mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.AffineSpace.Independent

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p₁ p₂ p₃ : P
h : AffineIndependent k ![p₁, p₂, p₃]
⊢ AffineIndependent k ![p₂, p₁, p₃]

Tactic:
by
  rw [← affineIndependent_equiv (Equiv.swap 0 1)]
  convert! h using 1
  ext x
  fin_cases x <;> rfl

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Independent

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
ι : Type u_4
p : ι → P
ha : AffineIndependent k p
fs : Finset ι
w : ι → k
hw : ∑ i ∈ fs, w i = 1
s : Set ι
hm : (affineCombination k fs p) w ∈ affineSpan k (p '' s)
i : ι
hifs : i ∈ fs
his : i ∉ s
fs' : Finset ι
w' : ι → k
hfs's : ↑fs' ⊆ s
hw' : ∑ i ∈ fs', w' i = 1
he : (affineCombination k fs p) w = (affineCombination k fs' p) w'
hi' : (↑fs).indicator w i = 0
⊢ w i = 0

Tactic:
rw [Set.indicator_apply_eq_zero] at hi'

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
ι : Type u_4
p : ι → P
ha : AffineIndependent k p
fs : Finset ι
w : ι → k
hw : ∑ i ∈ fs, w i = 1
s : Set ι
hm : (affineCombination k fs p) w ∈ affineSpan k (p '' s)
i : ι
hifs : i ∈ fs
his : i ∉ s
fs' : Finset ι
w' : ι → k
hfs's : ↑fs' ⊆ s
hw' : ∑ i ∈ fs', w' i = 1
he : (affineCombination k fs p) w = (affineCombination k fs' p) w'
hi' : i ∈ ↑fs → w i = 0
⊢ w i = 0
Module: Mathlib.Analysis.Convex.StdSimplex

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
f : ι → ℝ
hf : f ∈ stdSimplex ℝ ι
x : ι
⊢ dist (f x) (0 x) ≤ 1

Tactic:
rw [Pi.zero_apply, Real.dist_0_eq_abs, abs_of_nonneg <| hf.1 x]

Goal state after:
ι : Type u_1
inst✝ : Fintype ι
f : ι → ℝ
hf : f ∈ stdSimplex ℝ ι
x : ι
⊢ f x ≤ 1
Module: Mathlib.Analysis.Convex.Combination

Goal state before:
pp: R : Type u_1
E : Type u_3
ι : Type u_5
ι' : Type u_6
inst✝² : Field R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
s : Finset ι
t : Finset ι'
ws : ι → R
zs : ι → E
wt : ι' → R
zt : ι' → E
hws : ∑ i ∈ s, ws i = 1
hwt : ∑ i ∈ t, wt i = 1
a b : R
hab : a + b = 1
⊢ ∑ x ∈ s.disjSum t, Sum.elim (fun x ↦ a • ws x • zs x) (fun x ↦ b • wt x • zt x) x =
    ∑ i ∈ s.disjSum t, Sum.elim (fun i ↦ a * ws i) (fun j ↦ b * wt j) i • Sum.elim zs zt i

pp: case hw
R : Type u_1
E : Type u_3
ι : Type u_5
ι' : Type u_6
inst✝² : Field R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
s : Finset ι
t : Finset ι'
ws : ι → R
zs : ι → E
wt : ι' → R
zt : ι' → E
hws : ∑ i ∈ s, ws i = 1
hwt : ∑ i ∈ t, wt i = 1
a b : R
hab : a + b = 1
⊢ ∑ i ∈ s.disjSum t, Sum.elim (fun i ↦ a * ws i) (fun j ↦ b * wt j) i = 1

Tactic:
· congr with ⟨⟩ <;> simp only [Sum.elim_inl, Sum.elim_inr, mul_smul]

Goal state after:
case hw
R : Type u_1
E : Type u_3
ι : Type u_5
ι' : Type u_6
inst✝² : Field R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
s : Finset ι
t : Finset ι'
ws : ι → R
zs : ι → E
wt : ι' → R
zt : ι' → E
hws : ∑ i ∈ s, ws i = 1
hwt : ∑ i ∈ t, wt i = 1
a b : R
hab : a + b = 1
⊢ ∑ i ∈ s.disjSum t, Sum.elim (fun i ↦ a * ws i) (fun j ↦ b * wt j) i = 1
Module: Mathlib.Analysis.Convex.Combination

Goal state before:
pp: R : Type u_1
E : Type u_3
inst✝⁴ : Field R
inst✝³ : AddCommGroup E
inst✝² : Module R E
s : Set E
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
⊢ Convex R s ↔
    ∀ (t : Finset E) (w : E → R), (∀ i ∈ t, 0 ≤ w i) → ∑ i ∈ t, w i = 1 → (∀ x ∈ t, x ∈ s) → ∑ x ∈ t, w x • x ∈ s

Tactic:
refine ⟨fun hs t w hw₀ hw₁ hts => hs.sum_mem hw₀ hw₁ hts, ?_⟩

Goal state after:
R : Type u_1
E : Type u_3
inst✝⁴ : Field R
inst✝³ : AddCommGroup E
inst✝² : Module R E
s : Set E
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
⊢ (∀ (t : Finset E) (w : E → R), (∀ i ∈ t, 0 ≤ w i) → ∑ i ∈ t, w i = 1 → (∀ x ∈ t, x ∈ s) → ∑ x ∈ t, w x • x ∈ s) →
    Convex R s
Module: Mathlib.Topology.AlexandrovDiscrete

Goal state before:
pp: α : Type u_3
inst✝ : TopologicalSpace α
⊢ (∀ (x : Set (Set α)), (∀ ⦃x_1 : Set α⦄, x_1 ∈ x → IsOpen[inst✝] x_1ᶜ) → IsOpen[inst✝] (⋃₀ x)ᶜ) = ?m.14

Tactic:
isOpen_compl_iff

Goal state after:
α : Type u_3
inst✝ : TopologicalSpace α
⊢ (∀ (x : Set (Set α)), (∀ ⦃x_1 : Set α⦄, x_1 ∈ x → IsClosed[inst✝] x_1) → IsClosed[inst✝] (⋃₀ x)) = ?m.14
Module: Mathlib.Topology.AlexandrovDiscrete

Goal state before:
pp: α : Type u_3
inst✝¹ : TopologicalSpace α
inst✝ : AlexandrovDiscrete α
t : Set α
a : α
⊢ nhdsKer {a} ⊆ t ↔ t ∈ 𝓝 a

Tactic:
simp [nhdsKer_subset_iff_mem_nhdsSet]

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

Goal state before:
pp: 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 : TopologicalSpace E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousConstSMul 𝕜 E
s : Set E
hs : Convex 𝕜 s
h : (s \ interior s).Pairwise fun x y ↦ ([x -[𝕜] y] \ frontier s).Nonempty
⊢ StrictConvex 𝕜 s

Tactic:
refine hs.strictConvex' <| h.imp_on fun x hx y hy _ => ?_
  simp only [segment_eq_image_lineMap, ← self_sdiff_frontier]
  rintro ⟨_, ⟨⟨c, hc, rfl⟩, hcs⟩⟩
  refine ⟨c, hs.segment_subset hx.1 hy.1 ?_, hcs⟩
  exact lineMap_mem_segment 𝕜 x y hc

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

Goal state before:
pp: case neg.convert_3
R : Type u_1
E : Type u_3
inst✝⁴ : Field R
inst✝³ : AddCommGroup E
inst✝² : Module R E
s : Set E
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
h : ∀ (t : Finset E) (w : E → R), (∀ i ∈ t, 0 ≤ w i) → ∑ i ∈ t, w i = 1 → (∀ x ∈ t, x ∈ s) → ∑ x ∈ t, w x • x ∈ s
x : E
hx : x ∈ s
y : E
hy : y ∈ s
a b : R
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
h_cases : ¬x = y
i : E
hi : i = x ∨ i = y
⊢ i ∈ s

Tactic:
cases hi <;> subst i

Goal state after:
case neg.convert_3.inl
R : Type u_1
E : Type u_3
inst✝⁴ : Field R
inst✝³ : AddCommGroup E
inst✝² : Module R E
s : Set E
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
h : ∀ (t : Finset E) (w : E → R), (∀ i ∈ t, 0 ≤ w i) → ∑ i ∈ t, w i = 1 → (∀ x ∈ t, x ∈ s) → ∑ x ∈ t, w x • x ∈ s
x : E
hx : x ∈ s
y : E
hy : y ∈ s
a b : R
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
h_cases : ¬x = y
⊢ x ∈ s

case neg.convert_3.inr
R : Type u_1
E : Type u_3
inst✝⁴ : Field R
inst✝³ : AddCommGroup E
inst✝² : Module R E
s : Set E
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
h : ∀ (t : Finset E) (w : E → R), (∀ i ∈ t, 0 ≤ w i) → ∑ i ∈ t, w i = 1 → (∀ x ∈ t, x ∈ s) → ∑ x ∈ t, w x • x ∈ s
x : E
hx : x ∈ s
y : E
hy : y ∈ s
a b : R
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
h_cases : ¬x = y
⊢ y ∈ s
Module: Mathlib.Analysis.Convex.Combination

Goal state before:
pp: case mp
R : Type u_1
E : Type u_3
inst✝⁴ : Field R
inst✝³ : AddCommGroup E
inst✝² : Module R E
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
s : Set E
x : E
ι : Type
t : Finset ι
w : ι → R
z : ι → E
h : (∀ i ∈ t, 0 ≤ w i) ∧ ∑ i ∈ t, w i = 1 ∧ (∀ i ∈ t, z i ∈ s) ∧ t.centerMass w z = x
⊢ (∀ (i : ↥t), 0 ≤ (w ∘ Subtype.val) i) ∧
    ∑ i, (w ∘ Subtype.val) i = 1 ∧
      (∀ (i : ↥t), (z ∘ Subtype.val) i ∈ s) ∧ ∑ i, (w ∘ Subtype.val) i • (z ∘ Subtype.val) i = x

Tactic:
simpa [← sum_attach t, centerMass_eq_of_sum_1 _ _ h.2.1] using h

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

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : Set E
⊢ ediam ((convexHull ℝ) s) = ediam s

Tactic:
refine (ediam_le fun x hx y hy => ?_).antisymm (ediam_mono <| subset_convexHull ℝ s)
  rcases convexHull_exists_dist_ge2 hx hy with ⟨x', hx', y', hy', H⟩
  rw [edist_dist]
  apply le_trans (ENNReal.ofReal_le_ofReal H)
  rw [← edist_dist]
  exact edist_le_ediam_of_mem hx' hy'

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

Goal state before:
pp: case refine_1.refine_1
R : Type u_1
E : Type u_3
inst✝⁴ : Field R
inst✝³ : AddCommGroup E
inst✝² : Module R E
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
s : Set E
ι : Type
t : Finset ι
w : ι → R
z : ι → E
hw₀ : ∀ i ∈ t, 0 ≤ w i
hw₁ : ∑ i ∈ t, w i = 1
hz : ∀ i ∈ t, z i ∈ s
⊢ ↑(Finset.image z t) ⊆ s

Tactic:
rw [coe_image, Set.image_subset_iff]

Goal state after:
case refine_1.refine_1
R : Type u_1
E : Type u_3
inst✝⁴ : Field R
inst✝³ : AddCommGroup E
inst✝² : Module R E
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
s : Set E
ι : Type
t : Finset ι
w : ι → R
z : ι → E
hw₀ : ∀ i ∈ t, 0 ≤ w i
hw₁ : ∑ i ∈ t, w i = 1
hz : ∀ i ∈ t, z i ∈ s
⊢ ↑t ⊆ z ⁻¹' s
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: case mpr
𝕜 : Type u_2
𝕜₂ : Type u_3
E : Type u_6
F : Type u_7
ι' : Type u_10
inst✝⁷ : SeminormedRing 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : SeminormedRing 𝕜₂
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹ : RingHomIsometric σ₁₂
ι : Type u_11
inst✝ : Nonempty ι
p : Seminorm 𝕜 E
q : ι' → Seminorm 𝕜₂ F
f : E →ₛₗ[σ₁₂] F
h : ∀ (i : ι'), ∃ C, (q i).comp f ≤ C • p
i : ι'
⊢ ∃ s C, (q i).comp f ≤ C • s.sup fun x ↦ p

Tactic:
· use {Classical.arbitrary ι}
    simp only [h, Finset.sup_singleton]

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : TopologicalSpace E
p : SeminormFamily 𝕜 E ι
hp : WithSeminorms p
x : E
U : Set E
⊢ U ∈ 𝓝 x ↔ ∃ s, ∃ r > 0, (s.sup p).ball x r ⊆ U

Tactic:
hp.hasBasis_ball.mem_iff,

Goal state after:
𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : TopologicalSpace E
p : SeminormFamily 𝕜 E ι
hp : WithSeminorms p
x : E
U : Set E
⊢ (∃ i, 0 < i.2 ∧ (i.1.sup p).ball x i.2 ⊆ U) ↔ ∃ s, ∃ r > 0, (s.sup p).ball x r ⊆ U
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: 𝕜 : Type u_11
E : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
⊢ WithSeminorms fun x ↦ normSeminorm 𝕜 E

Tactic:
by
  rw [SeminormFamily.withSeminorms_iff_nhds_eq_iInf, iInf_const, coe_normSeminorm,
    comap_norm_nhds_zero]

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.NNNorm

Goal state before:
pp: case inr
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : SeminormedAddCommGroup F
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝ : RingHomIsometric σ₁₂
f : E →SL[σ₁₂] F
M : ℝ≥0∞
hM : ∀ (x : E), ‖f x‖ₑ ≤ M * ‖x‖ₑ
h'M : M < ∞
⊢ ‖f‖ₑ ≤ M

Tactic:
lift M to NNReal using h'M.ne

Goal state after:
case inr
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : SeminormedAddCommGroup F
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝ : RingHomIsometric σ₁₂
f : E →SL[σ₁₂] F
M : ℝ≥0
hM : ∀ (x : E), ‖f x‖ₑ ≤ ↑M * ‖x‖ₑ
h'M : ↑M < ∞
⊢ ‖f‖ₑ ≤ ↑M
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝⁶ : NormedField 𝕜
inst✝⁵ : NormedSpace ℝ 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : Module ℝ E
inst✝¹ : IsScalarTower ℝ 𝕜 E
inst✝ : TopologicalSpace E
p : SeminormFamily 𝕜 E ι
hp : WithSeminorms p
this : IsTopologicalAddGroup E
⊢ LocallyConvexSpace ℝ E

Tactic:
apply ofBasisZero ℝ E id fun s => s ∈ p.basisSets

Goal state after:
case hbasis
𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝⁶ : NormedField 𝕜
inst✝⁵ : NormedSpace ℝ 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : Module ℝ E
inst✝¹ : IsScalarTower ℝ 𝕜 E
inst✝ : TopologicalSpace E
p : SeminormFamily 𝕜 E ι
hp : WithSeminorms p
this : IsTopologicalAddGroup E
⊢ (𝓝 0).HasBasis (fun s ↦ s ∈ p.basisSets) id

case hconvex
𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝⁶ : NormedField 𝕜
inst✝⁵ : NormedSpace ℝ 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : Module ℝ E
inst✝¹ : IsScalarTower ℝ 𝕜 E
inst✝ : TopologicalSpace E
p : SeminormFamily 𝕜 E ι
hp : WithSeminorms p
this : IsTopologicalAddGroup E
⊢ ∀ i ∈ p.basisSets, Convex ℝ (id i)
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝⁶ : NormedField 𝕜
inst✝⁵ : NormedSpace ℝ 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : Module ℝ E
inst✝¹ : IsScalarTower ℝ 𝕜 E
inst✝ : TopologicalSpace E
p : SeminormFamily 𝕜 E ι
hp : WithSeminorms p
⊢ LocallyConvexSpace ℝ E

Tactic:
by
  have := hp.topologicalAddGroup
  apply ofBasisZero ℝ E id fun s => s ∈ p.basisSets
  · rw [hp.1, AddGroupFilterBasis.nhds_eq _, AddGroupFilterBasis.N_zero]
    exact FilterBasis.hasBasis _
  · intro s hs
    change s ∈ Set.iUnion _ at hs
    simp_rw [Set.mem_iUnion, Set.mem_singleton_iff] at hs
    rcases hs with ⟨I, r, _, rfl⟩
    exact convex_ball _ _ _

Goal state after:
Module: Mathlib.Topology.Compactness.CompactlyCoherentSpace

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → 𝐤Y
K : Set X
hK : IsCompact K
H : ContinuousOn (⇑(CompactCoherentification.mk Y).symm ∘ f) K
⊢ ContinuousOn f K

Tactic:
exact continuousOn_isCompact_mk (hK.image_of_continuousOn H) |>.comp H (mapsTo_image f K)

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : TopologicalSpace β
f : α →ₘ[μ] β
⊢ StronglyMeasurable ↑f

Tactic:
simp only [cast]

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : TopologicalSpace β
f : α →ₘ[μ] β
⊢ StronglyMeasurable
    (if h : ∃ b, f = mk (const α b) ⋯ then const α (Classical.choose h)
    else AEStronglyMeasurable.mk ↑(Quotient.out f) ⋯)
Module: Mathlib.MeasureTheory.Function.AEEqFun

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : TopologicalSpace β
f : α →ₘ[μ] β
| mk ↑f ⋯

Tactic:
simp only [cast]

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : TopologicalSpace β
f : α →ₘ[μ] β
| mk
    (if h : ∃ b, f = mk (const α b) ⋯ then const α (Classical.choose h)
    else AEStronglyMeasurable.mk ↑(Quotient.out f) ⋯)
    ⋯
Module: Mathlib.Probability.UniformOn

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t : Set Ω
hs : s.Finite
hs' : s.Nonempty
ht : s ⊆ t
this : IsProbabilityMeasure (uniformOn s)
⊢ (uniformOn s) t = 1

Tactic:
refine eq_of_le_of_not_lt prob_le_one ?_

Goal state after:
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t : Set Ω
hs : s.Finite
hs' : s.Nonempty
ht : s ⊆ t
this : IsProbabilityMeasure (uniformOn s)
⊢ ¬(uniformOn s) t < 1
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: case cons
ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝² : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝¹ : (i : ι) → MeasurableSpace (α i)
μ✝ : (i : ι) → Measure (α i)
δ : Type u_4
X : δ → Type u_5
inst✝ : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
i : δ
l : List δ
ih : Measure (TProd X l)
⊢ Measure (TProd X (i :: l))

Tactic:
exact (μ i).prod (α := X i) ih

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Defs

Goal state before:
pp: α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
inst✝ : ENorm ε
μ : Measure α
f : α → ε
p : ℝ≥0
hp : p ≠ 0
⊢ eLpNorm f (↑p) μ ^ ↑p = ∫⁻ (x : α), ‖f x‖ₑ ^ ↑p ∂μ

Tactic:
by
  simp [eLpNorm_eq_eLpNorm' (by exact_mod_cast hp) ENNReal.coe_ne_top,
    lintegral_rpow_enorm_eq_rpow_eLpNorm' ((NNReal.coe_pos.trans pos_iff_ne_zero).mpr hp)]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace β
inst✝ : NeZero μ
b : β
x : α
⊢ ↑(const α b) x = b

Tactic:
simp only [cast]

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace β
inst✝ : NeZero μ
b : β
x : α
⊢ (if h : ∃ b_1, const α b = mk (Function.const α b_1) ⋯ then Function.const α (Classical.choose h)
      else AEStronglyMeasurable.mk ↑(Quotient.out (const α b)) ⋯)
      x =
    b
Module: Mathlib.MeasureTheory.Function.AEEqFun

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : TopologicalSpace β
b : β
⊢ ∃ b', ↑(const α b) = fun x ↦ b'

Tactic:
simp only [cast]

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : TopologicalSpace β
b : β
⊢ ∃ b',
    (if h : ∃ b_1, const α b = mk (Function.const α b_1) ⋯ then Function.const α (Classical.choose h)
      else AEStronglyMeasurable.mk ↑(Quotient.out (const α b)) ⋯) =
      fun x ↦ b'
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : ENorm ε
f : α → ε
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
h : ∫⁻ (a : α), ‖f a‖ₑ ^ p.toReal ∂μ < ∞
⊢ eLpNorm f p μ < ∞

Tactic:
have hp' := ENNReal.toReal_pos hp_ne_zero hp_ne_top

Goal state after:
α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : ENorm ε
f : α → ε
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
h : ∫⁻ (a : α), ‖f a‖ₑ ^ p.toReal ∂μ < ∞
hp' : 0 < p.toReal
⊢ eLpNorm f p μ < ∞
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: case refine_1.refine_2
α : Type u_4
β : Type u_5
γ : Type u_6
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : Fintype γ
μ : γ → Measure α
ν : γ → Measure β
inst✝¹ : ∀ (i : γ), SigmaFinite (μ i)
inst✝ : ∀ (i : γ), SigmaFinite (ν i)
⊢ IsCountablySpanning (univ.pi '' univ.pi fun i ↦ {s | MeasurableSet s})

Tactic:
· exact (FiniteSpanningSetsIn.pi (fun i ↦ (ν i).toFiniteSpanningSetsIn)).isCountablySpanning

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: case e_a
α : Type u_1
F : Type u_5
m0 : MeasurableSpace α
q : ℝ
μ : Measure α
inst✝¹ : NormedAddCommGroup F
inst✝ : IsFiniteMeasure μ
c : F
hc_ne_zero : c ≠ 0
hq_ne_zero : q ≠ 0
⊢ ‖c‖ₑ ^ (q * (1 / q)) = ‖c‖ₑ

Tactic:
suffices hp_cancel : q * (1 / q) = 1 by rw [hp_cancel, ENNReal.rpow_one]

Goal state after:
case e_a
α : Type u_1
F : Type u_5
m0 : MeasurableSpace α
q : ℝ
μ : Measure α
inst✝¹ : NormedAddCommGroup F
inst✝ : IsFiniteMeasure μ
c : F
hc_ne_zero : c ≠ 0
hq_ne_zero : q ≠ 0
⊢ q * (1 / q) = 1
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: n : ℕ
α : Fin (n + 1) → Type u
m : (i : Fin (n + 1)) → MeasurableSpace (α i)
μ : (i : Fin (n + 1)) → Measure (α i)
inst✝ : ∀ (i : Fin (n + 1)), SigmaFinite (μ i)
i : Fin (n + 1)
⊢ MeasurePreserving (⇑(MeasurableEquiv.piFinSuccAbove α i)) (Measure.pi μ)
    ((μ i).prod (Measure.pi fun j ↦ μ (i.succAbove j)))

Tactic:
set e := (MeasurableEquiv.piFinSuccAbove α i).symm
  refine MeasurePreserving.symm e ?_
  refine ⟨e.measurable, (pi_eq fun s _ => ?_).symm⟩
  rw [e.map_apply, i.prod_univ_succAbove _, ← pi_pi, ← prod_prod]
  congr 1 with ⟨x, f⟩
  simp [e, i.forall_iff_succAbove]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hs : Convex 𝕜 s
hf : ∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z → (f z - f y) / (z - y) < (f y - f x) / (y - x)
⊢ StrictConcaveOn 𝕜 s f

Tactic:
rw [← neg_strictConvexOn_iff]
  refine strictConvexOn_of_slope_strict_mono_adjacent hs fun hx hz hxy hyz => ?_
  simp only [Pi.neg_apply]
  linear_combination hf hx hz hxy hyz

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ENormedAddMonoid ε
s : Set α
f : α → ε
hsf : Function.support f ⊆ s
hp0 : ¬p = 0
hp_top : ¬p = ∞
⊢ ¬p.toReal ≤ 0

Tactic:
simpa only [not_le] using ENNReal.toReal_pos hp0 hp_top

Goal state after:
Module: Mathlib.Data.Real.ConjExponents

Goal state before:
pp: p q : ℝ
h : p.HolderConjugate q
⊢ ENNReal.ofReal p⁻¹ + ENNReal.ofReal q⁻¹ = ENNReal.ofReal 1

Tactic:
← ENNReal.ofReal_add h.inv_nonneg h.symm.inv_nonneg,

Goal state after:
p q : ℝ
h : p.HolderConjugate q
⊢ ENNReal.ofReal (p⁻¹ + q⁻¹) = ENNReal.ofReal 1
Module: Mathlib.Analysis.Convex.SpecificFunctions.Basic

Goal state before:
pp: case neg
s : ℝ
hs : -1 ≤ s
p : ℝ
hp1✝ : 0 ≤ p
hp2✝ : p ≤ 1
hp1 : 0 < p
hp2 : p < 1
hs' : ¬s = 0
⊢ (1 + s) ^ p ≤ 1 + p * s

Tactic:
exact (rpow_one_add_lt_one_add_mul_self hs hs' hp1 hp2).le

Goal state after:
Module: Mathlib.Data.Real.ConjExponents

Goal state before:
pp: p q : ℝ≥0∞
hp : 1 < p.toReal
h : p.HolderConjugate q
⊢ p.toReal.HolderConjugate q.toReal

Tactic:
have hq : 0 < q.toReal := by
    rw [toReal_pos_iff]
    refine ⟨pos q p, lt_top_iff_one_lt q p |>.mpr ?_⟩
    contrapose! hp
    exact toReal_mono one_ne_top hp

Goal state after:
p q : ℝ≥0∞
hp : 1 < p.toReal
h : p.HolderConjugate q
hq : 0 < q.toReal
⊢ p.toReal.HolderConjugate q.toReal
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
β : Type u_8
mβ : MeasurableSpace β
f : α → β
g : β → ε
hg : AEStronglyMeasurable g (Measure.map f μ)
hf : AEMeasurable f μ
⊢ MemLp g p (Measure.map f μ) ↔ MemLp (g ∘ f) p μ

Tactic:
by
  simp [MemLp, eLpNorm_map_measure hg hf, hg.comp_aemeasurable hf, hg]

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case pos
ι : Type u
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 ≤ w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 ≤ z i
i : ι
his : i ∈ s
hzi : z i = 0
hwi : w i ≠ 0
⊢ z i ^ w i = 0

Tactic:
rw [hzi]
      exact zero_rpow hwi

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case pos
ι : Type u
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 < w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 ≤ z i
i : ι
his : i ∈ s
hzi : z i = 0
hwi : w i ≠ 0
⊢ z i ^ w i = 0

Tactic:
rw [hzi]
      exact zero_rpow hwi

Goal state after:
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: p : ℝ≥0∞
hp : 1 ≤ p
⊢ p.LpAddConst = 1

Tactic:
rw [LpAddConst, if_neg]
  intro h
  exact lt_irrefl _ (h.2.trans_le hp)

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSpace.Complete

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝¹ : NormedAddCommGroup E
ι : Type u_4
fi : Filter ι
inst✝ : Fact (1 ≤ p)
f : ι → α → E
f_ℒp : ∀ (n : ι), MemLp (f n) p μ
f_lim : α → E
f_lim_ℒp : MemLp f_lim p μ
⊢ Tendsto (fun n ↦ eLpNorm (↑↑(MemLp.toLp (f n) ⋯) - ↑↑(MemLp.toLp f_lim f_lim_ℒp)) p μ) fi (𝓝 0) ↔
    Tendsto (fun n ↦ eLpNorm (f n - f_lim) p μ) fi (𝓝 0)

Tactic:
refine Filter.tendsto_congr fun n => ?_

Goal state after:
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝¹ : NormedAddCommGroup E
ι : Type u_4
fi : Filter ι
inst✝ : Fact (1 ≤ p)
f : ι → α → E
f_ℒp : ∀ (n : ι), MemLp (f n) p μ
f_lim : α → E
f_lim_ℒp : MemLp f_lim p μ
n : ι
⊢ eLpNorm (↑↑(MemLp.toLp (f n) ⋯) - ↑↑(MemLp.toLp f_lim f_lim_ℒp)) p μ = eLpNorm (f n - f_lim) p μ
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
p : ℝ
hp : 1 ≤ p
w f : ι → ℝ
hw : ∀ (i : ι), 0 ≤ w i
hf : ∀ (i : ι), 0 ≤ f i
⊢ ∑ i ∈ s, w i * f i ≤ (∑ i ∈ s, w i) ^ (1 - p⁻¹) * (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹

Tactic:
lift w to ι → ℝ≥0 using hw
  lift f to ι → ℝ≥0 using hf
  beta_reduce at *
  norm_cast at *
  exact NNReal.inner_le_weight_mul_Lp _ hp _ _

Goal state after: