.count a ↑((s.finsuppAntidiagEquiv n) f) = ↑f ↑a

Tactic:
simp [finsuppAntidiagEquiv]

Goal state after:
Module: Mathlib.RingTheory.MvPolynomial.MonomialOrder

Goal state before:
pp: σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommSemiring R
f : MvPolynomial σ R
n : ℕ
hf : m.Monic f
a✝ : Nontrivial R
⊢ m.Monic (f ^ n)

Tactic:
rw [Monic, leadingCoeff_pow_of_pow_leadingCoeff_ne_zero, hf.leadingCoeff_eq_one, one_pow]

Goal state after:
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommSemiring R
f : MvPolynomial σ R
n : ℕ
hf : m.Monic f
a✝ : Nontrivial R
⊢ m.leadingCoeff f ^ n ≠ 0
Module: Mathlib.RingTheory.MvPolynomial.MonomialOrder

Goal state before:
pp: σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommSemiring R
ι : Type u_3
P : ι → MvPolynomial σ R
s : Finset ι
H : ∀ i ∈ s, m.Monic (P i)
i : ι
⊢ i ∈ s → IsRegular (m.leadingCoeff (P i))

Tactic:
hi

Goal state after:
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommSemiring R
ι : Type u_3
P : ι → MvPolynomial σ R
s : Finset ι
H : ∀ i ∈ s, m.Monic (P i)
i : ι
hi : i ∈ s
⊢ IsRegular (m.leadingCoeff (P i))
Module: Mathlib.Data.Nat.Squarefree

Goal state before:
pp: n : ℕ
⊢ (Squarefree n ∧ ∃ p k, Prime p ∧ 0 < k ∧ p ^ k = n) → Prime n

Tactic:
rintro ⟨h, p, k, hp, hk, rfl⟩

Goal state after:
p k : ℕ
hp : Prime p
hk : 0 < k
h : Squarefree (p ^ k)
⊢ Prime (p ^ k)