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

Goal state before:
pp: F : Type u_1
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : TopologicalSpace F
s t : Set F
hs : AmpleSet s
ht : AmpleSet t
x : F
hx : x ∈ s ∪ t
⊢ (convexHull ℝ) (connectedComponentIn (s ∪ t) x) = univ

Tactic:
rcases hx with (h | h) <;>
  -- The connected component of `x ∈ s` in `s ∪ t` contains the connected component of `x` in `s`,
  -- hence is also full; similarly for `t`.
  [have hx := hs x h; have hx := ht x h]

Goal state after:
case inl
F : Type u_1
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : TopologicalSpace F
s t : Set F
hs : AmpleSet s
ht : AmpleSet t
x : F
h : x ∈ s
hx : (convexHull ℝ) (connectedComponentIn s x) = univ
⊢ (convexHull ℝ) (connectedComponentIn (s ∪ t) x) = univ

case inr
F : Type u_1
inst✝² : AddCommGroup F
inst✝¹ : Module ℝ F
inst✝ : TopologicalSpace F
s t : Set F
hs : AmpleSet s
ht : AmpleSet t
x : F
h : x ∈ t
hx : (convexHull ℝ) (connectedComponentIn t x) = univ
⊢ (convexHull ℝ) (connectedComponentIn (s ∪ t) x) = univ
Module: Mathlib.Data.List.Triplewise

Goal state before:
pp: case nil
α : Type u_1
l : List α
p q : α → α → α → Prop
h : ∀ {a b c : α}, p a b c → q a b c
⊢ Triplewise q []

Tactic:
| nil =>

Goal state after:
Module: Mathlib.Data.List.Triplewise

Goal state before:
pp: case nil
α : Type u_1
β : Type u_2
l : List α
f : α → β
p' : β → β → β → Prop
⊢ Triplewise p' (map f []) ↔ Triplewise (fun a b c ↦ p' (f a) (f b) (f c)) []

Tactic:
| nil =>

Goal state after:
Module: Mathlib.Data.List.Triplewise

Goal state before:
pp: case nil
α : Type u_1
l : List α
p : α → α → α → Prop
⊢ Triplewise p [] ↔ ∀ (i j k : Nat) (hij : i < j) (hjk : j < k) (hk : k < [].length), p [][i] [][j] [][k]

Tactic:
| nil =>

Goal state after:
Module: Mathlib.Data.List.Triplewise

Goal state before:
pp: case nil
α : Type u_1
l : List α
p : α → α → α → Prop
⊢ Triplewise p [].reverse ↔ Triplewise (fun a b c ↦ p c b a) []

Tactic:
| nil =>

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

Goal state before:
pp: case nil
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : IsOrderedRing R
⊢ List.Wbtw R [] ∧ Pairwise (fun x1 x2 ↦ x1 ≠ x2) [] ↔ Triplewise (Sbtw R) [] ∧ ∀ (a : P), [] ≠ [a, a]

Tactic:
| nil =>

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

Goal state before:
pp: case cons.refine_1.refine_1.nil
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : IsOrderedRing R
head : P
hpne : Pairwise (fun x1 x2 ↦ x1 ≠ x2) [head]
hp : Pairwise (Wbtw R head) []
ht : List.Wbtw R []
⊢ Pairwise (Sbtw R head) []

Tactic:
| nil =>

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

Goal state before:
pp: case cons.refine_2.refine_1.nil
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : IsOrderedRing R
head : P
x✝ : (Pairwise (Sbtw R head) [] ∧ Triplewise (Sbtw R) []) ∧ ∀ (a : P), [head] ≠ [a, a]
hp : Pairwise (Sbtw R head) []
ht : Triplewise (Sbtw R) []
ha : ∀ (a : P), [head] ≠ [a, a]
ht' : List.Wbtw R []
ih : Pairwise (fun x1 x2 ↦ x1 ≠ x2) [] ↔ ∀ (a : P), [] ≠ [a, a]
ha' : head ∈ []
⊢ False

Tactic:
| nil =>

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

Goal state before:
pp: case cons.refine_2.refine_1.cons.inl.nil
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : IsOrderedRing R
head : P
x✝ : (Pairwise (Sbtw R head) [head] ∧ Triplewise (Sbtw R) [head]) ∧ ∀ (a : P), [head, head] ≠ [a, a]
hp : (∀ a' ∈ [], Sbtw R head head a') ∧ Pairwise (Sbtw R head) []
ht : Triplewise (Sbtw R) [head]
ha : ∀ (a : P), [head, head] ≠ [a, a]
ht' : List.Wbtw R [head]
ih : Pairwise (fun x1 x2 ↦ x1 ≠ x2) [head] ↔ ∀ (a : P), [head] ≠ [a, a]
ha' : head ∈ [head]
⊢ False

Tactic:
| nil =>

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

Goal state before:
pp: case refine_2.nil
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
hl : [] ≠ []
h : List.Wbtw R []
⊢ ∃ l', (∀ a ∈ l', 0 ≤ a) ∧ l'.SortedLE ∧ map (⇑(lineMap ([].head hl) ([].getLast hl))) l' = []

Tactic:
| nil =>

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

Goal state before:
pp: case neg.nil
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
head : P
l'' : List R
hl''0 : ∀ a ∈ l'', 0 ≤ a
hl''s : l''.SortedLE
hl : [head] ≠ []
h : Pairwise (Wbtw R head) [] ∧ List.Wbtw R []
ht : ¬[] = []
hl'' : map (⇑(lineMap ([].head ht) ([].getLast ht))) l'' = []
⊢ ∃ l', (∀ a ∈ l', 0 ≤ a) ∧ l'.SortedLE ∧ map (⇑(lineMap head ([].getLast ht))) l' = [head]

Tactic:
| nil =>

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

Goal state before:
pp: case refine_2.refine_2.nil
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
l' : List R
hl's : l'.SortedLE
hl : [] ≠ []
hp : Pairwise (fun x1 x2 ↦ x1 ≠ x2) []
hl'l : map (⇑(lineMap ([].head hl) ([].getLast hl))) l' = []
⊢ [].length = 1 ∨ [].head hl ≠ [].getLast hl

Tactic:
| nil =>

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

Goal state before:
pp: case refine_2.refine_2.cons.nil
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
l' : List R
hl's : l'.SortedLE
head : P
hl : [head] ≠ []
hp : Pairwise (fun x1 x2 ↦ x1 ≠ x2) [head]
hl'l : map (⇑(lineMap ([head].head hl) ([head].getLast hl))) l' = [head]
⊢ [] = [] ∨ head ≠ [head].getLast hl

Tactic:
| nil =>

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

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : IsOrderedRing R
w x y z : P
h₁ : Wbtw R w x z
h₂ : Wbtw R x y z
⊢ Wbtw R w y z

Tactic:
rw [wbtw_comm] at *

Goal state after:
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : IsOrderedRing R
w x y z : P
h₁ : Wbtw R z x w
h₂ : Wbtw R z y x
⊢ Wbtw R z y w
Module: Mathlib.Analysis.Convex.Between

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁷ : Ring R
inst✝⁶ : PartialOrder R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : AddTorsor V P
inst✝² : IsOrderedRing R
inst✝¹ : IsDomain R
inst✝ : IsTorsionFree R V
w x y z : P
h₁ : Wbtw R w x z
h₂ : Sbtw R x y z
⊢ Sbtw R w y z

Tactic:
rw [wbtw_comm] at *

Goal state after:
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁷ : Ring R
inst✝⁶ : PartialOrder R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : AddTorsor V P
inst✝² : IsOrderedRing R
inst✝¹ : IsDomain R
inst✝ : IsTorsionFree R V
w x y z : P
h₁ : Wbtw R z x w
h₂ : Sbtw R x y z
⊢ Sbtw R w y z
Module: Mathlib.Analysis.Convex.Between

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AffineSpace V P
inst✝ : IsOrderedRing R
s : Simplex R P 1
w : Fin (1 + 1) → R
hw : ∑ i, w i = 1
h01 : ∀ (i : Fin (1 + 1)), w i ∈ Set.Icc 0 1
⊢ w = Finset.affineCombinationLineMapWeights 0 1 (w 1)

Tactic:
rw [Fin.sum_univ_two] at hw

Goal state after:
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AffineSpace V P
inst✝ : IsOrderedRing R
s : Simplex R P 1
w : Fin (1 + 1) → R
hw : w 0 + w 1 = 1
h01 : ∀ (i : Fin (1 + 1)), w i ∈ Set.Icc 0 1
⊢ w = Finset.affineCombinationLineMapWeights 0 1 (w 1)
Module: Mathlib.CategoryTheory.CofilteredSystem

Goal state before:
pp: case inr
J : Type u
inst✝¹ : Category.{v_1, u} J
F : J ⥤ Type v
i : J
s : Set (F.obj i)
inst✝ : IsCofilteredOrEmpty J
hFn : ∀ (j : J), Nonempty (F.obj j)
Fsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective ⇑(ConcreteCategory.hom (F.map f))
j : J
ji : j ⟶ i
y : F.obj i
ys : y ∈ s
⊢ ∃ x, ∀ (i_1 : j ⟶ i), (ConcreteCategory.hom (F.map i_1)) x ∈ s

Tactic:
obtain ⟨x, rfl⟩ := Fsur ji y

Goal state after:
case inr
J : Type u
inst✝¹ : Category.{v_1, u} J
F : J ⥤ Type v
i : J
s : Set (F.obj i)
inst✝ : IsCofilteredOrEmpty J
hFn : ∀ (j : J), Nonempty (F.obj j)
Fsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective ⇑(ConcreteCategory.hom (F.map f))
j : J
ji : j ⟶ i
x : F.obj j
ys : (ConcreteCategory.hom (F.map ji)) x ∈ s
⊢ ∃ x, ∀ (i_1 : j ⟶ i), (ConcreteCategory.hom (F.map i_1)) x ∈ s
Module: Mathlib.Analysis.Convex.Between

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AffineSpace V P
inst✝ : IsOrderedRing R
s : Simplex R P 1
w : Fin (1 + 1) → R
hw : ∑ i, w i = 1
h01 : ∀ (i : Fin (1 + 1)), w i ∈ Set.Ioo 0 1
⊢ w = Finset.affineCombinationLineMapWeights 0 1 (w 1)

Tactic:
rw [Fin.sum_univ_two] at hw

Goal state after:
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AffineSpace V P
inst✝ : IsOrderedRing R
s : Simplex R P 1
w : Fin (1 + 1) → R
hw : w 0 + w 1 = 1
h01 : ∀ (i : Fin (1 + 1)), w i ∈ Set.Ioo 0 1
⊢ w = Finset.affineCombinationLineMapWeights 0 1 (w 1)
Module: Mathlib.LinearAlgebra.Matrix.Permutation

Goal state before:
pp: n : Type u_1
R : Type u_2
inst✝² : DecidableEq n
σ : Perm n
inst✝¹ : Fintype n
inst✝ : AddCommMonoidWithOne R
⊢ (Perm.permMatrix R σ).trace = ↑(Function.fixedPoints ⇑σ).ncard

Tactic:
delta trace

Goal state after:
n : Type u_1
R : Type u_2
inst✝² : DecidableEq n
σ : Perm n
inst✝¹ : Fintype n
inst✝ : AddCommMonoidWithOne R
⊢ ∑ i, (Perm.permMatrix R σ).diag i = ↑(Function.fixedPoints ⇑σ).ncard
Module: Mathlib.LinearAlgebra.Matrix.Stochastic

Goal state before:
pp: R : Type u_1
n : Type u_2
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : Semiring R
inst✝¹ : PartialOrder R
inst✝ : IsOrderedRing R
M : Matrix n n R
x : n → R
hM : M ∈ rowStochastic R n
hx : ∀ (i : n), 0 ≤ x i
j k : n
a✝ : k ∈ univ
⊢ 0 ≤ x k * M k j

Tactic:
apply mul_nonneg (hx k)

Goal state after:
R : Type u_1
n : Type u_2
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : Semiring R
inst✝¹ : PartialOrder R
inst✝ : IsOrderedRing R
M : Matrix n n R
x : n → R
hM : M ∈ rowStochastic R n
hx : ∀ (i : n), 0 ≤ x i
j k : n
a✝ : k ∈ univ
⊢ 0 ≤ M k j
Module: Mathlib.LinearAlgebra.Matrix.Stochastic

Goal state before:
pp: R : Type u_1
n : Type u_2
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : Semiring R
inst✝¹ : PartialOrder R
inst✝ : IsOrderedRing R
M : Matrix n n R
x : n → R
hM : M ∈ rowStochastic R n
hx : x ⬝ᵥ 1 = 1
⊢ x ᵥ* M ⬝ᵥ 1 = 1

Tactic:
rw [← dotProduct_mulVec, hM.2, hx]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u
inst✝⁵ : Field 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : DecidableEq E
t : Finset E
f : E → 𝕜
fpos : ∀ y ∈ t, 0 ≤ f y
fsum : ∑ y ∈ t, f y = 1
g : E → 𝕜
gcombo : ∑ e ∈ t, g e • e = 0
gsum : ∑ e ∈ t, g e = 0
gpos : ∃ i ∈ t, 0 < g i
s : Finset E := {z ∈ t | 0 < g z}
i₀ : E
mem : i₀ ∈ s
w : ∀ i ∈ s, f i₀ / g i₀ ≤ f i / g i
hg : 0 < g i₀
hi₀ : i₀ ∈ t
k : E → 𝕜 := fun z ↦ f z - f i₀ / g i₀ * g z
hk : k i₀ = 0
ksum : ∑ e ∈ t.erase i₀, k e = 1
⊢ ∃ y w, (∀ y_1 ∈ t.erase ↑y, 0 ≤ w y_1) ∧ ∑ y ∈ t.erase ↑y, w y = 1 ∧ (t.erase ↑y).centerMass w id = t.centerMass f id

Tactic:
refine ⟨⟨i₀, hi₀⟩, k, ?_, by convert! ksum, ?_⟩

Goal state after:
case refine_1
𝕜 : Type u_1
E : Type u
inst✝⁵ : Field 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : DecidableEq E
t : Finset E
f : E → 𝕜
fpos : ∀ y ∈ t, 0 ≤ f y
fsum : ∑ y ∈ t, f y = 1
g : E → 𝕜
gcombo : ∑ e ∈ t, g e • e = 0
gsum : ∑ e ∈ t, g e = 0
gpos : ∃ i ∈ t, 0 < g i
s : Finset E := {z ∈ t | 0 < g z}
i₀ : E
mem : i₀ ∈ s
w : ∀ i ∈ s, f i₀ / g i₀ ≤ f i / g i
hg : 0 < g i₀
hi₀ : i₀ ∈ t
k : E → 𝕜 := fun z ↦ f z - f i₀ / g i₀ * g z
hk : k i₀ = 0
ksum : ∑ e ∈ t.erase i₀, k e = 1
⊢ ∀ y ∈ t.erase ↑⟨i₀, hi₀⟩, 0 ≤ k y

case refine_2
𝕜 : Type u_1
E : Type u
inst✝⁵ : Field 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : DecidableEq E
t : Finset E
f : E → 𝕜
fpos : ∀ y ∈ t, 0 ≤ f y
fsum : ∑ y ∈ t, f y = 1
g : E → 𝕜
gcombo : ∑ e ∈ t, g e • e = 0
gsum : ∑ e ∈ t, g e = 0
gpos : ∃ i ∈ t, 0 < g i
s : Finset E := {z ∈ t | 0 < g z}
i₀ : E
mem : i₀ ∈ s
w : ∀ i ∈ s, f i₀ / g i₀ ≤ f i / g i
hg : 0 < g i₀
hi₀ : i₀ ∈ t
k : E → 𝕜 := fun z ↦ f z - f i₀ / g i₀ * g z
hk : k i₀ = 0
ksum : ∑ e ∈ t.erase i₀, k e = 1
⊢ (t.erase ↑⟨i₀, hi₀⟩).centerMass k id = t.centerMass f id
Module: Mathlib.Analysis.Convex.Between

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
w x y z : P
h₁ : Wbtw R w x z
h₂ : Wbtw R x y z
⊢ Wbtw R w x y

Tactic:
rw [wbtw_comm] at *

Goal state after:
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
w x y z : P
h₁ : Wbtw R z x w
h₂ : Wbtw R z y x
⊢ Wbtw R y x w
Module: Mathlib.Analysis.Convex.Between

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
x z : P
v : V
hz : ∃ r, z = r • v +ᵥ x
ty : R
h : ∀ (p : P), p = x ∨ p = ty • v +ᵥ x ∨ p = z → ∃ r, p = r • v +ᵥ x
⊢ Wbtw R x (ty • v +ᵥ x) z ∨ Wbtw R (ty • v +ᵥ x) z x ∨ Wbtw R z x (ty • v +ᵥ x)

Tactic:
rcases hz with ⟨tz, rfl⟩

Goal state after:
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
x : P
v : V
ty tz : R
h : ∀ (p : P), p = x ∨ p = ty • v +ᵥ x ∨ p = tz • v +ᵥ x → ∃ r, p = r • v +ᵥ x
⊢ Wbtw R x (ty • v +ᵥ x) (tz • v +ᵥ x) ∨ Wbtw R (ty • v +ᵥ x) (tz • v +ᵥ x) x ∨ Wbtw R (tz • v +ᵥ x) x (ty • v +ᵥ x)
Module: Mathlib.Analysis.Convex.Between

Goal state before:
pp: case inl
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
x : P
v : V
ty tz : R
h : ∀ (p : P), p = x ∨ p = ty • v +ᵥ x ∨ p = tz • v +ᵥ x → ∃ r, p = r • v +ᵥ x
hy0 : ty < 0
⊢ Wbtw R x (ty • v +ᵥ x) (tz • v +ᵥ x) ∨ Wbtw R (ty • v +ᵥ x) (tz • v +ᵥ x) x ∨ Wbtw R (tz • v +ᵥ x) x (ty • v +ᵥ x)

Tactic:
rcases lt_trichotomy tz 0 with (hz0 | rfl | hz0)

Goal state after:
case inl.inl
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
x : P
v : V
ty tz : R
h : ∀ (p : P), p = x ∨ p = ty • v +ᵥ x ∨ p = tz • v +ᵥ x → ∃ r, p = r • v +ᵥ x
hy0 : ty < 0
hz0 : tz < 0
⊢ Wbtw R x (ty • v +ᵥ x) (tz • v +ᵥ x) ∨ Wbtw R (ty • v +ᵥ x) (tz • v +ᵥ x) x ∨ Wbtw R (tz • v +ᵥ x) x (ty • v +ᵥ x)

case inl.inr.inl
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
x : P
v : V
ty : R
hy0 : ty < 0
h : ∀ (p : P), p = x ∨ p = ty • v +ᵥ x ∨ p = 0 • v +ᵥ x → ∃ r, p = r • v +ᵥ x
⊢ Wbtw R x (ty • v +ᵥ x) (0 • v +ᵥ x) ∨ Wbtw R (ty • v +ᵥ x) (0 • v +ᵥ x) x ∨ Wbtw R (0 • v +ᵥ x) x (ty • v +ᵥ x)

case inl.inr.inr
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
x : P
v : V
ty tz : R
h : ∀ (p : P), p = x ∨ p = ty • v +ᵥ x ∨ p = tz • v +ᵥ x → ∃ r, p = r • v +ᵥ x
hy0 : ty < 0
hz0 : 0 < tz
⊢ Wbtw R x (ty • v +ᵥ x) (tz • v +ᵥ x) ∨ Wbtw R (ty • v +ᵥ x) (tz • v +ᵥ x) x ∨ Wbtw R (tz • v +ᵥ x) x (ty • v +ᵥ x)
Module: Mathlib.Analysis.Convex.Between

Goal state before:
pp: case inr.inr
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
x : P
v : V
ty tz : R
h : ∀ (p : P), p = x ∨ p = ty • v +ᵥ x ∨ p = tz • v +ᵥ x → ∃ r, p = r • v +ᵥ x
hy0 : 0 < ty
⊢ Wbtw R x (ty • v +ᵥ x) (tz • v +ᵥ x) ∨ Wbtw R (ty • v +ᵥ x) (tz • v +ᵥ x) x ∨ Wbtw R (tz • v +ᵥ x) x (ty • v +ᵥ x)

Tactic:
rcases lt_trichotomy tz 0 with (hz0 | rfl | hz0)

Goal state after:
case inr.inr.inl
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
x : P
v : V
ty tz : R
h : ∀ (p : P), p = x ∨ p = ty • v +ᵥ x ∨ p = tz • v +ᵥ x → ∃ r, p = r • v +ᵥ x
hy0 : 0 < ty
hz0 : tz < 0
⊢ Wbtw R x (ty • v +ᵥ x) (tz • v +ᵥ x) ∨ Wbtw R (ty • v +ᵥ x) (tz • v +ᵥ x) x ∨ Wbtw R (tz • v +ᵥ x) x (ty • v +ᵥ x)

case inr.inr.inr.inl
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
x : P
v : V
ty : R
hy0 : 0 < ty
h : ∀ (p : P), p = x ∨ p = ty • v +ᵥ x ∨ p = 0 • v +ᵥ x → ∃ r, p = r • v +ᵥ x
⊢ Wbtw R x (ty • v +ᵥ x) (0 • v +ᵥ x) ∨ Wbtw R (ty • v +ᵥ x) (0 • v +ᵥ x) x ∨ Wbtw R (0 • v +ᵥ x) x (ty • v +ᵥ x)

case inr.inr.inr.inr
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
x : P
v : V
ty tz : R
h : ∀ (p : P), p = x ∨ p = ty • v +ᵥ x ∨ p = tz • v +ᵥ x → ∃ r, p = r • v +ᵥ x
hy0 : 0 < ty
hz0 : 0 < tz
⊢ Wbtw R x (ty • v +ᵥ x) (tz • v +ᵥ x) ∨ Wbtw R (ty • v +ᵥ x) (tz • v +ᵥ x) x ∨ Wbtw R (tz • v +ᵥ x) x (ty • v +ᵥ x)
Module: Mathlib.Analysis.Convex.GaugeRescale

Goal state before:
pp: E : Type u_1
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℝ E
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : T1Space E
s t : Set E
hsc : Convex ℝ s
hsne : (interior s).Nonempty
hsb : IsVonNBounded ℝ s
hst : Convex ℝ t
htne : (interior t).Nonempty
htb : IsVonNBounded ℝ t
e : E ≃ₜ E
h₁ : ⇑e '' interior s = interior t
h₂ : ⇑e '' closure[inst✝³] s = closure[inst✝³] t
⊢ ⇑e '' frontier s = frontier t

Tactic:
simp_rw [← closure_sdiff_interior, image_sdiff e.injective, h₁, h₂]

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
hs : Convex ℝ s
x : E
hx : x ∈ interior s
H : ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Bornology.IsBounded t → μ (frontier t) = 0
B : ℕ → Set E := fun n ↦ ball x (↑n + 1)
y : E
hy : y ∈ frontier s
N : ℕ := ⌊dist y x⌋₊
hN : y ∈ B N
⊢ y ∈ frontier (s ∩ B N) ∩ B N

Tactic:
frontier_inter_open_inter isOpen_ball

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
hs : Convex ℝ s
x : E
hx : x ∈ interior s
H : ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Bornology.IsBounded t → μ (frontier t) = 0
B : ℕ → Set E := fun n ↦ ball x (↑n + 1)
y : E
hy : y ∈ frontier s
N : ℕ := ⌊dist y x⌋₊
hN : y ∈ B N
⊢ y ∈ frontier s ∩ ball x (↑N + 1)
Module: Mathlib.Geometry.Convex.ConvexSpace.Defs

Goal state before:
pp: R : Type u
inst✝² : PartialOrder R
inst✝¹ : Semiring R
M : Type u_9
N : Type u_10
inst✝ : IsStrictOrderedRing R
f : StdSimplex R M
x a : N
⊢ (map (fun x_1 ↦ x) f).weights a = (single x).weights a

Tactic:
by_cases x = a <;> simp [*, mapDomain]

Goal state after:
Module: Mathlib.Geometry.Convex.ConvexSpace.Defs

Goal state before:
pp: R : Type u_1
M : Type u_3
I : Type u_6
J : Type u_7
inst✝³ : PartialOrder R
inst✝² : Semiring R
inst✝¹ : IsStrictOrderedRing R
inst✝ : ConvexSpace R M
s : StdSimplex R I
f : I ≃ J
g : I → M
⊢ iConvexComb s g = iConvexComb (map (⇑f) s) (g ∘ ⇑f.symm)

Tactic:
by
  simp [iConvexComb_map]

Goal state after:
Module: Mathlib.Geometry.Convex.Set

Goal state before:
pp: case inr
R : Type u_3
X : Type u_5
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : ConvexSpace R X
S : Set (Set X)
hS : DirectedOn (fun x1 x2 ↦ x1 ⊆ x2) S
hS' : ∀ s ∈ S, IsConvexSet R s
hS'' : S.Nonempty
w : StdSimplex R X
hw : ↑w.weights.support ⊆ ⋃₀ S
s : Set X
hsS : s ∈ S
hws : ↑w.weights.support ⊆ s
⊢ sConvexComb w ∈ ⋃₀ S

Tactic:
exact mem_sUnion_of_mem (hS' s hsS hws) hsS

Goal state after:
Module: Mathlib.Geometry.Convex.ConvexSpace.Defs

Goal state before:
pp: R : Type u_1
M : Type u_3
inst✝³ : PartialOrder R
inst✝² : Semiring R
inst✝¹ : IsStrictOrderedRing R
inst✝ : ConvexSpace R M
s t : R
hs : 0 ≤ s
ht : 0 ≤ t
h : s + t = 1
s' t' : R
hs' : 0 ≤ s'
ht' : 0 ≤ t'
h' : s' + t' = 1
s'' t'' : R
hs'' : 0 ≤ s''
ht'' : 0 ≤ t''
h'' : s'' + t'' = 1
H : s * t'' = t * s' * s''
m₁ m₂ m₃ : M
⊢ convexCombPair s t hs ht h m₁ (convexCombPair s' t' hs' ht' h' m₂ m₃) =
    convexCombPair (s + t * s') (t * t') ⋯ ⋯ ⋯ (convexCombPair s'' t'' hs'' ht'' h'' m₁ m₂) m₃

Tactic:
simp only [add_comm s]
  rw [convexCombPair_symm, convexCombPair_symm (x := m₂),
    convexCombPair_convexCombPair_assoc_left (hs'' := ht'') (ht'' := hs'')
      (h'' := (add_comm _ _).trans h'') (H := H),
    convexCombPair_symm, convexCombPair_symm (x := m₂)]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
β : Type u_3
inst✝⁴ : Semiring 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : AddCommMonoid E
inst✝¹ : LinearOrder β
inst✝ : SMul 𝕜 E
s : Set E
f : E → β
⊢ QuasilinearOn 𝕜 s f ↔
    Convex 𝕜 s ∧
      ∀ ⦃x : E⦄,
        x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ∈ uIcc (f x) (f y)

Tactic:
by
  rw [QuasilinearOn, quasiconvexOn_iff_le_max, quasiconcaveOn_iff_min_le, and_and_and_comm,
    and_self_iff]
  apply and_congr_right'
  simp_rw [← forall_and, ← Icc_min_max, mem_Icc, and_comm]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
β : Type u_3
inst✝⁴ : Semiring 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : AddCommMonoid E
inst✝¹ : LinearOrder β
inst✝ : SMul 𝕜 E
s : Set E
f : E → β
hf : QuasiconvexOn 𝕜 s f
r : β
⊢ Convex 𝕜 {x | x ∈ s ∧ f x < r}

Tactic:
intro x hx y hy a b ha hb hab

Goal state after:
𝕜 : Type u_1
E : Type u_2
β : Type u_3
inst✝⁴ : Semiring 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : AddCommMonoid E
inst✝¹ : LinearOrder β
inst✝ : SMul 𝕜 E
s : Set E
f : E → β
hf : QuasiconvexOn 𝕜 s f
r : β
x : E
hx : x ∈ {x | x ∈ s ∧ f x < r}
y : E
hy : y ∈ {x | x ∈ s ∧ f x < r}
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ a • x + b • y ∈ {x | x ∈ s ∧ f x < r}
Module: Mathlib.Analysis.Convex.MetricSpace

Goal state before:
pp: X : Type u_2
inst✝² : ConvexSpace ℝ X
inst✝¹ : MetricSpace X
inst✝ : IsConvexDist X
s t : ℝ
hs : 0 ≤ s
ht : 0 ≤ t
h : s + t = 1
x y : X
H :
  ∀ {s t : ℝ} (hs : 0 ≤ s) (ht : 0 ≤ t) (h : s + t = 1) (x y : X),
    dist (convexCombPair s t hs ht h x y) x ≤ t * dist x y
⊢ dist y x ≤ dist (convexCombPair s t hs ht h x y) x + s * dist y x

Tactic:
← H ht hs ((add_comm _ _).trans h) y x,

Goal state after:
X : Type u_2
inst✝² : ConvexSpace ℝ X
inst✝¹ : MetricSpace X
inst✝ : IsConvexDist X
s t : ℝ
hs : 0 ≤ s
ht : 0 ≤ t
h : s + t = 1
x y : X
H :
  ∀ {s t : ℝ} (hs : 0 ≤ s) (ht : 0 ≤ t) (h : s + t = 1) (x y : X),
    dist (convexCombPair s t hs ht h x y) x ≤ t * dist x y
⊢ dist y x ≤ dist (convexCombPair s t hs ht h x y) x + dist (convexCombPair t s ht hs ⋯ y x) y