mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Convex.ConvexSpace.Module

Goal state before:
pp: R : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : Semiring R
inst✝⁹ : PartialOrder R
inst✝⁸ : IsStrictOrderedRing R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
f g : M → N
inst✝³ : ConvexSpace R M
inst✝² : IsModuleConvexSpace R M
inst✝¹ : ConvexSpace R N
inst✝ : IsModuleConvexSpace R N
hf : IsAffineMap R f
hg : IsAffineMap R g
w : StdSimplex R M
⊢ (f + g) (sConvexComb w) = sConvexComb (StdSimplex.map (f + g) w)

Tactic:
by
    simp [hf.map_sum_weights, hg.map_sum_weights, Finsupp.sum_mapDomain_index, add_smul]

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

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
inst✝⁵ : Field 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
F : Finset (Set E)
h_convex : ∀ X ∈ F, Convex 𝕜 X
h_inter : ∀ G ⊆ F, #G ≤ finrank 𝕜 E + 1 → (⋂₀ ↑G).Nonempty
G : Finset (Set E)
⊢ G ⊆ F → #G ≤ finrank 𝕜 E + 1 → (⋂ i ∈ G, i).Nonempty

Tactic:
hG_ss

Goal state after:
𝕜 : Type u_2
E : Type u_3
inst✝⁵ : Field 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
F : Finset (Set E)
h_convex : ∀ X ∈ F, Convex 𝕜 X
h_inter : ∀ G ⊆ F, #G ≤ finrank 𝕜 E + 1 → (⋂₀ ↑G).Nonempty
G : Finset (Set E)
hG_ss : G ⊆ F
⊢ #G ≤ finrank 𝕜 E + 1 → (⋂ i ∈ G, i).Nonempty
Module: Mathlib.Analysis.Convex.Radon

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝⁷ : Field 𝕜
inst✝⁶ : LinearOrder 𝕜
inst✝⁵ : IsStrictOrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : T2Space E
F : ι → Set E
h_convex : ∀ (i : ι), Convex 𝕜 (F i)
h_compact : ∀ (i : ι), IsCompact (F i)
h_inter : ∀ (I : Finset ι), #I ≤ finrank 𝕜 E + 1 → (⋂ i ∈ I, F i).Nonempty
h_nonempty : Nonempty ι
I : Finset ι
⊢ ∀ i ∈ I, Convex 𝕜 (F i)

Tactic:
by simp [h_convex]

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

Goal state before:
pp: V : Type u
⊢ Injective fun x ↦ { Adj := fun v w ↦ ↑x v w = true, symm := ⋯, loopless := ⋯ }

Tactic:
by
    rintro ⟨adj, _⟩ ⟨adj', _⟩
    simp only [mk.injEq, Subtype.mk.injEq]
    intro h
    funext v w
    simpa [Bool.coe_iff_coe] using congr_fun₂ h v w

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

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁶ : Field R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AffineSpace V P
n : ℕ
inst✝ : NeZero n
s : Simplex R P n
w₁ w₂ : Fin (n + 1) → R
hw₁ : ∑ j, w₁ j = 1
hw₂ : ∑ j, w₂ j = 1
i : Fin (n + 1)
hs : SignType.sign (w₁ i) = SignType.sign (w₂ i)
h0 : w₁ i ≠ 0
⊢ (affineSpan R (Set.range (s.faceOpposite i).points)).SSameSide ((Finset.affineCombination R Finset.univ s.points) w₁)
    ((Finset.affineCombination R Finset.univ s.points) w₂)

Tactic:
have h0' : w₂ i ≠ 0 := by intro h; simp_all
  refine ⟨?_, (s.affineCombination_mem_affineSpan_faceOpposite_iff hw₁).not.2 h0,
    (s.affineCombination_mem_affineSpan_faceOpposite_iff hw₂).not.2 h0'⟩
  obtain ⟨j, hj⟩ : ∃ j, j ≠ i := exists_ne _
  have hj' : s.points j ∈ affineSpan R (Set.range (s.faceOpposite i).points) := by
    simpa using hj
  refine (wSameSide_iff_exists_left hj').2 (.inr ?_)
  rw [← Finset.univ.affineCombination_piSingle R s.points
    (Finset.mem_univ j), Finset.affineCombination_vsub]
  let w₃ : Fin (n + 1) → R :=
    w₂ - w₂ i • (w₁ i)⁻¹ • (w₁ - Pi.single j 1)
  have hw₃1 : ∑ k, w₃ k = 1 := by simp [w₃, hw₂, ← Finset.mul_sum, hw₁]
  have hw₃i : w₃ i = 0 := by simp [w₃, hj.symm, h0]
  refine ⟨Finset.univ.affineCombination R s.points w₃,
    (s.affineCombination_mem_affineSpan_faceOpposite_iff hw₃1).2 hw₃i, ?_⟩
  simp only [w₃, Finset.affineCombination_vsub, sub_sub_cancel, smul_smul, map_smul,
    sameRay_smul_right_iff]
  left
  rcases h0.lt_or_gt with h | h
  · rw [sign_neg h, eq_comm, sign_eq_neg_one_iff] at hs
    exact (mul_pos_of_neg_of_neg hs (inv_neg''.2 h)).le
  · rw [sign_pos h, eq_comm, sign_eq_one_iff] at hs
    positivity

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
r t : ℝ
hr : 0 < r
ht : 0 < t
x : E
⊢ t ≤ (1 + ‖x‖) ^ (-r) ↔ 1 + ‖x‖ ≤ t ^ (-r⁻¹)

Tactic:
neg_inv

Goal state after:
E : Type u_1
inst✝ : NormedAddCommGroup E
r t : ℝ
hr : 0 < r
ht : 0 < t
x : E
⊢ t ≤ (1 + ‖x‖) ^ (-r) ↔ 1 + ‖x‖ ≤ t ^ (-r)⁻¹
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E →L[ℝ] F
x : E
⊢ ‖f x‖ ≤ ‖f‖ * (1 + ‖x‖) ^ 1

Tactic:
· exact (f.le_opNorm x).trans (by simp [mul_add])

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

Goal state before:
pp: case mp
𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →L[𝕜] E
r : ℝ
hr : 0 < r
a : ℝ
x : E
hx : x ≠ 0
hxT : T.rayleighQuotient x = a
c : 𝕜 := ↑‖x‖⁻¹ * ↑r
this : c ≠ 0
⊢ a ∈ T.rayleighQuotient '' sphere 0 r

Tactic:
refine ⟨c • x, ?_, ?_⟩

Goal state after:
case mp.refine_1
𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →L[𝕜] E
r : ℝ
hr : 0 < r
a : ℝ
x : E
hx : x ≠ 0
hxT : T.rayleighQuotient x = a
c : 𝕜 := ↑‖x‖⁻¹ * ↑r
this : c ≠ 0
⊢ c • x ∈ sphere 0 r

case mp.refine_2
𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →L[𝕜] E
r : ℝ
hr : 0 < r
a : ℝ
x : E
hx : x ≠ 0
hxT : T.rayleighQuotient x = a
c : 𝕜 := ↑‖x‖⁻¹ * ↑r
this : c ≠ 0
⊢ T.rayleighQuotient (c • x) = a
Module: Mathlib.Analysis.Distribution.TestFunction

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
Ω : Opens E
F : Type u_4
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedSpace 𝕜 F
n k : ℕ∞
inst✝¹ : Algebra ℝ 𝕜
inst✝ : IsScalarTower ℝ 𝕜 F
v₁ v₂ : E
⊢ lineDerivCLM 𝕜 (v₁ + v₂) = lineDerivCLM 𝕜 v₁ + lineDerivCLM 𝕜 v₂

Tactic:
ext
  simp [-lineDerivCLM_apply, lineDerivCLM_eq_fderivCLM]

Goal state after:
Module: Mathlib.Analysis.Distribution.TestFunction

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
Ω : Opens E
F : Type u_4
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedSpace 𝕜 F
n k : ℕ∞
inst✝¹ : Algebra ℝ 𝕜
inst✝ : IsScalarTower ℝ 𝕜 F
c : ℝ
v : E
⊢ lineDerivCLM 𝕜 (c • v) = c • lineDerivCLM 𝕜 v

Tactic:
ext
  simp [-lineDerivCLM_apply, lineDerivCLM_eq_fderivCLM]

Goal state after:
Module: Mathlib.Analysis.Distribution.TestFunction

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
Ω : Opens E
n : ℕ∞
m : MeasurableSpace E
inst✝⁷ : OpensMeasurableSpace E
F₁ : Type u_6
F₂ : Type u_7
F₃ : Type u_8
inst✝⁶ : NormedAddCommGroup F₁
inst✝⁵ : NormedSpace 𝕜 F₁
inst✝⁴ : NormedSpace ℝ F₁
inst✝³ : NormedAddCommGroup F₂
inst✝² : NormedSpace 𝕜 F₂
inst✝¹ : NormedAddCommGroup F₃
inst✝ : NormedSpace 𝕜 F₃
B : F₁ →L[𝕜] F₂ →L[𝕜] F₃
μ : Measure E
φ : E → F₂
hφ : LocallyIntegrableOn φ (↑Ω) μ
f : 𝓓^{n}(Ω, F₁)
⊢ IntegrableOn (fun x ↦ (B (f x)) (φ x)) (tsupport ⇑f) μ

Tactic:
replace hφ := hφ.integrableOn_compact_subset f.tsupport_subset f.hasCompactSupport

Goal state after:
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
Ω : Opens E
n : ℕ∞
m : MeasurableSpace E
inst✝⁷ : OpensMeasurableSpace E
F₁ : Type u_6
F₂ : Type u_7
F₃ : Type u_8
inst✝⁶ : NormedAddCommGroup F₁
inst✝⁵ : NormedSpace 𝕜 F₁
inst✝⁴ : NormedSpace ℝ F₁
inst✝³ : NormedAddCommGroup F₂
inst✝² : NormedSpace 𝕜 F₂
inst✝¹ : NormedAddCommGroup F₃
inst✝ : NormedSpace 𝕜 F₃
B : F₁ →L[𝕜] F₂ →L[𝕜] F₃
μ : Measure E
φ : E → F₂
f : 𝓓^{n}(Ω, F₁)
hφ : IntegrableOn φ (tsupport ⇑f) μ
⊢ IntegrableOn (fun x ↦ (B (f x)) (φ x)) (tsupport ⇑f) μ
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: case h
E : Type u_5
F : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : μ.HasTemperateGrowth
f : E → F
C₁ C₂ : ℝ
k : ℕ
hf : ∀ (x : E), ‖f x‖ ≤ C₁
h'f : ∀ (x : E), ‖x‖ ^ (k + μ.integrablePower) * ‖f x‖ ≤ C₂
h''f : AEStronglyMeasurable f μ
v : E
⊢ ‖‖v‖ ^ k * ‖f v‖‖ ≤ 2 ^ μ.integrablePower * (C₁ + C₂) * (1 + ‖v‖) ^ (-↑μ.integrablePower)

Tactic:
simp only [norm_mul, norm_pow, norm_norm]

Goal state after:
case h
E : Type u_5
F : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : μ.HasTemperateGrowth
f : E → F
C₁ C₂ : ℝ
k : ℕ
hf : ∀ (x : E), ‖f x‖ ≤ C₁
h'f : ∀ (x : E), ‖x‖ ^ (k + μ.integrablePower) * ‖f x‖ ≤ C₂
h''f : AEStronglyMeasurable f μ
v : E
⊢ ‖v‖ ^ k * ‖f v‖ ≤ 2 ^ μ.integrablePower * (C₁ + C₂) * (1 + ‖v‖) ^ (-↑μ.integrablePower)
Module: Mathlib.Analysis.Matrix.Order

Goal state before:
pp: 𝕜 : Type u_1
n : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Finite n
m : Type u_3
inst✝ : Finite m
y : Matrix m m 𝕜
hy : y.PosSemidef
this✝ : Fintype n
this : Fintype m
a : Matrix n n 𝕜
hx : (star a * a).PosSemidef
⊢ (kroneckerMap (fun x1 x2 ↦ x1 * x2) (star a * a) y).PosSemidef

Tactic:
obtain ⟨b, rfl⟩ := CStarAlgebra.nonneg_iff_eq_star_mul_self.mp hy.nonneg

Goal state after:
𝕜 : Type u_1
n : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Finite n
m : Type u_3
inst✝ : Finite m
this✝ : Fintype n
this : Fintype m
a : Matrix n n 𝕜
hx : (star a * a).PosSemidef
b : Matrix m m 𝕜
hy : (star b * b).PosSemidef
⊢ (kroneckerMap (fun x1 x2 ↦ x1 * x2) (star a * a) (star b * b)).PosSemidef
Module: Mathlib.MeasureTheory.Function.LpSpace.ContinuousFunctions

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : BorelSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : SecondCountableTopologyEither α E
inst✝¹ : IsFiniteMeasure μ
inst✝ : Fact (1 ≤ p)
⊢ ((ContinuousMap.toAEEqFunAddHom μ).comp (toContinuousMapAddMonoidHom α E)).range ≤ Lp E p μ

Tactic:
by rintro - ⟨f, rfl⟩; exact mem_Lp f

Goal state after:
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: case pos
𝕜 : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : SMulCommClass ℝ 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedSpace 𝕜 F'
inst✝ : SMulCommClass ℝ 𝕜 F'
n n₁ n₂ k : ℕ∞
K K₁ K₂ : Compacts E
A : 𝓓^{n₁}_{K₁}(E, F) → E → F'
hadd : ∀ (f g : 𝓓^{n₁}_{K₁}(E, F)) (x : E), A (f + g) x = A f x + A g x
hsmul : ∀ (c : 𝕜) (f : 𝓓^{n₁}_{K₁}(E, F)) (x : E), A (c • f) x = c • A f x
hsmooth : ∀ (f : 𝓓^{n₁}_{K₁}(E, F)), ContDiff ℝ (↑n₂) (A f)
hsupp : ∀ (f : 𝓓^{n₁}_{K₁}(E, F)), EqOn (A f) 0 (↑K₂)ᶜ
hbound :
  ∀ (i : ℕ),
    ↑i ≤ n₂ →
      ∃ s C,
        0 ≤ C ∧
          ∀ (f : 𝓓^{n₁}_{K₁}(E, F)),
            ∀ x ∈ K₂, ‖iteratedFDeriv ℝ i (A f) x‖ ≤ C * (s.sup fun j ↦ ContDiffMapSupportedIn.seminorm 𝕜 E F n₁ K₁ j) f
Φ : 𝓓^{n₁}_{K₁}(E, F) →ₗ[𝕜] 𝓓^{n₂}_{K₂}(E, F') :=
  { toFun := fun f ↦ { toFun := A f, contDiff' := ⋯, zero_on_compl' := ⋯ }, map_add' := ⋯, map_smul' := ⋯ }
i : ℕ
hi : ↑i ≤ n₂
s : Finset ℕ
C : ℝ
hC : 0 ≤ C
h :
  ∀ (f : 𝓓^{n₁}_{K₁}(E, F)),
    ∀ x ∈ K₂, ‖iteratedFDeriv ℝ i (A f) x‖ ≤ C * (s.sup fun j ↦ ContDiffMapSupportedIn.seminorm 𝕜 E F n₁ K₁ j) f
⊢ ∃ s C,
    ∀ (x : 𝓓^{n₁}_{K₁}(E, F)),
      (ContDiffMapSupportedIn.seminorm 𝕜 E F' n₂ K₂ i) (Φ x) ≤
        C * (s.sup (ContDiffMapSupportedIn.seminorm 𝕜 E F n₁ K₁)) x

Tactic:
exact ⟨s, C, fun f ↦ ((Φ f).seminorm_le_iff 𝕜 (mul_nonneg hC (apply_nonneg _ _)) i).2
          fun _ x hx ↦ h f x hx⟩

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs

Goal state before:
pp: A : Type u_2
inst✝¹¹ : Ring A
inst✝¹⁰ : StarRing A
inst✝⁹ : TopologicalSpace A
inst✝⁸ : Algebra ℝ A
inst✝⁷ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁶ : PartialOrder A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : NonnegSpectrumClass ℝ A
inst✝³ : IsTopologicalRing A
inst✝² : T2Space A
inst✝¹ : StarModule ℝ A
n : ℕ
inst✝ : n.AtLeastTwo
⊢ abs (OfNat.ofNat n) = OfNat.ofNat n

Tactic:
simpa using! abs_natCast n

Goal state after:
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : ↥(Lp E 2 μ)
h_two : ENNReal.toReal 2 = 2
⊢ (eLpNorm (↑↑f) 2 μ).toReal ^ 2 = RCLike.re ↑(∫⁻ (a : α), ↑‖↑↑f a‖₊ ^ 2 ∂μ).toReal

Tactic:
← ENNReal.toReal_pow,

Goal state after:
α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : ↥(Lp E 2 μ)
h_two : ENNReal.toReal 2 = 2
⊢ (eLpNorm (↑↑f) 2 μ ^ 2).toReal = RCLike.re ↑(∫⁻ (a : α), ↑‖↑↑f a‖₊ ^ 2 ∂μ).toReal
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs

Goal state before:
pp: A : Type u_2
inst✝⁹ : Ring A
inst✝⁸ : TopologicalSpace A
inst✝⁷ : StarRing A
inst✝⁶ : PartialOrder A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : IsTopologicalRing A
inst✝¹ : T2Space A
inst✝ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
⊢ abs a ^ 2 = star a * a

Tactic:
sq,

Goal state after:
A : Type u_2
inst✝⁹ : Ring A
inst✝⁸ : TopologicalSpace A
inst✝⁷ : StarRing A
inst✝⁶ : PartialOrder A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : IsTopologicalRing A
inst✝¹ : T2Space A
inst✝ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
⊢ abs a * abs a = star a * a
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs

Goal state before:
pp: A : Type u_2
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : StarRing A
inst✝⁷ : CStarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : SMulCommClass ℝ A A
inst✝⁴ : IsScalarTower ℝ A A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : NonnegSpectrumClass ℝ A
a : A
⊢ ‖abs a‖ ^ 2 = ‖a‖ ^ 2

Tactic:
sq,

Goal state after:
A : Type u_2
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : StarRing A
inst✝⁷ : CStarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : SMulCommClass ℝ A A
inst✝⁴ : IsScalarTower ℝ A A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : NonnegSpectrumClass ℝ A
a : A
⊢ ‖abs a‖ * ‖abs a‖ = ‖a‖ ^ 2
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs

Goal state before:
pp: A : Type u_2
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : StarRing A
inst✝⁷ : CStarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : SMulCommClass ℝ A A
inst✝⁴ : IsScalarTower ℝ A A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : NonnegSpectrumClass ℝ A
a : A
⊢ ‖abs a‖ * ‖abs a‖ = ‖a‖ ^ 2

Tactic:
sq,

Goal state after:
A : Type u_2
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : StarRing A
inst✝⁷ : CStarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : SMulCommClass ℝ A A
inst✝⁴ : IsScalarTower ℝ A A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : NonnegSpectrumClass ℝ A
a : A
⊢ ‖abs a‖ * ‖abs a‖ = ‖a‖ * ‖a‖
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs

Goal state before:
pp: A : Type u_2
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : StarRing A
inst✝⁷ : CStarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : SMulCommClass ℝ A A
inst✝⁴ : IsScalarTower ℝ A A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : NonnegSpectrumClass ℝ A
a : A
⊢ ‖star (abs a) * abs a‖ = ‖a‖ * ‖a‖

Tactic:
(abs_nonneg _).star_eq,

Goal state after:
A : Type u_2
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : StarRing A
inst✝⁷ : CStarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : SMulCommClass ℝ A A
inst✝⁴ : IsScalarTower ℝ A A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : NonnegSpectrumClass ℝ A
a : A
⊢ ‖abs a * abs a‖ = ‖a‖ * ‖a‖
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
ι : Type u_5
inst✝ : Finite ι
s : ι → Set α
mv : ∀ (j : ι), MeasurableSet (s j)
hv : ∀ (j : ι), μ (s j) ≠ ∞
⊢ (Matrix.of fun i j ↦ μ.real (s i ∩ s j)).PosSemidef

Tactic:
simp only [mv, ne_eq, hv, not_false_eq_true,
    ← real_inner_indicatorConstLp_one_indicatorConstLp_one]
  exact Matrix.posSemidef_gram _ _

Goal state after:
Module: Mathlib.MeasureTheory.Function.ContinuousMapDense

Goal state before:
pp: α : Type u_1
inst✝⁶ : TopologicalSpace α
inst✝⁵ : NormalSpace α
inst✝⁴ : MeasurableSpace α
inst✝³ : BorelSpace α
E : Type u_2
inst✝² : NormedAddCommGroup E
μ : Measure α
inst✝¹ : NormedSpace ℝ E
inst✝ : μ.WeaklyRegular
f : α → E
hf : Integrable f μ
ε : ℝ
hε : 0 < ε
⊢ ∃ g, ∫ (x : α), ‖f x - g x‖ ∂μ ≤ ε ∧ Integrable (⇑g) μ

Tactic:
simp only [← memLp_one_iff_integrable, ← ENNReal.ofReal_one]
    at hf ⊢
  simpa using hf.exists_boundedContinuous_integral_rpow_sub_le zero_lt_one hε

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
G : ι → Type u_4
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝ : CompleteSpace E
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV✝ hV : OrthogonalFamily 𝕜 G V
f g : ↥(lp G 2)
⊢ ∑' (i : ι), (V i) (↑(f + g) i) = ∑' (i : ι), (V i) (↑f i) + ∑' (i : ι), (V i) (↑g i)

Tactic:
by
    simp only [(hV.summable_of_lp f).tsum_add (hV.summable_of_lp g), lp.coeFn_add, Pi.add_apply,
      LinearIsometry.map_add]

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
n : ℤ
hn : n ≠ 0
hT : 0 < T
x : AddCircle T
⊢ ↑(toCircle ↑(n • (T / 2 / ↑n))) = -1

Tactic:
rw [zsmul_eq_mul, toCircle, Function.Periodic.lift_coe, Circle.coe_exp]

Goal state after:
T : ℝ
n : ℤ
hn : n ≠ 0
hT : 0 < T
x : AddCircle T
⊢ Complex.exp (↑(2 * π / T * (↑n * (T / 2 / ↑n))) * Complex.I) = -1
Module: Mathlib.MeasureTheory.Integral.PeakFunction

Goal state before:
pp: α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst✝³ : TopologicalSpace α
inst✝² : BorelSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s t : Set α
φ : ι → α → ℝ
hs : MeasurableSet s
ht : MeasurableSet t
hts : t ⊆ s
h'ts : t ∈ 𝓝[s] x₀
hnφ : ∀ᶠ (i : ι) in l, ∀ x ∈ s, 0 ≤ φ i x
hlφ : ∀ (u : Set α), IsOpen[inst✝³] u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)
hiφ : Tendsto (fun i ↦ ∫ (x : α) in t, φ i x ∂μ) l (𝓝 1)
h'iφ : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (φ i) (μ.restrict s)
hmg : IntegrableOn g s μ
hcg : Tendsto g (𝓝[s] x₀) (𝓝 0)
ε : ℝ
εpos : ε > 0
δ : ℝ
hδ : δ * ∫ (x : α) in s, ‖g x‖ ∂μ + 2 * δ < ε
δpos : 0 < δ
δone : δ < 1
u : Set α
u_open : IsOpen[inst✝³] u
x₀u : x₀ ∈ u
ut : s ∩ u ⊆ t
hu : ∀ x ∈ u ∩ s, g x ∈ ball 0 δ
i : ι
hi : ∀ x ∈ s \ u, dist (0 x) (φ i x) < δ
h'i : ‖∫ (x : α) in t, φ i x ∂μ - 1‖ < δ
hφpos : ∀ x ∈ s, 0 ≤ φ i x
h''i : IntegrableOn (fun x ↦ φ i x • g x) s μ
I : IntegrableOn (φ i) t μ
x : α
hx : x ∈ t
⊢ ‖φ i x‖ * δ = φ i x * δ

Tactic:
Real.norm_of_nonneg (hφpos _ (hts hx))

Goal state after:
α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst✝³ : TopologicalSpace α
inst✝² : BorelSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s t : Set α
φ : ι → α → ℝ
hs : MeasurableSet s
ht : MeasurableSet t
hts : t ⊆ s
h'ts : t ∈ 𝓝[s] x₀
hnφ : ∀ᶠ (i : ι) in l, ∀ x ∈ s, 0 ≤ φ i x
hlφ : ∀ (u : Set α), IsOpen[inst✝³] u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)
hiφ : Tendsto (fun i ↦ ∫ (x : α) in t, φ i x ∂μ) l (𝓝 1)
h'iφ : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (φ i) (μ.restrict s)
hmg : IntegrableOn g s μ
hcg : Tendsto g (𝓝[s] x₀) (𝓝 0)
ε : ℝ
εpos : ε > 0
δ : ℝ
hδ : δ * ∫ (x : α) in s, ‖g x‖ ∂μ + 2 * δ < ε
δpos : 0 < δ
δone : δ < 1
u : Set α
u_open : IsOpen[inst✝³] u
x₀u : x₀ ∈ u
ut : s ∩ u ⊆ t
hu : ∀ x ∈ u ∩ s, g x ∈ ball 0 δ
i : ι
hi : ∀ x ∈ s \ u, dist (0 x) (φ i x) < δ
h'i : ‖∫ (x : α) in t, φ i x ∂μ - 1‖ < δ
hφpos : ∀ x ∈ s, 0 ≤ φ i x
h''i : IntegrableOn (fun x ↦ φ i x • g x) s μ
I : IntegrableOn (φ i) t μ
x : α
hx : x ∈ t
⊢ φ i x * δ = φ i x * δ
Module: Mathlib.Analysis.Fourier.FourierTransformDeriv

Goal state before:
pp: x y : ℝ
⊢ ↑(𝐞 y) = (fourier 1) ↑y

Tactic:
rw [fourierChar_apply, fourier_coe_apply]

Goal state after:
x y : ℝ
⊢ cexp (↑(2 * π * y) * I) = cexp (2 * ↑π * I * ↑1 * ↑y / ↑1)
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
f : 𝓢(E, F)
inst✝ : ProperSpace E
s : ℝ
⊢ ⇑f =O[cocompact E] fun x ↦ ‖x‖ ^ s

Tactic:
let k := ⌈-s⌉₊

Goal state after:
E : Type u_5
F : Type u_6
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
f : 𝓢(E, F)
inst✝ : ProperSpace E
s : ℝ
k : ℕ := ⌈-s⌉₊
⊢ ⇑f =O[cocompact E] fun x ↦ ‖x‖ ^ s
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_5
F : Type u_6
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
k n : ℕ
c : 𝕜
f : 𝓢(E, F)
x : E
⊢ ‖c‖ * ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖ ≤ ‖c‖ * seminormAux k n f

Tactic:
rw [mul_assoc]

Goal state after:
𝕜 : Type u_2
E : Type u_5
F : Type u_6
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
k n : ℕ
c : 𝕜
f : 𝓢(E, F)
x : E
⊢ ‖c‖ * (‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖) ≤ ‖c‖ * seminormAux k n f
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
𝕜' : Type u_3
D : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
H : Type u_8
V : Type u_9
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : NormedField 𝕜
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : SMulCommClass ℝ 𝕜 F
inst✝² : NormedField 𝕜'
inst✝¹ : NormedSpace 𝕜' F
inst✝ : SMulCommClass ℝ 𝕜' F
c : ℤ
f : 𝓢(E, F)
⊢ ∀ (k n : ℕ), ∃ C, ∀ (x : E), ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (c • ⇑f) x‖ ≤ C

Tactic:
by simpa [← Int.cast_smul_eq_zsmul ℝ] using! ((c : ℝ) • f).decay'

Goal state after: