mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Convex.Basic

Goal state before:
pp: R : Type u_5
inst✝⁹ : CommSemiring R
A : Type u_6
inst✝⁸ : Semiring A
inst✝⁷ : Algebra R A
M : Type u_7
inst✝⁶ : AddCommMonoid M
inst✝⁵ : Module A M
inst✝⁴ : Module R M
inst✝³ : IsScalarTower R A M
inst✝² : PartialOrder R
inst✝¹ : PartialOrder A
inst✝ : FaithfulSMul R A
s : Set M
halg : Ici 0 ⊆ ⇑(algebraMap R A) '' Ici 0
hs : ∀ ⦃x : M⦄, x ∈ s → ∀ ⦃y : M⦄, y ∈ s → ∀ ⦃a b : R⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • x + b • y ∈ s
u : M
hu : u ∈ s
v : M
hv : v ∈ s
a b : A
ha : 0 ≤ a
hb : 0 ≤ b
c : R
hc1 : c ∈ Ici 0
hc2 : (algebraMap R A) c = a
d : R
hab : ↑(c + d) = 1
hd1 : d ∈ Ici 0
hd2 : (algebraMap R A) d = b
⊢ c + d = 1

Tactic:
exact (FaithfulSMul.algebraMap_eq_one_iff R A).mp hab

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
inst✝³ : SeminormedRing 𝕜
inst✝² : SMul 𝕜 E
inst✝¹ : Zero E
inst✝ : TopologicalSpace E
S : Set (Set E)
hS : S.Finite
⊢ IsVonNBounded 𝕜 (⋃ i ∈ S, i) ↔ ∀ s ∈ S, IsVonNBounded 𝕜 s

Tactic:
isVonNBounded_biUnion hS

Goal state after:
𝕜 : Type u_1
E : Type u_3
inst✝³ : SeminormedRing 𝕜
inst✝² : SMul 𝕜 E
inst✝¹ : Zero E
inst✝ : TopologicalSpace E
S : Set (Set E)
hS : S.Finite
⊢ (∀ i ∈ S, IsVonNBounded 𝕜 i) ↔ ∀ s ∈ S, IsVonNBounded 𝕜 s
Module: Mathlib.Analysis.LocallyConvex.Bounded

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
inst✝⁵ : NormedField 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : UniformSpace E
inst✝¹ : IsUniformAddGroup E
inst✝ : ContinuousSMul 𝕜 E
s : Set E
hs : ∀ U ∈ 𝓝 0, ∃ t, t.Finite ∧ s ⊆ ⋃ y ∈ t, y +ᵥ U
h✝ : ∃ x, 1 < ‖x‖
this : NontriviallyNormedField 𝕜 := { toNormedField := inst✝⁵, non_trivial := h✝ }
U : Set E
hU : U ∈ 𝓝 0
h : Tendsto (fun x ↦ x.1 + x.2) (𝓝 0) (𝓝 0)
h' : (𝓝 (0, 0)).HasBasis (fun i ↦ (i.1 ∈ 𝓝 0 ∧ Balanced 𝕜 i.1) ∧ i.2 ∈ 𝓝 0 ∧ Balanced 𝕜 i.2) fun i ↦ id i.1 ×ˢ id i.2
⊢ Absorbs 𝕜 U s

Tactic:
id

Goal state after:
𝕜 : Type u_1
E : Type u_3
inst✝⁵ : NormedField 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : UniformSpace E
inst✝¹ : IsUniformAddGroup E
inst✝ : ContinuousSMul 𝕜 E
s : Set E
hs : ∀ U ∈ 𝓝 0, ∃ t, t.Finite ∧ s ⊆ ⋃ y ∈ t, y +ᵥ U
h✝ : ∃ x, 1 < ‖x‖
this : NontriviallyNormedField 𝕜 := { toNormedField := inst✝⁵, non_trivial := h✝ }
U : Set E
hU : U ∈ 𝓝 0
h : Tendsto (fun x ↦ x.1 + x.2) (𝓝 0) (𝓝 0)
h' : (𝓝 (0, 0)).HasBasis (fun i ↦ (i.1 ∈ 𝓝 0 ∧ Balanced 𝕜 i.1) ∧ i.2 ∈ 𝓝 0 ∧ Balanced 𝕜 i.2) fun i ↦ i.1 ×ˢ i.2
⊢ Absorbs 𝕜 U s
Module: Mathlib.Topology.MetricSpace.Equicontinuity

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : PseudoMetricSpace α
ι : Type u_4
inst✝ : PseudoMetricSpace β
b : ℝ → ℝ
b_lim : Tendsto b (𝓝 0) (𝓝 0)
F : ι → β → α
H : ∀ (x y : β) (i : ι), dist (F i x) (F i y) ≤ b (dist x y)
ε : ℝ
ε0 : ε > 0
δ : ℝ
δ0 : δ > 0
hδ : ∀ ⦃x : ℝ⦄, dist x 0 < δ → dist (b x) 0 < ε
x y : β
hxy : dist x y < δ
i : ι
⊢ |b (dist x y)| = dist (b (dist x y)) 0

Tactic:
by simp [Real.dist_eq]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
inst✝⁵ : Ring 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : TopologicalSpace E
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
s : Set E
x : E
inst✝ : AddRightStrictMono 𝕜
hs : StrictConvex 𝕜 s
zero_mem : 0 ∈ s
hx : x ∈ s
hx₀ : x ≠ 0
t : 𝕜
ht₀ : 0 < t
ht₁ : t < 1
⊢ 0 + x ∈ s

Tactic:
by simpa using hx

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
s : Set P
h₁ : s.Subsingleton
p : P
h₂ : ↑(affineSpan k {p}) = ↑⊤
hp : p ∈ s
this : s = {p}
⊢ Subsingleton P

Tactic:
AffineSubspace.coe_affineSpan_singleton,

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
s : Set P
h₁ : s.Subsingleton
p : P
h₂ : {p} = ↑⊤
hp : p ∈ s
this : s = {p}
⊢ Subsingleton P
Module: Mathlib.LinearAlgebra.AffineSpace.Combination

Goal state before:
pp: k : Type u_1
V : Type u_2
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
ι : Type u_6
s : Finset ι
w : ι → k
p : ι → V
hw : s.sum w = 0
⊢ (s.weightedVSub p) w = ∑ i ∈ s, w i • p i

Tactic:
by
  simp [s.weightedVSub_apply, vsub_eq_sub, smul_sub, ← Finset.sum_smul, hw]

Goal state after:
Module: Mathlib.Analysis.Seminorm

Goal state before:
pp: 𝕜 : Type u_3
𝕜₂ : Type u_4
E : Type u_7
E₂ : Type u_8
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : SeminormedCommRing 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝⁴ : RingHomIsometric σ₁₂
inst✝³ : AddCommGroup E
inst✝² : AddCommGroup E₂
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜₂ E₂
p : Seminorm 𝕜₂ E₂
f : E →ₛₗ[σ₁₂] E₂
c : 𝕜₂
⊢ p.comp (c • f) = ‖c‖₊ • p.comp f

Tactic:
ext; simp [NNReal.smul_def, map_smul_eq_mul]

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic

Goal state before:
pp: k : Type u_1
V : Type u_2
W : Type u_3
P : Type u_4
Q : Type u_5
inst✝⁶ : Ring k
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module k V
inst✝³ : AffineSpace V P
inst✝² : AddCommGroup W
inst✝¹ : Module k W
inst✝ : AffineSpace W Q
s : Set P
t : Set Q
hs : s.Nonempty
ht : t.Nonempty
⊢ Submodule.span k ((s -ᵥ s) ×ˢ (t -ᵥ t)) = (vectorSpan k s).prod (vectorSpan k t)

Tactic:
exact Submodule.span_prod_eq k hs.zero_mem_vsub_self ht.zero_mem_vsub_self

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Ring k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
inst✝¹ : IsDomain k
inst✝ : Module.IsTorsionFree k V
p₁ q₁ p₂ q₂ : P
⊢ (∃ z, z • (q₁ -ᵥ p₁) = q₂ -ᵥ p₂) ↔ ∃ z, z • (q₂ -ᵥ p₂) = q₁ -ᵥ p₁

Tactic:
exact ⟨fun ⟨z, hz⟩ ↦ ⟨z⁻¹, by simp [← hz]⟩, fun ⟨z, hz⟩ ↦ ⟨z⁻¹, by simp [← hz]⟩⟩

Goal state after:
Module: Mathlib.Analysis.Seminorm

Goal state before:
pp: 𝕜 : Type u_3
E : Type u_7
ι : Type u_11
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : ι → Seminorm 𝕜 E
s : Finset ι
x : E
r : ℝ≥0
⊢ {y | (s.sup fun i ↦ NNReal.mk ((p i) (y - x)) ⋯) ≤ r} = {x_1 | ∀ i ∈ s, (p i) (x_1 - x) ≤ ↑r}

Tactic:
Finset.sup_le_iff,

Goal state after:
𝕜 : Type u_3
E : Type u_7
ι : Type u_11
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : ι → Seminorm 𝕜 E
s : Finset ι
x : E
r : ℝ≥0
⊢ {y | ∀ b ∈ s, NNReal.mk ((p b) (y - x)) ⋯ ≤ r} = {x_1 | ∀ i ∈ s, (p i) (x_1 - x) ≤ ↑r}
Module: Mathlib.LinearAlgebra.AffineSpace.Independent

Goal state before:
pp: case mpr
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
i1 : ι
h : ∀ (s : Finset { x // x ≠ i1 }) (g : { x // x ≠ i1 } → k), ∑ i ∈ s, g i • (p ↑i -ᵥ p i1) = 0 → ∀ i ∈ s, g i = 0
s : Finset ι
w : ι → k
hw : ∑ i ∈ s, w i = 0
hs : (s.weightedVSub p) w = 0
i : ι
hi : i ∈ s
⊢ w i = 0

Tactic:
rw [Finset.weightedVSub_eq_weightedVSubOfPoint_of_sum_eq_zero s w p hw (p i1), ←
        s.weightedVSubOfPoint_erase w p i1, Finset.weightedVSubOfPoint_apply] at hs

Goal state after:
case mpr
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
i1 : ι
h : ∀ (s : Finset { x // x ≠ i1 }) (g : { x // x ≠ i1 } → k), ∑ i ∈ s, g i • (p ↑i -ᵥ p i1) = 0 → ∀ i ∈ s, g i = 0
s : Finset ι
w : ι → k
hw : ∑ i ∈ s, w i = 0
hs : ∑ i ∈ s.erase i1, w i • (p i -ᵥ p i1) = 0
i : ι
hi : i ∈ s
⊢ w i = 0
Module: Mathlib.LinearAlgebra.AffineSpace.Combination

Goal state before:
pp: case mpr
ι : Type u_1
k : Type u_2
V : Type u_3
P : Type u_4
inst✝⁴ : Ring k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
inst✝ : Nontrivial k
p1 : P
p : ι → P
⊢ (∃ s w, ∑ i ∈ s, w i = 1 ∧ p1 = (Finset.affineCombination k s p) w) → p1 ∈ affineSpan k (Set.range p)

Tactic:
rintro ⟨s, w, hw, rfl⟩
    exact affineCombination_mem_affineSpan hw p

Goal state after:
Module: Mathlib.Analysis.Seminorm

Goal state before:
pp: 𝕝 : Type u_6
E : Type u_7
inst✝⁴ : SeminormedRing 𝕝
inst✝³ : AddCommGroup E
inst✝² : Module 𝕝 E
inst✝¹ : UniformSpace E
inst✝ : IsUniformAddGroup E
p : Seminorm 𝕝 E
hp✝ : ContinuousAt (⇑p) 0
hp : Tendsto (⇑p) (𝓝 0) (𝓝 0)
⊢ UniformContinuous ⇑p

Tactic:
rw [UniformContinuous, uniformity_eq_comap_nhds_zero_swapped,
    Metric.uniformity_eq_comap_nhds_zero, Filter.tendsto_comap_iff]

Goal state after:
𝕝 : Type u_6
E : Type u_7
inst✝⁴ : SeminormedRing 𝕝
inst✝³ : AddCommGroup E
inst✝² : Module 𝕝 E
inst✝¹ : UniformSpace E
inst✝ : IsUniformAddGroup E
p : Seminorm 𝕝 E
hp✝ : ContinuousAt (⇑p) 0
hp : Tendsto (⇑p) (𝓝 0) (𝓝 0)
⊢ Tendsto ((fun p ↦ dist p.1 p.2) ∘ fun x ↦ (p x.1, p x.2)) (comap (fun x ↦ x.1 - x.2) (𝓝 0)) (𝓝 0)
Module: Mathlib.Analysis.Convex.StdSimplex

Goal state before:
pp: case refine_2
𝕜 : Type u_2
ι : Type u_1
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : Fintype ι
inst✝ : IsOrderedRing 𝕜
f : ι → 𝕜
hf : f ∈ stdSimplex 𝕜 ι
g : ι → 𝕜
hg : g ∈ stdSimplex 𝕜 ι
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ ∑ x, (a • f x + b • g x) = 1

Tactic:
rwa [Finset.sum_add_distrib, ← Finset.smul_sum, ← Finset.smul_sum, hf.2, hg.2, smul_eq_mul,
      smul_eq_mul, mul_one, mul_one]

Goal state after:
Module: Mathlib.Topology.AlexandrovDiscrete

Goal state before:
pp: α : Type u_3
inst✝¹ : TopologicalSpace α
inst✝ : AlexandrovDiscrete α
β : Type u_5
f : α → β
S : Set (Set β)
hS : ∀ s ∈ S, IsOpen[coinduced f inst✝¹] s
⊢ IsOpen[inst✝¹] (f ⁻¹' ⋂₀ S)

Tactic:
preimage_sInter

Goal state after:
α : Type u_3
inst✝¹ : TopologicalSpace α
inst✝ : AlexandrovDiscrete α
β : Type u_5
f : α → β
S : Set (Set β)
hS : ∀ s ∈ S, IsOpen[coinduced f inst✝¹] s
⊢ IsOpen[inst✝¹] (⋂ t ∈ S, f ⁻¹' t)
Module: Mathlib.Analysis.Convex.Combination

Goal state before:
pp: R : Type u_1
E : Type u_3
ι : Type u_5
inst✝⁴ : Field R
inst✝³ : AddCommGroup E
inst✝² : Module R E
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
s : Finset ι
v : ι → E
w : ι → R
hw₀ : ∀ i ∈ s, 0 ≤ w i
hw₁ : s.sum w = 1
⊢ (affineCombination R s v) w ∈ (convexHull R) (Set.range v)

Tactic:
rw [affineCombination_eq_centerMass hw₁]
  apply s.centerMass_mem_convexHull hw₀
  · simp [hw₁]
  · simp

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

Goal state before:
pp: case refine_2.refine_1
R : Type u_1
E : Type u_3
ι : Type u_5
inst✝⁴ : Field R
inst✝³ : AddCommGroup E
inst✝² : Module R E
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
v : ι → E
s : Finset ι
w : ι → R
hw₀ : ∀ i ∈ s, 0 ≤ w i
hw₁ : s.sum w = 1
s' : Finset ι
w' : ι → R
hw₀' : ∀ i ∈ s', 0 ≤ w' i
hw₁' : s'.sum w' = 1
a b : R
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
W : ι → R := fun i ↦ (if i ∈ s then a * w i else 0) + if i ∈ s' then b * w' i else 0
hW₁ : (s ∪ s').sum W = 1
⊢ ∀ i ∈ s ∪ s', 0 ≤ W i

pp: case refine_2.refine_2
R : Type u_1
E : Type u_3
ι : Type u_5
inst✝⁴ : Field R
inst✝³ : AddCommGroup E
inst✝² : Module R E
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
v : ι → E
s : Finset ι
w : ι → R
hw₀ : ∀ i ∈ s, 0 ≤ w i
hw₁ : s.sum w = 1
s' : Finset ι
w' : ι → R
hw₀' : ∀ i ∈ s', 0 ≤ w' i
hw₁' : s'.sum w' = 1
a b : R
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
W : ι → R := fun i ↦ (if i ∈ s then a * w i else 0) + if i ∈ s' then b * w' i else 0
hW₁ : (s ∪ s').sum W = 1
⊢ (affineCombination R (s ∪ s') v) W = a • (affineCombination R s v) w + b • (affineCombination R s' v) w'

Tactic:
· rintro i -
      by_cases hi : i ∈ s <;> by_cases hi' : i ∈ s' <;>
        simp [W, hi, hi', add_nonneg, mul_nonneg ha (hw₀ i _), mul_nonneg hb (hw₀' i _)]

Goal state after:
case refine_2.refine_2
R : Type u_1
E : Type u_3
ι : Type u_5
inst✝⁴ : Field R
inst✝³ : AddCommGroup E
inst✝² : Module R E
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
v : ι → E
s : Finset ι
w : ι → R
hw₀ : ∀ i ∈ s, 0 ≤ w i
hw₁ : s.sum w = 1
s' : Finset ι
w' : ι → R
hw₀' : ∀ i ∈ s', 0 ≤ w' i
hw₁' : s'.sum w' = 1
a b : R
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
W : ι → R := fun i ↦ (if i ∈ s then a * w i else 0) + if i ∈ s' then b * w' i else 0
hW₁ : (s ∪ s').sum W = 1
⊢ (affineCombination R (s ∪ s') v) W = a • (affineCombination R s v) w + b • (affineCombination R s' v) w'
Module: Mathlib.Analysis.Convex.Combination

Goal state before:
pp: case refine_3
R : Type u_1
E : Type u_3
ι : Type u_5
inst✝⁴ : Field R
inst✝³ : AddCommGroup E
inst✝² : Module R E
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
v : ι → E
⊢ {x | ∃ s w, (∀ i ∈ s, 0 ≤ w i) ∧ s.sum w = 1 ∧ (affineCombination R s v) w = x} ⊆ (convexHull R) (Set.range v)

Tactic:
· rintro x ⟨s, w, hw₀, hw₁, rfl⟩
    exact affineCombination_mem_convexHull hw₀ hw₁

Goal state after:
Module: Mathlib.Topology.Algebra.Module.LocallyConvex

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
ts : Set (TopologicalSpace E)
h : ∀ t ∈ ts, LocallyConvexSpace 𝕜 E
⊢ LocallyConvexSpace 𝕜 E

Tactic:
let : TopologicalSpace E := sInf ts

Goal state after:
𝕜 : Type u_2
E : Type u_3
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
ts : Set (TopologicalSpace E)
h : ∀ t ∈ ts, LocallyConvexSpace 𝕜 E
this : TopologicalSpace E := sInf ts
⊢ LocallyConvexSpace 𝕜 E
Module: Mathlib.Analysis.Normed.Module.Convex

Goal state before:
pp: case neg
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : Nontrivial F
x✝ : F
r : ℝ
hr : 0 ≤ r
x : F
h : x ∈ closedBall 0 r
U : Set F
hU_sub : sphere 0 r ⊆ U
hU : Convex ℝ U
zero_mem : 0 ∈ U
hr₀ : ¬r = 0
x_zero : ¬x = 0
z : F := (r * ‖x‖⁻¹) • x
hz_def : z = (r * ‖x‖⁻¹) • x
hr₁ : r⁻¹ * ‖x‖ ≤ 1
hz : z ∈ U
this : (‖x‖⁻¹ * (r⁻¹ * ‖x‖ * r)) • x ∈ U
⊢ x ∈ U

Tactic:
mul_comm r⁻¹,

Goal state after:
case neg
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : Nontrivial F
x✝ : F
r : ℝ
hr : 0 ≤ r
x : F
h : x ∈ closedBall 0 r
U : Set F
hU_sub : sphere 0 r ⊆ U
hU : Convex ℝ U
zero_mem : 0 ∈ U
hr₀ : ¬r = 0
x_zero : ¬x = 0
z : F := (r * ‖x‖⁻¹) • x
hz_def : z = (r * ‖x‖⁻¹) • x
hr₁ : r⁻¹ * ‖x‖ ≤ 1
hz : z ∈ U
this : (‖x‖⁻¹ * (‖x‖ * r⁻¹ * r)) • x ∈ U
⊢ x ∈ U
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: 𝕝 : Type u_4
𝕝₂ : Type u_5
E : Type u_6
F : Type u_7
ι' : Type u_10
inst✝⁹ : AddCommGroup E
inst✝⁸ : NormedField 𝕝
inst✝⁷ : Module 𝕝 E
inst✝⁶ : AddCommGroup F
inst✝⁵ : NormedField 𝕝₂
inst✝⁴ : Module 𝕝₂ F
τ₁₂ : 𝕝 →+* 𝕝₂
inst✝³ : RingHomIsometric τ₁₂
q : SeminormFamily 𝕝₂ F ι'
inst✝² : TopologicalSpace E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : TopologicalSpace F
hq : WithSeminorms q
f : E →ₛₗ[τ₁₂] F
hf : ∀ (i : ι'), Continuous[inst✝², _] ⇑((q i).comp f)
this : IsTopologicalAddGroup F
⊢ Tendsto (⇑f) (𝓝 0) (⨅ i, comap (⇑(q i)) (𝓝 0))

Tactic:
Filter.tendsto_iInf,

Goal state after:
𝕝 : Type u_4
𝕝₂ : Type u_5
E : Type u_6
F : Type u_7
ι' : Type u_10
inst✝⁹ : AddCommGroup E
inst✝⁸ : NormedField 𝕝
inst✝⁷ : Module 𝕝 E
inst✝⁶ : AddCommGroup F
inst✝⁵ : NormedField 𝕝₂
inst✝⁴ : Module 𝕝₂ F
τ₁₂ : 𝕝 →+* 𝕝₂
inst✝³ : RingHomIsometric τ₁₂
q : SeminormFamily 𝕝₂ F ι'
inst✝² : TopologicalSpace E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : TopologicalSpace F
hq : WithSeminorms q
f : E →ₛₗ[τ₁₂] F
hf : ∀ (i : ι'), Continuous[inst✝², _] ⇑((q i).comp f)
this : IsTopologicalAddGroup F
⊢ ∀ (i : ι'), Tendsto (⇑f) (𝓝 0) (comap (⇑(q i)) (𝓝 0))
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: 𝕜 : Type u_2
𝕜₂ : Type u_3
E : Type u_6
F : Type u_7
ι' : Type u_10
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module 𝕜 E
inst✝⁶ : NormedField 𝕜₂
inst✝⁵ : AddCommGroup F
inst✝⁴ : Module 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝³ : RingHomIsometric σ₁₂
κ : Type u_11
q : SeminormFamily 𝕜₂ F ι'
inst✝² : UniformSpace E
inst✝¹ : IsUniformAddGroup E
inst✝ : ContinuousSMul 𝕜 E
f : κ → E →ₛₗ[σ₁₂] F
i : ι'
x✝ : SeminormedAddCommGroup F := (q i).toSeminormedAddCommGroup
⊢ (List.map (fun p ↦ p i)
      [fun k ↦ EquicontinuousAt (DFunLike.coe ∘ f) 0, fun k ↦ Equicontinuous (DFunLike.coe ∘ f), fun k ↦
        UniformEquicontinuous (DFunLike.coe ∘ f), fun i ↦
        ∃ p, Continuous[inst✝².toTopologicalSpace, _] ⇑p ∧ ∀ (k : κ), (q i).comp (f k) ≤ p, fun i ↦
        BddAbove (Set.range fun k ↦ (q i).comp (f k)) ∧
          Continuous[inst✝².toTopologicalSpace, _] (⨆ k, ⇑((q i).comp (f k)))]).TFAE

Tactic:
simp only [List.map]

Goal state after:
𝕜 : Type u_2
𝕜₂ : Type u_3
E : Type u_6
F : Type u_7
ι' : Type u_10
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module 𝕜 E
inst✝⁶ : NormedField 𝕜₂
inst✝⁵ : AddCommGroup F
inst✝⁴ : Module 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝³ : RingHomIsometric σ₁₂
κ : Type u_11
q : SeminormFamily 𝕜₂ F ι'
inst✝² : UniformSpace E
inst✝¹ : IsUniformAddGroup E
inst✝ : ContinuousSMul 𝕜 E
f : κ → E →ₛₗ[σ₁₂] F
i : ι'
x✝ : SeminormedAddCommGroup F := (q i).toSeminormedAddCommGroup
⊢ [EquicontinuousAt (DFunLike.coe ∘ f) 0, Equicontinuous (DFunLike.coe ∘ f), UniformEquicontinuous (DFunLike.coe ∘ f),
      ∃ p, Continuous[inst✝².toTopologicalSpace, _] ⇑p ∧ ∀ (k : κ), (q i).comp (f k) ≤ p,
      BddAbove (Set.range fun k ↦ (q i).comp (f k)) ∧
        Continuous[inst✝².toTopologicalSpace, _] (⨆ k, ⇑((q i).comp (f k)))].TFAE
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: 𝕜 : Type u_2
𝕜₂ : Type u_3
E : Type u_6
F : Type u_7
ι' : Type u_10
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module 𝕜 E
inst✝⁶ : NormedField 𝕜₂
inst✝⁵ : AddCommGroup F
inst✝⁴ : Module 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝³ : RingHomIsometric σ₁₂
κ : Type u_11
q : SeminormFamily 𝕜₂ F ι'
inst✝² : UniformSpace E
inst✝¹ : IsUniformAddGroup E
inst✝ : ContinuousSMul 𝕜 E
f : κ → E →ₛₗ[σ₁₂] F
i : ι'
x✝¹ : SeminormedAddCommGroup F := (q i).toSeminormedAddCommGroup
tfae_1_to_3 : EquicontinuousAt (DFunLike.coe ∘ f) 0 → UniformEquicontinuous (DFunLike.coe ∘ f)
tfae_3_to_2 : UniformEquicontinuous (DFunLike.coe ∘ f) → Equicontinuous (DFunLike.coe ∘ f)
tfae_2_to_1 : Equicontinuous (DFunLike.coe ∘ f) → EquicontinuousAt (DFunLike.coe ∘ f) 0
x✝ : UniformEquicontinuous (DFunLike.coe ∘ f)
H : UniformEquicontinuous (DFunLike.coe ∘ f) := x✝
this : ∀ᶠ (x : E) in 𝓝 0, ∀ (k : κ), (q i) ((f k) x) ≤ 1
bdd : BddAbove (Set.range fun k ↦ (q i).comp (f k))
⊢ BddAbove (Set.range fun k ↦ (q i).comp (f k)) ∧ Continuous[inst✝².toTopologicalSpace, _] (⨆ k, ⇑((q i).comp (f k)))

Tactic:
rw [← Seminorm.coe_iSup_eq bdd]

Goal state after:
𝕜 : Type u_2
𝕜₂ : Type u_3
E : Type u_6
F : Type u_7
ι' : Type u_10
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module 𝕜 E
inst✝⁶ : NormedField 𝕜₂
inst✝⁵ : AddCommGroup F
inst✝⁴ : Module 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝³ : RingHomIsometric σ₁₂
κ : Type u_11
q : SeminormFamily 𝕜₂ F ι'
inst✝² : UniformSpace E
inst✝¹ : IsUniformAddGroup E
inst✝ : ContinuousSMul 𝕜 E
f : κ → E →ₛₗ[σ₁₂] F
i : ι'
x✝¹ : SeminormedAddCommGroup F := (q i).toSeminormedAddCommGroup
tfae_1_to_3 : EquicontinuousAt (DFunLike.coe ∘ f) 0 → UniformEquicontinuous (DFunLike.coe ∘ f)
tfae_3_to_2 : UniformEquicontinuous (DFunLike.coe ∘ f) → Equicontinuous (DFunLike.coe ∘ f)
tfae_2_to_1 : Equicontinuous (DFunLike.coe ∘ f) → EquicontinuousAt (DFunLike.coe ∘ f) 0
x✝ : UniformEquicontinuous (DFunLike.coe ∘ f)
H : UniformEquicontinuous (DFunLike.coe ∘ f) := x✝
this : ∀ᶠ (x : E) in 𝓝 0, ∀ (k : κ), (q i) ((f k) x) ≤ 1
bdd : BddAbove (Set.range fun k ↦ (q i).comp (f k))
⊢ BddAbove (Set.range fun k ↦ (q i).comp (f k)) ∧ Continuous[inst✝².toTopologicalSpace, _] ⇑(⨆ i_1, (q i).comp (f i_1))
Module: Mathlib.Analysis.Normed.Operator.NNNorm

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : DenselyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜₂
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝ : RingHomIsometric σ₁₂
f : E →SL[σ₁₂] F
⊢ sSup ((fun x ↦ ‖f x‖₊) '' closedBall 0 1) = ‖f‖₊

Tactic:
have hbdd : ∀ y ∈ (fun x => ‖f x‖₊) '' closedBall 0 1, y ≤ ‖f‖₊ := by
    rintro - ⟨x, hx, rfl⟩
    exact f.unit_le_opNorm x (mem_closedBall_zero_iff.1 hx)

Goal state after:
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : DenselyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜₂
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝ : RingHomIsometric σ₁₂
f : E →SL[σ₁₂] F
hbdd : ∀ y ∈ (fun x ↦ ‖f x‖₊) '' closedBall 0 1, y ≤ ‖f‖₊
⊢ sSup ((fun x ↦ ‖f x‖₊) '' closedBall 0 1) = ‖f‖₊
Module: Mathlib.Analysis.Normed.Operator.Bilinear

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
F : Type u_6
G : Type u_8
inst✝¹⁰ : SeminormedAddCommGroup E
inst✝⁹ : SeminormedAddCommGroup F
inst✝⁸ : SeminormedAddCommGroup G
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NontriviallyNormedField 𝕜₂
inst✝⁵ : NontriviallyNormedField 𝕜₃
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedSpace 𝕜₂ F
inst✝² : NormedSpace 𝕜₃ G
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₁₃ : 𝕜 →+* 𝕜₃
inst✝¹ : RingHomIsometric σ₂₃
inst✝ : RingHomIsometric σ₁₃
f : E →SL[σ₁₃] F →SL[σ₂₃] G
x : E
y : F
⊢ ‖(f x) y‖ ≤ ‖f.flip‖ * ‖x‖ * ‖y‖

Tactic:
by
    rw [mul_right_comm]
    exact (flip f).le_opNorm₂ y x

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

Goal state before:
pp: case refine_2.inr
𝕜 : Type u_1
E : Type u_4
Fₗ : Type u_7
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : SeminormedAddCommGroup Fₗ
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 Fₗ
c : StrongDual 𝕜 E
f : Fₗ
hf : 0 < ‖f‖
⊢ ‖c‖ * ‖f‖ ≤ ‖smulRight c f‖

Tactic:
· rw [← le_div_iff₀ hf]
      refine opNorm_le_bound _ (by positivity) fun x => ?_
      rw [div_mul_eq_mul_div, le_div_iff₀ hf]
      calc
        ‖c x‖ * ‖f‖ = ‖c x • f‖ := (norm_smul _ _).symm
        _ = ‖smulRight c f x‖ := rfl
        _ ≤ ‖smulRight c f‖ * ‖x‖ := le_opNorm _ _

Goal state after:
Module: Mathlib.Topology.UniformSpace.CompactConvergence

Goal state before:
pp: α : Type u₁
β : Type u₂
inst✝³ : TopologicalSpace α
inst✝² : UniformSpace β
K : Set α
V : Set (β × β)
f : C(α, β)
ι : Type u₃
p : Filter ι
F : ι → C(α, β)
inst✝¹ : CompleteSpace β
inst✝ : CompactlyCoherentSpace α
⊢ IsComplete {f | Continuous[inst✝³, inst✝².toTopologicalSpace] f}

Tactic:
← completeSpace_coe_iff_isComplete

Goal state after:
α : Type u₁
β : Type u₂
inst✝³ : TopologicalSpace α
inst✝² : UniformSpace β
K : Set α
V : Set (β × β)
f : C(α, β)
ι : Type u₃
p : Filter ι
F : ι → C(α, β)
inst✝¹ : CompleteSpace β
inst✝ : CompactlyCoherentSpace α
⊢ CompleteSpace ↑{f | Continuous[inst✝³, inst✝².toTopologicalSpace] f}
Module: Mathlib.Probability.ConditionalProbability

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
s t : Set Ω
hms : MeasurableSet s
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
⊢ μ (t ∩ s) + μ (t ∩ sᶜ) = μ t

Tactic:
exact measure_inter_add_sdiff t hms

Goal state after:
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: case pos
α : Type u_1
f : Filter α
inst✝ : CountableInterFilter f
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a = ∞
hu : u =ᶠ[f] 0
⊢ limsup (fun x ↦ a * u x) f = a * limsup u f

Tactic:
have hau : (a * u ·) =ᶠ[f] 0 := hu.mono fun x hx => by simp [hx]
    simp only [limsup_congr hu, limsup_congr hau, Pi.zero_def, ← ENNReal.bot_eq_zero,
      limsup_const_bot]
    simp

Goal state after:
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: α : Type u_1
f : Filter α
u : α → ℝ
h₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
r : ℝ≥0
h : ∀ y > ↑r, ∀ᶠ (a : α) in f, u a < y
x : ℝ≥0
hx : some x > ↑r
⊢ ↑x > ↑r

Tactic:
by simpa using hx

Goal state after:
Module: Mathlib.MeasureTheory.MeasurableSpace.Pi

Goal state before:
pp: case intro
ι : Type u_1
α : ι → Type u_2
inst✝ : Finite ι
C : (i : ι) → Set (Set (α i))
s : (i : ι) → ℕ → Set (α i)
h1s : ∀ (i : ι) (n : ℕ), s i n ∈ C i
h2s : ∀ (i : ι), ⋃ n, s i n = univ
val✝ : Encodable (ι → ℕ)
⊢ IsCountablySpanning (univ.pi '' univ.pi C)

Tactic:
let e : ℕ → ι → ℕ := fun n => (@decode (ι → ℕ) _ n).getD default

Goal state after:
case intro
ι : Type u_1
α : ι → Type u_2
inst✝ : Finite ι
C : (i : ι) → Set (Set (α i))
s : (i : ι) → ℕ → Set (α i)
h1s : ∀ (i : ι) (n : ℕ), s i n ∈ C i
h2s : ∀ (i : ι), ⋃ n, s i n = univ
val✝ : Encodable (ι → ℕ)
e : ℕ → ι → ℕ := fun n ↦ (decode n).getD default
⊢ IsCountablySpanning (univ.pi '' univ.pi C)