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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
s : Set E
hs : Convex 𝕜 s
x : E
hx : x ∉ (convexHull 𝕜) (s \ {x})
h : s \ {x} = (convexHull 𝕜) (s \ {x})
⊢ Convex 𝕜 (s \ {x})

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
s : Set E
hs : Convex 𝕜 s
x : E
hx : x ∉ (convexHull 𝕜) (s \ {x})
y : E
hy : y ∈ (convexHull 𝕜) (s \ {x})
⊢ y ∉ {x}

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : Semiring 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
x₁ x₂ : E
y : F
⊢ (fun x ↦ (x, y)) '' [x₁ -[𝕜] x₂] = [(x₁, y) -[𝕜] (x₂, y)]

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
s : Set E
hs : Convex 𝕜 s
x : E
⊢ Convex 𝕜 (s \ {x}) ↔ x ∉ (convexHull 𝕜) (s \ {x})

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁶ : Semiring 𝕜
inst✝⁵ : PartialOrder 𝕜
inst✝⁴ : AddCommMonoid E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : Module 𝕜 E
inst✝ : PosSMulMono 𝕜 E
s : Set E
hs : s.OrdConnected
h : IsChain (fun x1 x2 ↦ x1 ≤ x2) s
⊢ Convex 𝕜 s

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : Semiring 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
x : E
y₁ y₂ : F
⊢ (fun y ↦ (x, y)) '' [y₁ -[𝕜] y₂] = [(x, y₁) -[𝕜] (x, y₂)]

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
s : Set 𝕜
⊢ Convex 𝕜 s ↔ s.OrdConnected

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : Ring 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : AddCommGroup E
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
f : E →ᵃ[𝕜] F
s : Set E
⊢ ⇑f '' (convexHull 𝕜) s = (convexHull 𝕜) (⇑f '' s)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : Ring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
⊢ affineSpan 𝕜 ((convexHull 𝕜) s) = affineSpan 𝕜 s

Tactic:
by

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

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
S : AffineSpace V P
s : Set P
inst✝ : Nontrivial P
⊢ affineSpan k s = ⊤ ↔ vectorSpan k s = ⊤

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : Ring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
⊢ (convexHull 𝕜) (-s) = -(convexHull 𝕜) s

Tactic:
by

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

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
S : AffineSpace V P
ι : Type u_5
inst✝ : Fintype ι
p : ι → P
h : affineSpan k (range p) = ⊤
⊢ 0 < Fintype.card ι

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : Semiring 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
x₁ x₂ : E
y : F
⊢ (fun x ↦ (x, y)) '' openSegment 𝕜 x₁ x₂ = openSegment 𝕜 (x₁, y) (x₂, y)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bornology.Absorbs

Goal state before:
pp: M : Type u_1
α : Type u_2
inst✝¹ : Bornology M
inst✝ : SMul M α
s : Set α
⊢ Absorbs M s ∅

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bornology.Absorbs

Goal state before:
pp: M : Type u_1
α : Type u_2
inst✝² : Bornology M
inst✝¹ : SMul M α
s t : Set α
inst✝ : BoundedSpace M
⊢ Absorbs M s t

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bornology.Absorbs

Goal state before:
pp: M : Type u_1
α : Type u_2
inst✝¹ : Bornology M
inst✝ : SMul M α
s t₁ t₂ : Set α
⊢ Absorbs M s (t₁ ∪ t₂) ↔ Absorbs M s t₁ ∧ Absorbs M s t₂

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bornology.Absorbs

Goal state before:
pp: M : Type u_1
α : Type u_2
inst✝¹ : Bornology M
inst✝ : SMul M α
s : Set α
T : Set (Set α)
hT : T.Finite
⊢ Absorbs M s (⋃₀ T) ↔ ∀ t ∈ T, Absorbs M s t

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bornology.Absorbs

Goal state before:
pp: M : Type u_1
α : Type u_2
inst✝¹ : Bornology M
inst✝ : SMul M α
s : Set α
ι : Type u_3
t : ι → Set α
I : Set ι
hI : I.Finite
⊢ Absorbs M s (⋃ i ∈ I, t i) ↔ ∀ i ∈ I, Absorbs M s (t i)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bornology.Absorbs

Goal state before:
pp: M : Type u_1
E : Type u_2
inst✝² : Bornology M
s₁ s₂ t₁ t₂ : Set E
inst✝¹ : AddZeroClass E
inst✝ : DistribSMul M E
h₁ : Absorbs M s₁ t₁
h₂ : Absorbs M s₂ t₂
x : M
hx₁ : t₁ ⊆ x • s₁
hx₂ : t₂ ⊆ x • s₂
⊢ t₁ + t₂ ⊆ x • (s₁ + s₂)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bornology.Absorbs

Goal state before:
pp: G₀ : Type u_1
α : Type u_2
inst✝² : GroupWithZero G₀
inst✝¹ : Bornology G₀
inst✝ : MulAction G₀ α
s : Set α
a : G₀
ha : a ≠ 0
⊢ s ⊆ a • univ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bornology.Absorbs

Goal state before:
pp: G₀ : Type u_1
α : Type u_2
inst✝² : GroupWithZero G₀
inst✝¹ : Bornology G₀
inst✝ : MulAction G₀ α
s t : Set α
c : G₀
hc : c ≠ 0
⊢ t ⊆ c • s ↔ MapsTo (fun x ↦ c⁻¹ • x) t s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bornology.Absorbs

Goal state before:
pp: G₀ : Type u_1
α : Type u_2
inst✝² : GroupWithZero G₀
inst✝¹ : Bornology G₀
inst✝ : MulAction G₀ α
s t u : Set α
⊢ Absorbs G₀ (s ∩ t) u ↔ Absorbs G₀ s u ∧ Absorbs G₀ t u

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bornology.Absorbs

Goal state before:
pp: G₀ : Type u_1
inst✝⁴ : GroupWithZero G₀
inst✝³ : Bornology G₀
inst✝² : (cobounded G₀).NeBot
E : Type u_3
inst✝¹ : AddMonoid E
inst✝ : DistribMulAction G₀ E
s : Set E
⊢ Absorbs G₀ s 0 ↔ 0 ∈ s

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : Semiring 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
x : E
y₁ y₂ : F
⊢ (fun y ↦ (x, y)) '' openSegment 𝕜 y₁ y₂ = openSegment 𝕜 (x, y₁) (x, y₂)

Tactic:
by

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

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
⊢ ⊥.direction = ⊥

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bornology.Absorbs

Goal state before:
pp: M : Type u_1
E : Type u_2
inst✝³ : Monoid M
inst✝² : AddGroup E
inst✝¹ : DistribMulAction M E
inst✝ : Bornology M
s t : Set E
⊢ Absorbs M (-s) (-t) ↔ Absorbs M s t

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bornology.Absorbs

Goal state before:
pp: M : Type u_1
E : Type u_2
inst✝³ : Monoid M
inst✝² : AddGroup E
inst✝¹ : DistribMulAction M E
inst✝ : Bornology M
s₁ s₂ t₁ t₂ : Set E
h₁ : Absorbs M s₁ t₁
h₂ : Absorbs M s₂ t₂
⊢ Absorbs M (s₁ - s₂) (t₁ - t₂)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_5
M : ι → Type u_6
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : (i : ι) → AddCommMonoid (M i)
inst✝ : (i : ι) → Module 𝕜 (M i)
s : Set ι
x y : (i : ι) → M i
⊢ [x -[𝕜] y] ⊆ s.pi fun i ↦ [x i -[𝕜] y i]

Tactic:
by

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

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
Q : AffineSubspace k P
⊢ (↑Q).Nonempty ↔ Q ≠ ⊥

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bornology.Absorbs

Goal state before:
pp: M : Type u_1
α : Type u_2
inst✝¹ : Bornology M
inst✝ : SMul M α
s t : Set α
hs : Absorbent M s
ht : t.Finite
⊢ Absorbs M s t

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_5
M : ι → Type u_6
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : (i : ι) → AddCommMonoid (M i)
inst✝ : (i : ι) → Module 𝕜 (M i)
s : Set ι
x y : (i : ι) → M i
⊢ openSegment 𝕜 x y ⊆ s.pi fun i ↦ openSegment 𝕜 (x i) (y i)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bornology.Absorbs

Goal state before:
pp: M : Type u_1
E : Type u_2
inst✝² : Bornology M
inst✝¹ : AddZeroClass E
inst✝ : DistribSMul M E
s₁ s₂ t : Set E
x : E
h₁ : Absorbent M s₁
h₂ : Absorbs M s₂ t
⊢ Absorbs M (s₁ + s₂) (x +ᵥ t)

Tactic:
by

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

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
Q : AffineSubspace k P
⊢ Q = ⊥ ∨ (↑Q).Nonempty

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bornology.Absorbs

Goal state before:
pp: G₀ : Type u_1
α : Type u_2
inst✝² : GroupWithZero G₀
inst✝¹ : Bornology G₀
inst✝ : MulAction G₀ α
s : Set α
x : α
⊢ Absorbs G₀ s {x} ↔ ∀ᶠ (c : G₀) in cobounded G₀, c⁻¹ • x ∈ s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bornology.Absorbs

Goal state before:
pp: M : Type u_1
N : Type u_2
α : Type u_3
inst✝⁶ : Monoid N
inst✝⁵ : SMul M N
inst✝⁴ : SMul M α
inst✝³ : MulAction N α
inst✝² : IsScalarTower M N α
inst✝¹ : Bornology M
inst✝ : Bornology N
s t : Set α
h : Absorbs N s t
hbdd : Tendsto (fun x ↦ x • 1) (cobounded M) (cobounded N)
x : M
hx : t ⊆ (x • 1) • s
⊢ t ⊆ x • s

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_5
M : ι → Type u_6
inst✝⁴ : Semiring 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : (i : ι) → AddCommMonoid (M i)
inst✝¹ : (i : ι) → Module 𝕜 (M i)
inst✝ : DecidableEq ι
i : ι
x₁ x₂ : M i
y : (i : ι) → M i
⊢ update y i '' [x₁ -[𝕜] x₂] = [update y i x₁ -[𝕜] update y i x₂]

Tactic:
by

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

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
S : AffineSpace V P
ι : Sort u_4
p₁ p₂ : P
inst✝ : Subsingleton P
s : AffineSubspace k P
⊢ s = ⊥ ∨ s = ⊤

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
K : Submodule 𝕜 E
⊢ Convex 𝕜 ↑K

Tactic:
by

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

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 : AffineSubspace k P
h : (↑s).Nonempty
⊢ s.direction = ⊤ ↔ s = ⊤

Tactic:
by

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

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
s₁ s₂ : AffineSubspace k P
⊢ (s₁ ⊓ s₂).direction ≤ s₁.direction ⊓ s₂.direction

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_5
M : ι → Type u_6
inst✝⁴ : Semiring 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : (i : ι) → AddCommMonoid (M i)
inst✝¹ : (i : ι) → Module 𝕜 (M i)
inst✝ : DecidableEq ι
i : ι
x₁ x₂ : M i
y : (i : ι) → M i
⊢ update y i '' openSegment 𝕜 x₁ x₂ = openSegment 𝕜 (update y i x₁) (update y i x₂)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝¹ : SeminormedRing 𝕜
inst✝ : SMul 𝕜 E
s : Set E
x : E
⊢ x ∈ balancedCore 𝕜 s ↔ ∃ t, Balanced 𝕜 t ∧ t ⊆ s ∧ x ∈ t

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝¹ : SeminormedRing 𝕜
inst✝ : SMul 𝕜 E
s : Set E
a : 𝕜
ha : ‖a‖ ≤ 1
⊢ a • balancedCore 𝕜 s ⊆ balancedCore 𝕜 s

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝¹ : SeminormedRing 𝕜
inst✝ : SMul 𝕜 E
s : Set E
x : E
⊢ x ∈ balancedHull 𝕜 s ↔ ∃ r, ‖r‖ ≤ 1 ∧ x ∈ r • s

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝¹ : SeminormedRing 𝕜
inst✝ : SMul 𝕜 E
s t : Set E
ht : Balanced 𝕜 t
h : s ⊆ t
⊢ balancedHull 𝕜 s ⊆ t

Tactic:
by

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

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
s₁ s₂ : AffineSubspace k P
p : P
h₁ : p ∈ s₁
h₂ : p ∈ s₂
⊢ (s₁ ⊓ s₂).direction = s₁.direction ⊓ s₂.direction

Tactic:
by

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

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
t : Set (AffineSubspace k P)
⊢ (sInf t).direction ≤ ⨅ s ∈ t, s.direction

Tactic:
by

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

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
t : Set (AffineSubspace k P)
p : P
h : ∀ s ∈ t, p ∈ s
⊢ (sInf t).direction = ⨅ s ∈ t, s.direction

Tactic:
apply (direction_sInf t).antisymm

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
t : Set (AffineSubspace k P)
p : P
h : ∀ s ∈ t, p ∈ s
⊢ ⨅ s ∈ t, s.direction ≤ (sInf t).direction
Module: Mathlib.Analysis.LocallyConvex.BalancedCoreHull

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝¹ : SeminormedRing 𝕜
inst✝ : SMul 𝕜 E
s t : Set E
hst : s ⊆ t
x : E
hx : x ∈ balancedHull 𝕜 s
⊢ x ∈ balancedHull 𝕜 t

Tactic:
rw [mem_balancedHull_iff] at *

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝¹ : SeminormedRing 𝕜
inst✝ : SMul 𝕜 E
s t : Set E
hst : s ⊆ t
x : E
hx : ∃ r, ‖r‖ ≤ 1 ∧ x ∈ r • s
⊢ ∃ r, ‖r‖ ≤ 1 ∧ x ∈ r • t
Module: Mathlib.Analysis.LocallyConvex.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
inst✝¹ : SeminormedRing 𝕜
inst✝ : SMul 𝕜 E
A B : Set E
⊢ (∃ i, True ∧ ∀ ⦃x : 𝕜⦄, x ∈ norm ⁻¹' Ici i → B ⊆ x • A) ↔ ∃ r, ∀ (c : 𝕜), r ≤ ‖c‖ → B ⊆ c • A

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝¹ : SeminormedRing 𝕜
inst✝ : SMul 𝕜 E
s t : Set E
hst : s ⊆ t
⊢ balancedHull 𝕜 s ⊆ balancedHull 𝕜 t

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
inst✝¹ : SeminormedRing 𝕜
inst✝ : SMul 𝕜 E
s : Set E
⊢ Balanced 𝕜 s ↔ Metric.closedBall 0 1 • s ⊆ s

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
inst✝¹ : SeminormedRing 𝕜
inst✝ : SMul 𝕜 E
x✝¹ : 𝕜
x✝ : ‖x✝¹‖ ≤ 1
⊢ x✝¹ • ∅ ⊆ ∅

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
inst✝¹ : SeminormedRing 𝕜
inst✝ : SMul 𝕜 E
S : Set (Set E)
h : ∀ s ∈ S, Balanced 𝕜 s
x✝³ : 𝕜
x✝² : ‖x✝³‖ ≤ 1
x✝¹ : E
x✝ : x✝¹ ∈ ⋂ s ∈ S, x✝³ • s
⊢ x✝¹ ∈ ⋂₀ S

Tactic:
by

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

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
t : Set (AffineSubspace k P)
p : P
h : ∀ s ∈ t, p ∈ s
⊢ (sInf t).direction = ⨅ s ∈ t, s.direction

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
hs : Balanced 𝕜 s
hs_nonempty : s.Nonempty
⊢ 0 ∈ s

Tactic:
by

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

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
ι : Sort u_4
s : ι → AffineSubspace k P
⊢ ↑(iInf s) = ⋂ i, ↑(s i)

Tactic:
by

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

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
ι : Sort u_4
s : ι → AffineSubspace k P
p : P
⊢ p ∈ iInf s ↔ ∀ (i : ι), p ∈ s i

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
F : Type u_4
inst✝² : SeminormedRing 𝕜
inst✝¹ : SMul 𝕜 E
inst✝ : SMul 𝕜 F
s : Set F
hs : Balanced 𝕜 s
f : E →ₑ[id] F
a : 𝕜
ha : ‖a‖ ≤ 1
x : E
x✝ : x ∈ a • ⇑f ⁻¹' s
y : E
hy₁ : y ∈ ⇑f ⁻¹' s
hy₂ : (fun x ↦ a • x) y = x
⊢ x ∈ ⇑f ⁻¹' s

Tactic:
by

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

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
ι : Sort u_4
s : ι → AffineSubspace k P
⊢ (iInf s).direction ≤ ⨅ i, (s i).direction

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
inst✝³ : SeminormedRing 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
s : Set E
inst✝ : NormOneClass 𝕜
h : Balanced 𝕜 s
x : E
hx : -x ∈ s
⊢ ‖-1‖ ≤ 1

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
inst✝³ : SeminormedRing 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
s : Set E
inst✝ : NormOneClass 𝕜
h : Balanced 𝕜 s
x : E
hx : -x ∈ s
⊢ x ∈ s

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
inst✝³ : SeminormedRing 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
s : Set E
inst✝ : NormOneClass 𝕜
h : Balanced 𝕜 s
x : E
hx : x ∈ s
⊢ ‖-1‖ ≤ 1

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
inst✝³ : SeminormedRing 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
s : Set E
inst✝ : NormOneClass 𝕜
h : Balanced 𝕜 s
x : E
hx : x ∈ s
⊢ -x ∈ s

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
hs : Balanced 𝕜 s
ht : Balanced 𝕜 t
⊢ Balanced 𝕜 (s - t)

Tactic:
simp_rw [sub_eq_add_neg]
  exact hs.add ht.neg

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
hs : Balanced 𝕜 s
ht : Balanced 𝕜 t
⊢ Balanced 𝕜 (s - t)

Tactic:
by

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

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
ι : Sort u_4
s : ι → AffineSubspace k P
p : P
h : ∀ (i : ι), p ∈ s i
⊢ (iInf s).direction = ⨅ i, (s i).direction

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
inst✝² : NormedDivisionRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
⊢ Absorbs 𝕜 s t ↔ ∀ᶠ (c : 𝕜) in 𝓝[≠] 0, MapsTo (fun x ↦ c • x) t s

Tactic:
by

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

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
ι : Sort u_4
s : ι → AffineSubspace k P
p : P
h : p ∈ iInf s
⊢ (iInf s).direction = ⨅ i, (s i).direction

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
inst✝² : NormedDivisionRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
x : E
⊢ Absorbs 𝕜 s {x} ↔ ∀ᶠ (c : 𝕜) in 𝓝[≠] 0, c • x ∈ s

Tactic:
by

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

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
s₁ s₂ : AffineSubspace k P
h : s₁ ≤ s₂
⊢ s₁.direction ≤ s₂.direction

Tactic:
by

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

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
s₁ s₂ : AffineSubspace k P
⊢ s₁.direction ⊔ s₂.direction ≤ (s₁ ⊔ s₂).direction

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
inst✝² : NormedDivisionRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
h₀ : 0 ∈ s
⊢ Absorbs 𝕜 s t ↔ ∀ᶠ (c : 𝕜) in 𝓝 0, MapsTo (fun x ↦ c • x) t s

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
⊢ Balanced 𝕜 (balancedHull 𝕜 s)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
𝕝 : Type u_2
E : Type u_3
inst✝⁷ : NormedDivisionRing 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
s : Set E
inst✝⁴ : NormedRing 𝕝
inst✝³ : Module 𝕜 𝕝
inst✝² : NormSMulClass 𝕜 𝕝
inst✝¹ : SMulWithZero 𝕝 E
inst✝ : IsScalarTower 𝕜 𝕝 E
b : 𝕜
hs : Balanced 𝕝 s
a : 𝕝
h : ‖a‖ ≤ ‖b‖
hb : b ≠ 0
⊢ a • s = b • (b⁻¹ • a) • s

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
𝕝 : Type u_2
E : Type u_3
inst✝⁷ : NormedDivisionRing 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
s : Set E
inst✝⁴ : NormedRing 𝕝
inst✝³ : Module 𝕜 𝕝
inst✝² : NormSMulClass 𝕜 𝕝
inst✝¹ : SMulWithZero 𝕝 E
inst✝ : IsScalarTower 𝕜 𝕝 E
b : 𝕜
hs : Balanced 𝕝 s
a : 𝕝
h : ‖a‖ ≤ ‖b‖
hb : b ≠ 0
⊢ ‖b⁻¹ • a‖ ≤ 1

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
𝕝 : Type u_2
E : Type u_3
inst✝⁷ : NormedDivisionRing 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
s : Set E
inst✝⁴ : NormedRing 𝕝
inst✝³ : Module 𝕜 𝕝
inst✝² : NormSMulClass 𝕜 𝕝
inst✝¹ : SMulWithZero 𝕝 E
inst✝ : IsScalarTower 𝕜 𝕝 E
b : 𝕜
hs : Balanced 𝕝 s
a : 𝕝
h : ‖a‖ ≤ ‖b‖
⊢ a • s ⊆ b • s

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedDivisionRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
⊢ balancedCoreAux 𝕜 ∅ = ∅

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedDivisionRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
x : E
hx : x ∈ balancedCoreAux 𝕜 s
⊢ x ∈ s

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
𝕝 : Type u_2
E : Type u_3
inst✝⁸ : NormedDivisionRing 𝕜
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
s : Set E
inst✝⁵ : NormedRing 𝕝
inst✝⁴ : Module 𝕜 𝕝
inst✝³ : NormSMulClass 𝕜 𝕝
inst✝² : SMulWithZero 𝕝 E
inst✝¹ : IsScalarTower 𝕜 𝕝 E
a : 𝕜
x : E
inst✝ : SMulCommClass 𝕝 𝕜 E
hs : Balanced 𝕝 s
b : 𝕝
ha : a • x ∈ s
hba : ‖b‖ ≤ ‖a‖
ha₀ : a ≠ 0
⊢ (a⁻¹ • b) • a • x ∈ s

Tactic:
by

Goal state after: