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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q u x y : E
hu : u ∈ segment 𝕜 x p
az bz : 𝕜
haz : 0 ≤ az
hbz : 0 ≤ bz
habz : az + bz = 1
haz' : 0 < az
bv : 𝕜
hbv : 0 ≤ bv
hav : 0 ≤ 0
habv : bv = 1
⊢ q ∈ {p, q, az • x + bz • y}

Tactic:
by

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

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y p₂ : P
h : p₂ ∈ s
⊢ s.WSameSide x y ↔ y ∈ s ∨ ∃ p₁ ∈ s, SameRay R (x -ᵥ p₁) (y -ᵥ p₂)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q x y : E
az bz : 𝕜
haz : 0 ≤ az
hbz : 0 ≤ bz
habz : az + bz = 1
haz' : 0 < az
av bv : 𝕜
hav : 0 ≤ av
hbv : 0 ≤ bv
habv : av + bv = 1
hav' : 0 < av
au bu : 𝕜
hau : 0 ≤ au
hbu : 0 ≤ bu
habu : au + bu = 1
⊢ 0 < az * av + bz * au

Tactic:
by

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

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y p₁ : P
h : p₁ ∈ s
⊢ s.SSameSide x y ↔ x ∉ s ∧ y ∉ s ∧ ∃ p₂ ∈ s, SameRay R (x -ᵥ p₁) (y -ᵥ p₂)

Tactic:
by

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : NormedSpace ℝ V
inst✝² : StrictConvexSpace ℝ V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
p p₁ p₂ p₃ : P
r : ℝ
h : Collinear ℝ {p₁, p₂, p₃}
hp₁ : dist p₁ p = r
hp₂ : dist p₂ p ≤ r
hp₃ : dist p₃ p = r
hp₁p₃ : p₁ ≠ p₃
⊢ Wbtw ℝ p₁ p₂ p₃

Tactic:
by

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

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y p₂ : P
h : p₂ ∈ s
⊢ s.SSameSide x y ↔ x ∉ s ∧ y ∉ s ∧ ∃ p₁ ∈ s, SameRay R (x -ᵥ p₁) (y -ᵥ p₂)

Tactic:
by

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

Goal state before:
pp: R : Type u_1
V : Type u_2
ι : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : TopologicalSpace V
inst✝¹ : AddCommGroup V
inst✝ : Module R V
s : Set V
w : ι → R
z : ι → V
hs : StrictConvex R s
i : ι
t : Finset ι
hi✝ : i ∉ t
ht :
  (∀ i ∈ t, 0 ≤ w i) →
    ∀ (i j : ι), i ∈ t → j ∈ t → z i ≠ z j → w i ≠ 0 → w j ≠ 0 → (∀ i ∈ t, z i ∈ s) → t.centerMass w z ∈ interior s
h₀ : ∀ i_1 ∈ insert i t, 0 ≤ w i_1
i' j' : ι
hi' : i' ∈ insert i t
hj' : j' ∈ insert i t
hi'j' : z i' ≠ z j'
hi'0 : w i' ≠ 0
hj'0 : w j' ≠ 0
hmem : ∀ i_1 ∈ insert i t, z i_1 ∈ s
zi : z i ∈ s
hs₀ : ∀ j ∈ t, 0 ≤ w j
hsum_t : ¬∑ j ∈ t, w j = 0
hi : ¬w i = 0
hzi : z i = t.centerMass w z
⊢ 0 < w i

Tactic:
by

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : NormedSpace ℝ V
inst✝² : StrictConvexSpace ℝ V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
p p₁ p₂ p₃ : P
r : ℝ
h : Collinear ℝ {p₁, p₂, p₃}
hp₁ : dist p₁ p = r
hp₂ : dist p₂ p < r
hp₃ : dist p₃ p = r
hp₁p₃ : p₁ ≠ p₃
⊢ Sbtw ℝ p₁ p₂ p₃

Tactic:
by

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : NormedSpace ℝ V
inst✝² : StrictConvexSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c : P
⊢ dist (a -ᵥ a) (b -ᵥ a) + dist (b -ᵥ a) (c -ᵥ a) = dist (a -ᵥ a) (c -ᵥ a) ↔ b -ᵥ a ∈ [a -ᵥ a -[ℝ] c -ᵥ a]

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q x y : E
az bz : 𝕜
haz : 0 ≤ az
hbz : 0 ≤ bz
habz : az + bz = 1
haz' : 0 < az
av bv : 𝕜
hav : 0 ≤ av
hbv : 0 ≤ bv
habv : av + bv = 1
hav' : 0 < av
au bu : 𝕜
hau : 0 ≤ au
hbu : 0 ≤ bu
habu : au + bu = 1
hab : 0 < az * av + bz * au
w : Fin 3 → 𝕜 := ![az * av * bu, bz * au * bv, au * av]
z : Fin 3 → E := ![p, q, az • x + bz • y]
⊢ ∀ (i : Fin 3), 0 ≤ w i

Tactic:
by

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

Goal state before:
pp: R : Type u_1
V : Type u_2
ι : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : TopologicalSpace V
inst✝¹ : AddCommGroup V
inst✝ : Module R V
s : Set V
w : ι → R
z : ι → V
hs : StrictConvex R s
i : ι
t : Finset ι
hi✝ : i ∉ t
ht :
  (∀ i ∈ t, 0 ≤ w i) →
    ∀ (i j : ι), i ∈ t → j ∈ t → z i ≠ z j → w i ≠ 0 → w j ≠ 0 → (∀ i ∈ t, z i ∈ s) → t.centerMass w z ∈ interior s
h₀ : ∀ i_1 ∈ insert i t, 0 ≤ w i_1
i' j' : ι
hi' : i' ∈ insert i t
hj' : j' ∈ insert i t
hi'j' : z i' ≠ z j'
hi'0 : w i' ≠ 0
hj'0 : w j' ≠ 0
hmem : ∀ i_1 ∈ insert i t, z i_1 ∈ s
zi : z i ∈ s
hs₀ : ∀ j ∈ t, 0 ≤ w j
hsum_t : ¬∑ j ∈ t, w j = 0
hi : ¬w i = 0
hzi : z i = t.centerMass w z
⊢ w i + ∑ j ∈ t, w j ≠ 0

Tactic:
by

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : NormedSpace ℝ V
inst✝² : StrictConvexSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c : P
⊢ dist a b + dist b c = dist a c ↔ Wbtw ℝ a b c

Tactic:
by

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : NormedSpace ℝ V
inst✝² : StrictConvexSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c : P
⊢ dist a c < dist a b + dist b c ↔ ¬Wbtw ℝ a b c

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q x y : E
az bz : 𝕜
haz : 0 ≤ az
hbz : 0 ≤ bz
habz : az + bz = 1
haz' : 0 < az
av bv : 𝕜
hav : 0 ≤ av
hbv : 0 ≤ bv
habv : av + bv = 1
hav' : 0 < av
au bu : 𝕜
hau : 0 ≤ au
hbu : 0 ≤ bu
habu : au + bu = 1
hab : 0 < az * av + bz * au
w : Fin 3 → 𝕜 := ![az * av * bu, bz * au * bv, au * av]
z : Fin 3 → E := ![p, q, az • x + bz • y]
hw₀ : ∀ (i : Fin 3), 0 ≤ w i
⊢ ∑ i, w i = az * av + bz * au

Tactic:
by

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

Goal state before:
pp: E : Type u_3
PE : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : StrictConvexSpace ℝ E
inst✝¹ : MetricSpace PE
inst✝ : NormedAddTorsor E PE
r : ℝ
x y z : PE
hxy : dist x y = r * dist x z
hyz : dist y z = (1 - r) * dist x z
⊢ y -ᵥ x ∈ [0 -[ℝ] z -ᵥ x]

Tactic:
by

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

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y p₁ : P
h : p₁ ∈ s
⊢ s.WOppSide x y ↔ x ∈ s ∨ ∃ p₂ ∈ s, SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)

Tactic:
by

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

Goal state before:
pp: E : Type u_3
PE : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : StrictConvexSpace ℝ E
inst✝¹ : MetricSpace PE
inst✝ : NormedAddTorsor E PE
r : ℝ
x y : PE
hxy : dist x y = r * dist x x
hyz : dist y x = (1 - r) * dist x x
this : y -ᵥ x ∈ [0 -[ℝ] x -ᵥ x]
⊢ y = x

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q x y : E
az bz : 𝕜
haz : 0 ≤ az
hbz : 0 ≤ bz
habz : az + bz = 1
haz' : 0 < az
av bv : 𝕜
hav : 0 ≤ av
hbv : 0 ≤ bv
habv : av + bv = 1
hav' : 0 < av
au bu : 𝕜
hau : 0 ≤ au
hbu : 0 ≤ bu
habu : au + bu = 1
hab : 0 < az * av + bz * au
w : Fin 3 → 𝕜 := ![az * av * bu, bz * au * bv, au * av]
z : Fin 3 → E := ![p, q, az • x + bz • y]
hw₀ : ∀ (i : Fin 3), 0 ≤ w i
hw : ∑ i, w i = az * av + bz * au
i : Fin 3
⊢ z i ∈ {p, q, az • x + bz • y}

Tactic:
by

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

Goal state before:
pp: R : Type u_1
V : Type u_2
ι : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : TopologicalSpace V
inst✝¹ : AddCommGroup V
inst✝ : Module R V
s : Set V
w : ι → R
z : ι → V
hs : StrictConvex R s
i : ι
t : Finset ι
hi✝ : i ∉ t
ht :
  (∀ i ∈ t, 0 ≤ w i) →
    ∀ (i j : ι), i ∈ t → j ∈ t → z i ≠ z j → w i ≠ 0 → w j ≠ 0 → (∀ i ∈ t, z i ∈ s) → t.centerMass w z ∈ interior s
h₀ : ∀ i_1 ∈ insert i t, 0 ≤ w i_1
i' j' : ι
hi' : i' ∈ insert i t
hj' : j' ∈ insert i t
hi'j' : z i' ≠ z j'
hi'0 : w i' ≠ 0
hj'0 : w j' ≠ 0
hmem : ∀ i_1 ∈ insert i t, z i_1 ∈ s
zi : z i ∈ s
hs₀ : ∀ j ∈ t, 0 ≤ w j
hsum_t : ¬∑ j ∈ t, w j = 0
hi : ¬w i = 0
hzi : z i = t.centerMass w z
hwi : w i + ∑ j ∈ t, w j ≠ 0
hijt : ∀ (i'' j'' : ι), i'' ∈ t → j'' ∈ t → z i'' ≠ z j'' → w i'' ≠ 0 → w j'' = 0
⊢ ∃ i'' ∈ t, w i'' ≠ 0

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q x y : E
az bz : 𝕜
haz : 0 ≤ az
hbz : 0 ≤ bz
habz : az + bz = 1
haz' : 0 < az
av bv : 𝕜
hav : 0 ≤ av
hbv : 0 ≤ bv
habv : av + bv = 1
hav' : 0 < av
au bu : 𝕜
hau : 0 ≤ au
hbu : 0 ≤ bu
habu : au + bu = 1
hab : 0 < az * av + bz * au
w : Fin 3 → 𝕜 := ![az * av * bu, bz * au * bv, au * av]
z : Fin 3 → E := ![p, q, az • x + bz • y]
hw₀ : ∀ (i : Fin 3), 0 ≤ w i
hw : ∑ i, w i = az * av + bz * au
hz : ∀ (i : Fin 3), z i ∈ {p, q, az • x + bz • y}
⊢ 0 < ∑ i, w i

Tactic:
by

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

Goal state before:
pp: R : Type u_1
V : Type u_2
ι : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : TopologicalSpace V
inst✝¹ : AddCommGroup V
inst✝ : Module R V
s : Set V
w : ι → R
z : ι → V
hs : StrictConvex R s
i : ι
t : Finset ι
hi✝ : i ∉ t
ht :
  (∀ i ∈ t, 0 ≤ w i) →
    ∀ (i j : ι), i ∈ t → j ∈ t → z i ≠ z j → w i ≠ 0 → w j ≠ 0 → (∀ i ∈ t, z i ∈ s) → t.centerMass w z ∈ interior s
h₀ : ∀ i_1 ∈ insert i t, 0 ≤ w i_1
i' j' : ι
hi' : i' ∈ insert i t
hj' : j' ∈ insert i t
hi'j' : z i' ≠ z j'
hi'0 : w i' ≠ 0
hj'0 : w j' ≠ 0
hmem : ∀ i_1 ∈ insert i t, z i_1 ∈ s
zi : z i ∈ s
hs₀ : ∀ j ∈ t, 0 ≤ w j
hsum_t : ¬∑ j ∈ t, w j = 0
hi : ¬w i = 0
hzi : z i = t.centerMass w z
hwi : w i + ∑ j ∈ t, w j ≠ 0
hijt : ∀ (i'' j'' : ι), i'' ∈ t → j'' ∈ t → z i'' ≠ z j'' → w i'' ≠ 0 → w j'' = 0
i'' : ι
hi'' : i'' ∈ t
hwi'' : w i'' ≠ 0
⊢ ∀ j'' ∈ t, w j'' ≠ 0 → z j'' = Function.const ι (z i'') j''

Tactic:
by

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

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y p₂ : P
h : p₂ ∈ s
⊢ s.WOppSide x y ↔ y ∈ s ∨ ∃ p₁ ∈ s, SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)

Tactic:
by

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

Goal state before:
pp: R : Type u_1
V : Type u_2
ι : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : TopologicalSpace V
inst✝¹ : AddCommGroup V
inst✝ : Module R V
s : Set V
w : ι → R
z : ι → V
hs : StrictConvex R s
i : ι
t : Finset ι
hi✝ : i ∉ t
ht :
  (∀ i ∈ t, 0 ≤ w i) →
    ∀ (i j : ι), i ∈ t → j ∈ t → z i ≠ z j → w i ≠ 0 → w j ≠ 0 → (∀ i ∈ t, z i ∈ s) → t.centerMass w z ∈ interior s
h₀ : ∀ i_1 ∈ insert i t, 0 ≤ w i_1
i' j' : ι
hi' : i' ∈ insert i t
hj' : j' ∈ insert i t
hi'j' : z i' ≠ z j'
hi'0 : w i' ≠ 0
hj'0 : w j' ≠ 0
hmem : ∀ i_1 ∈ insert i t, z i_1 ∈ s
zi : z i ∈ s
hs₀ : ∀ j ∈ t, 0 ≤ w j
hsum_t : ¬∑ j ∈ t, w j = 0
hi : ¬w i = 0
hzi : z i = t.centerMass w z
hwi : w i + ∑ j ∈ t, w j ≠ 0
hijt : ∀ (i'' j'' : ι), i'' ∈ t → j'' ∈ t → z i'' ≠ z j'' → w i'' ≠ 0 → w j'' = 0
i'' : ι
hi'' : i'' ∈ t
hwi'' : w i'' ≠ 0
hijt' : ∀ j'' ∈ t, w j'' ≠ 0 → z j'' = Function.const ι (z i'') j''
⊢ i = i' ∨ i = j'

Tactic:
by

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

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y p₁ : P
h : p₁ ∈ s
⊢ s.SOppSide x y ↔ x ∉ s ∧ y ∉ s ∧ ∃ p₂ ∈ s, SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)

Tactic:
by

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

Goal state before:
pp: E : Type u_3
PE : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : StrictConvexSpace ℝ E
inst✝¹ : MetricSpace PE
inst✝ : NormedAddTorsor E PE
r : ℝ
x y z : PE
hxy : dist x y = r * dist x z
hyz : dist y z = (1 - r) * dist x z
⊢ y = (AffineMap.lineMap x z) r

Tactic:
by

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

Goal state before:
pp: R : Type u_1
V : Type u_2
ι : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : TopologicalSpace V
inst✝¹ : AddCommGroup V
inst✝ : Module R V
s : Set V
w : ι → R
z : ι → V
hs : StrictConvex R s
i : ι
t : Finset ι
hi✝¹ : i ∉ t
ht :
  (∀ i ∈ t, 0 ≤ w i) →
    ∀ (i j : ι), i ∈ t → j ∈ t → z i ≠ z j → w i ≠ 0 → w j ≠ 0 → (∀ i ∈ t, z i ∈ s) → t.centerMass w z ∈ interior s
h₀ : ∀ i_1 ∈ insert i t, 0 ≤ w i_1
i' j' : ι
hi' : i' ∈ insert i t
hj' : j' ∈ insert i t
hi'j' : z i' ≠ z j'
hi'0 : w i' ≠ 0
hj'0 : w j' ≠ 0
hmem : ∀ i_1 ∈ insert i t, z i_1 ∈ s
zi : z i ∈ s
hs₀ : ∀ j ∈ t, 0 ≤ w j
hsum_t : ¬∑ j ∈ t, w j = 0
hi✝ : ¬w i = 0
hzi : z i = t.centerMass w z
hwi : w i + ∑ j ∈ t, w j ≠ 0
hijt : ∀ (i'' j'' : ι), i'' ∈ t → j'' ∈ t → z i'' ≠ z j'' → w i'' ≠ 0 → w j'' = 0
i'' : ι
hi'' : i'' ∈ t
hwi'' : w i'' ≠ 0
hijt' : ∀ j'' ∈ t, w j'' ≠ 0 → z j'' = Function.const ι (z i'') j''
hi : i = i' ∨ i = j'
⊢ t.centerMass w z = z i''

Tactic:
by

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

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y p₂ : P
h : p₂ ∈ s
⊢ s.SOppSide x y ↔ x ∉ s ∧ y ∉ s ∧ ∃ p₁ ∈ s, SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q u v x y z : E
hz : z ∈ segment 𝕜 x y
hu : u ∈ segment 𝕜 x p
hv : v ∈ segment 𝕜 y q
⊢ ¬Disjoint (segment 𝕜 u v) ((convexHull 𝕜) {p, q, z})

Tactic:
by

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

Goal state before:
pp: E : Type u_3
PE : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : StrictConvexSpace ℝ E
inst✝¹ : MetricSpace PE
inst✝ : NormedAddTorsor E PE
x y z : PE
hx : dist x y = dist x z / 2
hy : dist y z = dist x z / 2
⊢ y = midpoint ℝ x z

Tactic:
by

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

Goal state before:
pp: R : Type u_1
V : Type u_2
ι : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : TopologicalSpace V
inst✝¹ : AddCommGroup V
inst✝ : Module R V
s : Set V
w : ι → R
z : ι → V
hs : StrictConvex R s
i : ι
t : Finset ι
hi✝ : i ∉ t
ht :
  (∀ i ∈ t, 0 ≤ w i) →
    ∀ (i j : ι), i ∈ t → j ∈ t → z i ≠ z j → w i ≠ 0 → w j ≠ 0 → (∀ i ∈ t, z i ∈ s) → t.centerMass w z ∈ interior s
h₀ : ∀ i_1 ∈ insert i t, 0 ≤ w i_1
i' j' : ι
hi' : i' ∈ insert i t
hj' : j' ∈ insert i t
hi'j' : z i' ≠ z j'
hi'0 : w i' ≠ 0
hj'0 : w j' ≠ 0
hmem : ∀ i_1 ∈ insert i t, z i_1 ∈ s
zi : z i ∈ s
hs₀ : ∀ j ∈ t, 0 ≤ w j
hsum_t : ¬∑ j ∈ t, w j = 0
hi : ¬w i = 0
hzi : ¬z i = t.centerMass w z
⊢ 0 < w i

Tactic:
by

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

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y z : P
hxy : s.WSameSide x y
hyz : s.WSameSide y z
hy : y ∉ s
⊢ s.WSameSide x z

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
hs : Convex 𝕜 s
ht : Convex 𝕜 t
hst : Disjoint s t
S : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}
C : Set E
hsC : s ⊆ C
hmax : Maximal (fun x ↦ x ∈ S) C
hC : Convex 𝕜 C ∧ Disjoint C t
x : E
hx : x ∈ Cᶜ
y : E
hy : y ∈ Cᶜ
z : E
hz : z ∈ segment 𝕜 x y
hzC : z ∈ C
h : ∀ c ∈ Cᶜ, ∃ a ∈ C, (segment 𝕜 c a ∩ t).Nonempty
⊢ False

Tactic:
by

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

Goal state before:
pp: R : Type u_1
V : Type u_2
ι : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : TopologicalSpace V
inst✝¹ : AddCommGroup V
inst✝ : Module R V
s : Set V
t : Finset ι
w : ι → R
z : ι → V
hs : StrictConvex R s
⊢ (∀ i ∈ t, 0 ≤ w i) →
    ∀ (i j : ι), i ∈ t → j ∈ t → z i ≠ z j → w i ≠ 0 → w j ≠ 0 → (∀ i ∈ t, z i ∈ s) → t.centerMass w z ∈ interior s

Tactic:
by

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

Goal state before:
pp: case hy
V : Type u_1
P : Type u_2
inst✝¹¹ : NormedAddCommGroup V
inst✝¹⁰ : NormedSpace ℝ V
inst✝⁹ : StrictConvexSpace ℝ V
E : Type u_3
F : Type u_4
PE : Type u_5
PF : Type u_6
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : StrictConvexSpace ℝ E
inst✝³ : MetricSpace PE
inst✝² : MetricSpace PF
inst✝¹ : NormedAddTorsor E PE
inst✝ : NormedAddTorsor F PF
r : ℝ
f : PF → PE
x✝ y✝ z : PE
hi : Isometry f
x y : PF
⊢ dist (midpoint ℝ x y) y = dist x y / 2

Tactic:
simp only [dist_midpoint_right, Real.norm_of_nonneg zero_le_two, div_eq_inv_mul]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
hs : Convex 𝕜 s
ht : Convex 𝕜 t
hst : Disjoint s t
S : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}
C : Set E
hsC : s ⊆ C
hmax : Maximal (fun x ↦ x ∈ S) C
hC : Convex 𝕜 C ∧ Disjoint C t
x : E
hx : x ∈ Cᶜ
y : E
hy : y ∈ Cᶜ
z : E
hz : z ∈ segment 𝕜 x y
hzC : z ∈ C
c : E
hc : c ∈ Cᶜ
h✝ : ∀ a ∈ C, segment 𝕜 c a ∩ t = ∅
h : Disjoint ((convexHull 𝕜) (insert c C)) t
⊢ False

Tactic:
by

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

Goal state before:
pp: R : Type u_1
V : Type u_2
ι : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : TopologicalSpace V
inst✝¹ : AddCommGroup V
inst✝ : Module R V
s : Set V
t : Finset ι
w : ι → R
z : ι → V
hs : StrictConvex R s
h0 : ∀ i ∈ t, 0 ≤ w i
h1 : ∑ i ∈ t, w i = 1
i j : ι
hi : i ∈ t
hj : j ∈ t
hij : z i ≠ z j
hi0 : w i ≠ 0
hj0 : w j ≠ 0
hz : ∀ i ∈ t, z i ∈ s
⊢ ∑ k ∈ t, w k • z k ∈ interior s

Tactic:
by

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

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y z : P
hxy : s.WSameSide x y
hyz : s.WOppSide y z
hy : y ∉ s
⊢ s.WOppSide x z

Tactic:
by

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝¹¹ : NormedAddCommGroup V
inst✝¹⁰ : NormedSpace ℝ V
inst✝⁹ : StrictConvexSpace ℝ V
E : Type u_3
F : Type u_4
PE : Type u_5
PF : Type u_6
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : StrictConvexSpace ℝ E
inst✝³ : MetricSpace PE
inst✝² : MetricSpace PF
inst✝¹ : NormedAddTorsor E PE
inst✝ : NormedAddTorsor F PF
r : ℝ
f : PF → PE
x✝ y✝ z : PE
hi : Isometry f
x y : PF
⊢ f (midpoint ℝ x y) = midpoint ℝ (f x) (f y)

Tactic:
by

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝¹¹ : NormedAddCommGroup V
inst✝¹⁰ : NormedSpace ℝ V
inst✝⁹ : StrictConvexSpace ℝ V
E : Type u_3
F : Type u_4
PE : Type u_5
PF : Type u_6
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : StrictConvexSpace ℝ E
inst✝³ : MetricSpace PE
inst✝² : MetricSpace PF
inst✝¹ : NormedAddTorsor E PE
inst✝ : NormedAddTorsor F PF
r : ℝ
f : PF → PE
x✝ y z : PE
hi : Isometry f
x : F
⊢ ‖(AffineMap.ofMapMidpoint f ⋯ ⋯).linear x‖ = ‖x‖

Tactic:
by

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

Goal state before:
pp: V : Type u_2
ι : Type u_4
inst✝² : NormedAddCommGroup V
inst✝¹ : NormedSpace ℝ V
inst✝ : StrictConvexSpace ℝ V
t : Finset ι
w : ι → ℝ
p : V
r : ℝ
z : ι → V
h0 : ∀ i ∈ t, 0 ≤ w i
i j : ι
hi : i ∈ t
hj : j ∈ t
hij : z i ≠ z j
hi0 : w i ≠ 0
hj0 : w j ≠ 0
hz : ∀ i ∈ t, z i ∈ closedBall p r
⊢ t.centerMass w z ∈ ball p r

Tactic:
by

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

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y z : P
hxy : s.WOppSide x y
hyz : s.WOppSide y z
hy : y ∉ s
⊢ s.WSameSide x z

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
hs : Convex 𝕜 s
ht : Convex 𝕜 t
hst : Disjoint s t
⊢ ∃ C, Convex 𝕜 C ∧ Convex 𝕜 Cᶜ ∧ s ⊆ C ∧ t ⊆ Cᶜ

Tactic:
by

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

Goal state before:
pp: V : Type u_2
ι : Type u_4
inst✝² : NormedAddCommGroup V
inst✝¹ : NormedSpace ℝ V
inst✝ : StrictConvexSpace ℝ V
t : Finset ι
w : ι → ℝ
p : V
r : ℝ
z : ι → V
h0 : ∀ i ∈ t, 0 ≤ w i
h1 : ∑ i ∈ t, w i = 1
i j : ι
hi : i ∈ t
hj : j ∈ t
hij : z i ≠ z j
hi0 : w i ≠ 0
hj0 : w j ≠ 0
hz : ∀ i ∈ t, z i ∈ closedBall p r
⊢ ∑ k ∈ t, w k • z k ∈ ball p r

Tactic:
by

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

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y : P
⊢ s.WSameSide x y ∧ s.WOppSide x y ↔ x ∈ s ∨ y ∈ s

Tactic:
by

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

Goal state before:
pp: V : Type u_2
ι : Type u_4
inst✝² : NormedAddCommGroup V
inst✝¹ : NormedSpace ℝ V
inst✝ : StrictConvexSpace ℝ V
t : Finset ι
w : ι → ℝ
r : ℝ
z : ι → V
h0 : ∀ i ∈ t, 0 ≤ w i
h1 : ∑ i ∈ t, w i = 1
i j : ι
hi : i ∈ t
hj : j ∈ t
hij : z i ≠ z j
hi0 : w i ≠ 0
hj0 : w j ≠ 0
hz : ∀ i ∈ t, ‖z i‖ ≤ r
⊢ ‖∑ k ∈ t, w k • z k‖ < r

Tactic:
simp_rw [← mem_closedBall_zero_iff] at hz
  rw [← mem_ball_zero_iff]
  exact sum_mem_ball_of_strictConvexSpace h0 h1 hi hj hij hi0 hj0 hz

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

Goal state before:
pp: V : Type u_2
ι : Type u_4
inst✝² : NormedAddCommGroup V
inst✝¹ : NormedSpace ℝ V
inst✝ : StrictConvexSpace ℝ V
t : Finset ι
w : ι → ℝ
r : ℝ
z : ι → V
h0 : ∀ i ∈ t, 0 ≤ w i
h1 : ∑ i ∈ t, w i = 1
i j : ι
hi : i ∈ t
hj : j ∈ t
hij : z i ≠ z j
hi0 : w i ≠ 0
hj0 : w j ≠ 0
hz : ∀ i ∈ t, ‖z i‖ ≤ r
⊢ ‖∑ k ∈ t, w k • z k‖ < r

Tactic:
by

Goal state after: