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

Goal state before:
pp: 𝕜 : Type u_2
ι : Type u_1
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : Fintype ι
inst✝ : IsOrderedAddMonoid 𝕜
⊢ stdSimplex 𝕜 ι ⊆ Icc 0 1

Tactic:
intro f h
  rw [← pi_univ_Icc, univ_pi_eq_iInter, mem_iInter]
  simpa using fun i ↦ mem_Icc_of_mem_stdSimplex h i

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✝⁴ : Ring k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
ι : Type u_4
inst✝ : Nontrivial k
p : ι → P
ha : AffineIndependent k p
s₁ s₂ : Set ι
i : ι
his₁ : i ∈ s₁
his₂ : i ∉ s₂
h₁ : s₁.Nontrivial
j : ι
hj : j ∈ s₁
hne : j ≠ i
he : vectorSpan k (p '' s₁) = vectorSpan k (p '' s₂)
fs : Finset ↑s₂
w : ↑s₂ → k
hw : ∑ i ∈ fs, w i = 0
hs : p i -ᵥ p j = (fs.weightedVSub fun x ↦ p ↑x) w
w' : ι → k := extend Subtype.val w 0
hw' : ∑ t ∈ map (Embedding.subtype fun x ↦ x ∈ s₂) fs, w' t = 0
hs' : p i -ᵥ p j = ((map (Embedding.subtype fun x ↦ x ∈ s₂) fs).weightedVSub p) w'
fs' : Finset ι := insert i (insert j (map (Embedding.subtype fun x ↦ x ∈ s₂) fs))
hfsfs' : map (Embedding.subtype fun x ↦ x ∈ s₂) fs ⊆ fs'
w'' : ι → k := (↑(map (Embedding.subtype fun x ↦ x ∈ s₂) fs)).indicator w'
⊢ map (Embedding.subtype fun x ↦ x ∈ s₂) fs ⊆ fs'

Tactic:
by grind

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✝⁴ : Ring k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
ι : Type u_4
inst✝ : Nontrivial k
p : ι → P
ha : AffineIndependent k p
s₁ s₂ : Set ι
i : ι
his₁ : i ∈ s₁
his₂ : i ∉ s₂
h₁ : s₁.Nontrivial
j : ι
hj : j ∈ s₁
hne : j ≠ i
he : vectorSpan k (p '' s₁) = vectorSpan k (p '' s₂)
fs : Finset ↑s₂
w : ↑s₂ → k
hw : ∑ i ∈ fs, w i = 0
hs : p i -ᵥ p j = (fs.weightedVSub fun x ↦ p ↑x) w
w' : ι → k := extend Subtype.val w 0
hw' : ∑ t ∈ map (Embedding.subtype fun x ↦ x ∈ s₂) fs, w' t = 0
hs' : p i -ᵥ p j = ((map (Embedding.subtype fun x ↦ x ∈ s₂) fs).weightedVSub p) w'
fs' : Finset ι := insert i (insert j (map (Embedding.subtype fun x ↦ x ∈ s₂) fs))
hfsfs' : map (Embedding.subtype fun x ↦ x ∈ s₂) fs ⊆ fs'
w'' : ι → k := (↑(map (Embedding.subtype fun x ↦ x ∈ s₂) fs)).indicator w'
hs'' : p i -ᵥ p j = (fs'.weightedVSub p) w''
⊢ map (Embedding.subtype fun x ↦ x ∈ s₂) fs ⊆ fs'

Tactic:
by grind

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

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_4
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
I : Set k
n : ℕ
s : Simplex k P n
S : AffineSubspace k P
hS : affineSpan k (Set.range s.points) ≤ S
this : Nonempty ↥S := Nonempty.map (⇑(AffineSubspace.inclusion hS)) inferInstance
⊢ ↑(affineSpan k (Set.range s.points)) ⊆ Set.range ⇑S.subtype

Tactic:
simpa using! hS

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

Goal state before:
pp: R : Type u_1
E : Type u_3
ι : Type u_5
inst✝² : Field R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
t : Finset ι
w : ι → R
z : ι → E
t' : Finset ι
ht : t ⊆ t'
h : ∀ i ∈ t', i ∉ t → w i = 0
⊢ t.centerMass w z = t'.centerMass w z

Tactic:
by
  rw [centerMass, sum_subset ht h, smul_sum, centerMass, smul_sum]
  apply sum_subset ht
  intro i hit' hit
  rw [h i hit' hit, zero_smul, smul_zero]

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

Goal state before:
pp: R : Type u_1
E : Type u_3
ι : Type u_5
inst✝³ : Field R
inst✝² : AddCommGroup E
inst✝¹ : Module R E
t : Finset ι
w : ι → R
z : ι → E
inst✝ : DecidableEq ι
t' : Finset ι
w' : ι → R
z' : ι → E
h : ∀ (i : ι), i ∈ t ∧ w i ≠ 0 ∨ i ∈ t' ∧ w' i ≠ 0 → i ∈ t ∩ t' ∧ w i = w' i ∧ z i = z' i
⊢ {i ∈ t | w i ≠ 0} = {i ∈ t' | w' i ≠ 0}

Tactic:
by grind

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

Goal state before:
pp: R : Type u_1
E : Type u_3
ι : Type u_5
inst✝³ : Field R
inst✝² : AddCommGroup E
inst✝¹ : Module R E
t : Finset ι
w : ι → R
z : ι → E
inst✝ : DecidableEq ι
t' : Finset ι
w' : ι → R
z' : ι → E
h : ∀ (i : ι), i ∈ t ∧ w i ≠ 0 ∨ i ∈ t' ∧ w' i ≠ 0 → i ∈ t ∩ t' ∧ w i = w' i ∧ z i = z' i
⊢ ∀ x ∈ {i ∈ t' | w' i ≠ 0}, w x = w' x

Tactic:
by grind

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

Goal state before:
pp: R : Type u_1
E : Type u_3
ι : Type u_5
inst✝³ : Field R
inst✝² : AddCommGroup E
inst✝¹ : Module R E
t : Finset ι
w : ι → R
z : ι → E
inst✝ : DecidableEq ι
t' : Finset ι
w' : ι → R
z' : ι → E
h : ∀ (i : ι), i ∈ t ∧ w i ≠ 0 ∨ i ∈ t' ∧ w' i ≠ 0 → i ∈ t ∩ t' ∧ w i = w' i ∧ z i = z' i
⊢ ∀ x ∈ {i ∈ t' | w' i ≠ 0}, w x • z x = w' x • z' x

Tactic:
by grind

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

Goal state before:
pp: R : Type u_1
E : Type u_3
ι : Type u_5
inst✝³ : Field R
inst✝² : AddCommGroup E
inst✝¹ : Module R E
t : Finset ι
w : ι → R
z : ι → E
inst✝ : DecidableEq ι
t' : Finset ι
h : ∀ i ∈ t ∪ t', w i ≠ 0 → i ∈ t ∩ t'
⊢ ∀ (i : ι), i ∈ t ∧ w i ≠ 0 ∨ i ∈ t' ∧ w i ≠ 0 → i ∈ t ∩ t' ∧ w i = w i ∧ z i = z i

Tactic:
by grind

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

Goal state before:
pp: R : Type u_1
E : Type u_3
ι : Type u_5
inst✝² : Field R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
t : Finset ι
w : ι → R
z : ι → E
w' : ι → R
h : ∀ i ∈ t, w i = w' i
⊢ ∀ (i : ι), i ∈ t ∧ w i ≠ 0 ∨ i ∈ t ∧ w' i ≠ 0 → i ∈ t ∩ t ∧ w i = w' i ∧ z i = z i

Tactic:
by grind

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

Goal state before:
pp: R : Type u_1
E : Type u_3
ι : Type u_5
inst✝² : Field R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
t : Finset ι
w : ι → R
z z' : ι → E
h : ∀ i ∈ t, w i ≠ 0 → z i = z' i
⊢ ∀ (i : ι), i ∈ t ∧ w i ≠ 0 ∨ i ∈ t ∧ w i ≠ 0 → i ∈ t ∩ t ∧ w i = w i ∧ z i = z' i

Tactic:
by grind

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

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_4
inst✝⁶ : Ring k
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module k V
inst✝³ : AffineSpace V P
inst✝² : PartialOrder k
inst✝¹ : IsOrderedAddMonoid k
n : ℕ
s : Simplex k P n
fs : Finset (Fin (n + 1))
m : ℕ
inst✝ : NeZero m
h : #fs = m + 1
w : Fin (n + 1) → k
hw : ∑ i, w i = 1
⊢ ((∀ i ∈ fs, w i ∈ Set.Ioo 0 1) ∧ ∀ i ∉ fs, w i = 0) → (∀ i ∈ fs, 0 < w i) ∧ ∀ i ∉ fs, w i = 0

Tactic:
by grind

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.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 : PartialOrder k
inst✝ : IsOrderedAddMonoid k
n : ℕ
s : Simplex k P n
fs : Finset (Fin (n + 1))
m : ℕ
h : #fs = m + 1
w : Fin (n + 1) → k
hw : ∑ i, w i = 1
⊢ ((∀ i ∈ fs, w i ∈ Set.Icc 0 1) ∧ ∀ i ∉ fs, w i = 0) → (∀ i ∈ fs, 0 ≤ w i) ∧ ∀ i ∉ fs, w i = 0

Tactic:
by grind

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.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 : PartialOrder k
inst✝ : ZeroLEOneClass k
n : ℕ
s : Simplex k P n
fs : Finset (Fin (n + 1))
m : ℕ
h : #fs = m + 1
w : Fin (n + 1) → k
hw1 : ∑ i, w i = 1
hp : (∀ i ∈ fs, w i ∈ Set.Icc 0 1) ∧ ∀ i ∉ fs, w i = 0
hp' : (affineCombination k univ s.points) w ∈ affineSpan k (Set.range s.points)
i : Fin (n + 1)
⊢ w i ∈ Set.Icc 0 1

Tactic:
by_cases hi : i ∈ fs <;> aesop

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

Goal state before:
pp: case hz
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
⊢ ∀ i ∈ s, v i ∈ Set.range v

Tactic:
· simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁸ : Field 𝕜
inst✝⁷ : PartialOrder 𝕜
inst✝⁶ : ZeroLEOneClass 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousConstSMul 𝕜 E
inst✝ : LocallyConvexSpace 𝕜 E
s t : Set E
disj : Disjoint s t
hs₁ : Convex 𝕜 s
hs₂ : IsCompact s
ht₁ : Convex 𝕜 t
ht₂ : IsClosed[inst✝³] t
this✝ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
this : IsUniformAddGroup E
⊢ ∃ u v, IsOpen[inst✝³] u ∧ IsOpen[inst✝³] v ∧ Convex 𝕜 u ∧ Convex 𝕜 v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v

Tactic:
have := (LocallyConvexSpace.convex_open_basis_zero 𝕜 E).comap fun x : E × E => x.2 - x.1

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁸ : Field 𝕜
inst✝⁷ : PartialOrder 𝕜
inst✝⁶ : ZeroLEOneClass 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousConstSMul 𝕜 E
inst✝ : LocallyConvexSpace 𝕜 E
s t : Set E
disj : Disjoint s t
hs₁ : Convex 𝕜 s
hs₂ : IsCompact s
ht₁ : Convex 𝕜 t
ht₂ : IsClosed[inst✝³] t
this✝¹ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
this✝ : IsUniformAddGroup E
this :
  (comap (fun x ↦ x.2 - x.1) (𝓝 0)).HasBasis (fun s ↦ 0 ∈ s ∧ IsOpen[inst✝³] s ∧ Convex 𝕜 s) fun i ↦
    (fun x ↦ x.2 - x.1) ⁻¹' id i
⊢ ∃ u v, IsOpen[inst✝³] u ∧ IsOpen[inst✝³] v ∧ Convex 𝕜 u ∧ Convex 𝕜 v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v
Module: Mathlib.Topology.Algebra.Module.LocallyConvex

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁸ : Field 𝕜
inst✝⁷ : PartialOrder 𝕜
inst✝⁶ : ZeroLEOneClass 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousConstSMul 𝕜 E
inst✝ : LocallyConvexSpace 𝕜 E
s t : Set E
disj : Disjoint s t
hs₁ : Convex 𝕜 s
hs₂ : IsCompact s
ht₁ : Convex 𝕜 t
ht₂ : IsClosed[inst✝³] t
this✝¹ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
this✝ : IsUniformAddGroup E
this : (uniformity E).HasBasis (fun s ↦ 0 ∈ s ∧ IsOpen[inst✝³] s ∧ Convex 𝕜 s) fun i ↦ (fun x ↦ x.2 - x.1) ⁻¹' id i
V : Set E
hV :
  Disjoint (⋃ x ∈ s, UniformSpace.ball x ((fun x ↦ x.2 - x.1) ⁻¹' id V))
    (⋃ x ∈ t, UniformSpace.ball x ((fun x ↦ x.2 - x.1) ⁻¹' id V))
hV0 : 0 ∈ V
hVopen : IsOpen[inst✝³] V
hVconvex : Convex 𝕜 V
⊢ Disjoint (⋃ a ∈ s, (fun x ↦ a + x) '' V) (⋃ a ∈ t, (fun x ↦ a + x) '' V)

Tactic:
image_add_left

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁸ : Field 𝕜
inst✝⁷ : PartialOrder 𝕜
inst✝⁶ : ZeroLEOneClass 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousConstSMul 𝕜 E
inst✝ : LocallyConvexSpace 𝕜 E
s t : Set E
disj : Disjoint s t
hs₁ : Convex 𝕜 s
hs₂ : IsCompact s
ht₁ : Convex 𝕜 t
ht₂ : IsClosed[inst✝³] t
this✝¹ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
this✝ : IsUniformAddGroup E
this : (uniformity E).HasBasis (fun s ↦ 0 ∈ s ∧ IsOpen[inst✝³] s ∧ Convex 𝕜 s) fun i ↦ (fun x ↦ x.2 - x.1) ⁻¹' id i
V : Set E
hV :
  Disjoint (⋃ x ∈ s, UniformSpace.ball x ((fun x ↦ x.2 - x.1) ⁻¹' id V))
    (⋃ x ∈ t, UniformSpace.ball x ((fun x ↦ x.2 - x.1) ⁻¹' id V))
hV0 : 0 ∈ V
hVopen : IsOpen[inst✝³] V
hVconvex : Convex 𝕜 V
⊢ Disjoint (⋃ a ∈ s, (fun x ↦ -a + x) ⁻¹' V) (⋃ a ∈ t, (fun x ↦ -a + x) ⁻¹' V)
Module: Mathlib.Analysis.Normed.Module.Convex

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a : E
r : ENNReal
⊢ Convex ℝ (eball a r)

Tactic:
cases r with

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
a : E
r : ENNReal
⊢ Convex ℝ (closedEBall a r)

Tactic:
cases r with

Goal state after:
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
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
p₁ p₂ : E
⊢ (affineSpan R ((convexHull R) {p₁, p₂})).direction = R ∙ (p₂ -ᵥ p₁)

Tactic:
affineSpan_convexHull,

Goal state after:
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
p₁ p₂ : E
⊢ line[R, p₁, p₂].direction = R ∙ (p₂ -ᵥ p₁)
Module: Mathlib.Analysis.Convex.Jensen

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
β : Type u_4
ι : Type u_5
inst✝⁹ : Field 𝕜
inst✝⁸ : LinearOrder 𝕜
inst✝⁷ : IsStrictOrderedRing 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : AddCommGroup β
inst✝⁴ : PartialOrder β
inst✝³ : IsOrderedAddMonoid β
inst✝² : Module 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : IsStrictOrderedModule 𝕜 β
s : Set E
f : E → β
t : Finset ι
w : ι → 𝕜
p : ι → E
hf : StrictConvexOn 𝕜 s f
h₀ : ∀ i ∈ t, 0 ≤ w i
h₁ : ∑ i ∈ t, w i = 1
hmem : ∀ i ∈ t, p i ∈ s
⊢ ∀ i ∈ {i ∈ t | w i ≠ 0}, 0 < w i

Tactic:
by grind

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

Goal state before:
pp: case refine_1.inl
𝕜 : Type u_1
E : Type u_2
β : Type u_4
ι : Type u_5
inst✝⁹ : Field 𝕜
inst✝⁸ : LinearOrder 𝕜
inst✝⁷ : IsStrictOrderedRing 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : AddCommGroup β
inst✝⁴ : PartialOrder β
inst✝³ : IsOrderedAddMonoid β
inst✝² : Module 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : IsStrictOrderedModule 𝕜 β
s : Set E
f : E → β
w : ι → 𝕜
p : ι → E
hf : StrictConvexOn 𝕜 s f
h₀ : ∀ i ∈ ∅, 0 < w i
h₁ : ∑ i ∈ ∅, w i = 1
hmem : ∀ i ∈ ∅, p i ∈ s
⊢ f (∑ i ∈ ∅, w i • p i) = ∑ i ∈ ∅, w i • f (p i) → ∀ j ∈ ∅, p j = ∑ i ∈ ∅, w i • p i

pp: case refine_1.inr
𝕜 : Type u_1
E : Type u_2
β : Type u_4
ι : Type u_5
inst✝⁹ : Field 𝕜
inst✝⁸ : LinearOrder 𝕜
inst✝⁷ : IsStrictOrderedRing 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : AddCommGroup β
inst✝⁴ : PartialOrder β
inst✝³ : IsOrderedAddMonoid β
inst✝² : Module 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : IsStrictOrderedModule 𝕜 β
s : Set E
f : E → β
t : Finset ι
w : ι → 𝕜
p : ι → E
hf : StrictConvexOn 𝕜 s f
h₀ : ∀ i ∈ t, 0 < w i
h₁ : ∑ i ∈ t, w i = 1
hmem : ∀ i ∈ t, p i ∈ s
i₀ : ι
hi₀ : i₀ ∈ t
⊢ f (∑ i ∈ t, w i • p i) = ∑ i ∈ t, w i • f (p i) → ∀ j ∈ t, p j = ∑ i ∈ t, w i • p i

Tactic:
· simp

Goal state after:
case refine_1.inr
𝕜 : Type u_1
E : Type u_2
β : Type u_4
ι : Type u_5
inst✝⁹ : Field 𝕜
inst✝⁸ : LinearOrder 𝕜
inst✝⁷ : IsStrictOrderedRing 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : AddCommGroup β
inst✝⁴ : PartialOrder β
inst✝³ : IsOrderedAddMonoid β
inst✝² : Module 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : IsStrictOrderedModule 𝕜 β
s : Set E
f : E → β
t : Finset ι
w : ι → 𝕜
p : ι → E
hf : StrictConvexOn 𝕜 s f
h₀ : ∀ i ∈ t, 0 < w i
h₁ : ∑ i ∈ t, w i = 1
hmem : ∀ i ∈ t, p i ∈ s
i₀ : ι
hi₀ : i₀ ∈ t
⊢ f (∑ i ∈ t, w i • p i) = ∑ i ∈ t, w i • f (p i) → ∀ j ∈ t, p j = ∑ i ∈ t, w i • p i
Module: Mathlib.Analysis.Convex.Combination

Goal state before:
pp: case neg
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
inst✝⁵ : Finite ι
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : (i : ι) → AddCommGroup (E i)
inst✝ : (i : ι) → Module 𝕜 (E i)
s : Set ι
t : (i : ι) → Set (E i)
x : (i : ι) → E i
h : ∀ i ∈ s, x i ∈ (convexHull 𝕜) (t i)
val✝ : Fintype ι
this :
  ∀ {s : Set ι} {t : (i : ι) → Set (E i)},
    (∀ i ∈ s, x i ∈ (convexHull 𝕜) (t i)) → s = Set.univ → x ∈ (convexHull 𝕜) (s.pi t)
hs : ¬s = Set.univ
i : ι
x✝ : i ∈ Set.univ
hi : i ∉ s
⊢ x i ∈ (convexHull 𝕜) Set.univ

Tactic:
· simp

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

Goal state before:
pp: case refine_2
𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
t : TopologicalSpace E
p : SeminormFamily 𝕜 E ι
hp : 𝓝 0 = ⨅ i, comap (⇑(p i)) (𝓝 0)
this : IsTopologicalAddGroup E
hp' : ∀ (i : ι), Continuous[t, _] ⇑(p i)
⊢ ⨅ i, comap (⇑↑i) (𝓝 0) ≤ 𝓝 0

Tactic:
hp,

Goal state after:
case refine_2
𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
t : TopologicalSpace E
p : SeminormFamily 𝕜 E ι
hp : 𝓝 0 = ⨅ i, comap (⇑(p i)) (𝓝 0)
this : IsTopologicalAddGroup E
hp' : ∀ (i : ι), Continuous[t, _] ⇑(p i)
⊢ ⨅ i, comap (⇑↑i) (𝓝 0) ≤ ⨅ i, comap (⇑(p i)) (𝓝 0)
Module: Mathlib.Topology.Algebra.Module.Spaces.UniformConvergenceCLM

Goal state before:
pp: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝⁸ : NormedField 𝕜₁
inst✝⁷ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_3
F : Type u_4
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜₁ E
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜₂ F
inst✝¹ : TopologicalSpace F
inst✝ : IsTopologicalAddGroup F
ι : Type u_6
𝔖 : Set (Set E)
h𝔖₁ : 𝔖.Nonempty
h𝔖₂ : DirectedOn (fun x1 x2 ↦ x1 ⊆ x2) 𝔖
p : ι → Prop
b : ι → Set F
h : (𝓝 0).HasBasis p b
⊢ (𝓝 0).HasBasis (fun Si ↦ Si.1 ∈ 𝔖 ∧ p Si.2) fun Si ↦ {f | ∀ x ∈ Si.1, f x ∈ b Si.2}

Tactic:
let : UniformSpace F := IsTopologicalAddGroup.rightUniformSpace F
  have : IsUniformAddGroup F := isUniformAddGroup_of_addCommGroup
  rw [(isEmbedding_coeFn σ F 𝔖).isInducing.nhds_eq_comap]
  exact (UniformOnFun.hasBasis_nhds_zero_of_basis 𝔖 h𝔖₁ h𝔖₂ h).comap DFunLike.coe

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_4
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedSpace 𝕜 E
ι : Type u_9
inst✝² : Fintype ι
M : ι → Type u_10
inst✝¹ : (i : ι) → SeminormedAddCommGroup (M i)
inst✝ : (i : ι) → NormedSpace 𝕜 (M i)
C : ℝ
L : (i : ι) → E →L[𝕜] M i
hL : ∀ (i : ι), ‖L i‖ ≤ C
hC : 0 ≤ C
⊢ ‖pi L‖ ≤ C

Tactic:
refine opNorm_le_bound _ hC (fun x ↦ ?_)

Goal state after:
𝕜 : Type u_1
E : Type u_4
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedSpace 𝕜 E
ι : Type u_9
inst✝² : Fintype ι
M : ι → Type u_10
inst✝¹ : (i : ι) → SeminormedAddCommGroup (M i)
inst✝ : (i : ι) → NormedSpace 𝕜 (M i)
C : ℝ
L : (i : ι) → E →L[𝕜] M i
hL : ∀ (i : ι), ‖L i‖ ≤ C
hC : 0 ≤ C
x : E
⊢ ‖(pi L) x‖ ≤ C * ‖x‖
Module: Mathlib.Analysis.Normed.Operator.NNNorm

Goal state before:
pp: case inl
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
hM : ∀ (x : E), ‖f x‖ₑ ≤ ∞ * ‖x‖ₑ
⊢ ‖f‖ₑ ≤ ∞

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:
· simp

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.Topology.Algebra.Module.Spaces.UniformConvergenceCLM

Goal state before:
pp: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝¹⁵ : NormedField 𝕜₁
inst✝¹⁴ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E✝ : Type u_3
F✝ : Type u_4
G : Type u_5
inst✝¹³ : AddCommGroup E✝
inst✝¹² : Module 𝕜₁ E✝
inst✝¹¹ : TopologicalSpace E✝
inst✝¹⁰ : AddCommGroup F✝
inst✝⁹ : Module 𝕜₂ F✝
𝕜 : Type u_6
inst✝⁸ : NormedField 𝕜
E : Type u_7
ι : Type u_8
F : ι → Type u_9
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : (i : ι) → AddCommGroup (F i)
inst✝³ : (i : ι) → Module 𝕜 (F i)
inst✝² : (i : ι) → TopologicalSpace (F i)
inst✝¹ : ∀ (i : ι), IsTopologicalAddGroup (F i)
inst✝ : ∀ (i : ι), ContinuousConstSMul 𝕜 (F i)
𝔖 : Set (Set E)
this✝ : (i : ι) → UniformSpace (F i) := fun i ↦ IsTopologicalAddGroup.rightUniformSpace (F i)
this : ∀ (i : ι), IsUniformAddGroup (F i)
⊢ Continuous[Pi.topologicalSpace, (Pi.uniformSpace fun i ↦ E →ᵤ[𝔖] F i).toTopologicalSpace]
    (⇑(UniformOnFun.uniformEquivPiComm 𝔖 F) ∘
      (⇑(UniformOnFun.ofFun 𝔖) ∘ DFunLike.coe) ∘ fun F_1 ↦ ContinuousLinearMap.pi F_1)

Tactic:
refine continuous_pi fun i ↦ ?_

Goal state after:
𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝¹⁵ : NormedField 𝕜₁
inst✝¹⁴ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E✝ : Type u_3
F✝ : Type u_4
G : Type u_5
inst✝¹³ : AddCommGroup E✝
inst✝¹² : Module 𝕜₁ E✝
inst✝¹¹ : TopologicalSpace E✝
inst✝¹⁰ : AddCommGroup F✝
inst✝⁹ : Module 𝕜₂ F✝
𝕜 : Type u_6
inst✝⁸ : NormedField 𝕜
E : Type u_7
ι : Type u_8
F : ι → Type u_9
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : (i : ι) → AddCommGroup (F i)
inst✝³ : (i : ι) → Module 𝕜 (F i)
inst✝² : (i : ι) → TopologicalSpace (F i)
inst✝¹ : ∀ (i : ι), IsTopologicalAddGroup (F i)
inst✝ : ∀ (i : ι), ContinuousConstSMul 𝕜 (F i)
𝔖 : Set (Set E)
this✝ : (i : ι) → UniformSpace (F i) := fun i ↦ IsTopologicalAddGroup.rightUniformSpace (F i)
this : ∀ (i : ι), IsUniformAddGroup (F i)
i : ι
⊢ Continuous[Pi.topologicalSpace, ((fun i ↦ UniformOnFun.uniformSpace E (F i) 𝔖) i).toTopologicalSpace] fun a ↦
    (⇑(UniformOnFun.uniformEquivPiComm 𝔖 F) ∘
        (⇑(UniformOnFun.ofFun 𝔖) ∘ DFunLike.coe) ∘ fun F_1 ↦ ContinuousLinearMap.pi F_1)
      a i
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✝⁹ : SeminormedAddCommGroup E
inst✝⁸ : SeminormedAddCommGroup F
inst✝⁷ : SeminormedAddCommGroup G
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
inst✝⁴ : NontriviallyNormedField 𝕜₃
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace 𝕜₂ F
inst✝¹ : NormedSpace 𝕜₃ G
σ₁₂ : 𝕜 →+* 𝕜₂
σ₁₃ : 𝕜 →+* 𝕜₃
inst✝ : RingHomIsometric σ₁₃
f : E →SL[σ₁₂] F
g : E →SL[σ₁₃] G
h : ∀ (x : E), ‖f x‖ = ‖g x‖
x : E
⊢ ‖f x‖ ≤ ‖g‖ * ‖x‖

Tactic:
rw [h x]

Goal state after:
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✝⁹ : SeminormedAddCommGroup E
inst✝⁸ : SeminormedAddCommGroup F
inst✝⁷ : SeminormedAddCommGroup G
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
inst✝⁴ : NontriviallyNormedField 𝕜₃
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace 𝕜₂ F
inst✝¹ : NormedSpace 𝕜₃ G
σ₁₂ : 𝕜 →+* 𝕜₂
σ₁₃ : 𝕜 →+* 𝕜₃
inst✝ : RingHomIsometric σ₁₃
f : E →SL[σ₁₂] F
g : E →SL[σ₁₃] G
h : ∀ (x : E), ‖f x‖ = ‖g x‖
x : E
⊢ ‖g x‖ ≤ ‖g‖ * ‖x‖
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: f : Filter ℝ
hf : ¬IsCobounded (fun x1 x2 ↦ x1 ≤ x2) f
⊢ f.limsSup = 0

Tactic:
rwa [limsSup, sInf_of_not_bddBelow]

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

Goal state before:
pp: case mp.none
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
hx : none > ↑r
⊢ ∀ᶠ (a : α) in f, ENNReal.ofReal (u a) < none

pp: case mp.some
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
⊢ ∀ᶠ (a : α) in f, ENNReal.ofReal (u a) < some x

Tactic:
· simp

Goal state after:
case mp.some
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
⊢ ∀ᶠ (a : α) in f, ENNReal.ofReal (u a) < some x