mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Convex.ConvexSpace.AffineSpace

Goal state before:
pp: case e_a
R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AffineSpace V P
f : StdSimplex R (StdSimplex R P)
b : P
hL :
  (Finset.affineCombination R (StdSimplex.map convexCombination f).weights.support id)
      ⇑(StdSimplex.map convexCombination f).weights =
    ((StdSimplex.map convexCombination f).weights.support.weightedVSubOfPoint id b)
        ⇑(StdSimplex.map convexCombination f).weights +ᵥ
      b
hR :
  (Finset.affineCombination R f.join.weights.support id) ⇑f.join.weights =
    (f.join.weights.support.weightedVSubOfPoint id b) ⇑f.join.weights +ᵥ b
⊢ ∑ d ∈ f.weights.support, f.weights d • ∑ x ∈ d.weights.support, d.weights x • (x -ᵥ b) =
    ∑ x ∈ (∑ x ∈ f.weights.support, f.weights x • x.weights).support,
      (∑ x ∈ f.weights.support, f.weights x • x.weights) x • (x -ᵥ b)

Tactic:
[

Goal state after:
case e_a
R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AffineSpace V P
f : StdSimplex R (StdSimplex R P)
b : P
hL :
  (Finset.affineCombination R (StdSimplex.map convexCombination f).weights.support id)
      ⇑(StdSimplex.map convexCombination f).weights =
    ((StdSimplex.map convexCombination f).weights.support.weightedVSubOfPoint id b)
        ⇑(StdSimplex.map convexCombination f).weights +ᵥ
      b
hR :
  (Finset.affineCombination R f.join.weights.support id) ⇑f.join.weights =
    (f.join.weights.support.weightedVSubOfPoint id b) ⇑f.join.weights +ᵥ b
⊢ ∑ d ∈ f.weights.support, f.weights d • ∑ x ∈ d.weights.support, d.weights x • (x -ᵥ b) =
    ∑ x ∈ (∑ x ∈ f.weights.support, f.weights x • x.weights).support,
      (∑ x ∈ f.weights.support, f.weights x • x.weights) x • (x -ᵥ b)
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
J₁ : Type u₁
J₂ : Type u₂
g₁ : StdSimplex R J₁
g₂ : StdSimplex R J₂
m₁ : J₁ → M
m₂ : J₂ → M
this :
  sConvexComb (duple (sConvexComb (map (fun x ↦ m₁ x) g₁)) (sConvexComb (map (fun x ↦ m₂ x) g₂)) hs ht h) =
    sConvexComb (sConvexComb (duple (map (fun x ↦ m₁ x) g₁) (map (fun x ↦ m₂ x) g₂) hs ht h))
⊢ convexCombPair s t hs ht h (iConvexComb g₁ m₁) (iConvexComb g₂ m₂) =
    sConvexComb (convexCombPair s t hs ht h (map m₁ g₁) (map m₂ g₂))

Tactic:
simpa [convexCombPair, ← convexCombPair_def]

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
inst✝³ : PartialOrder R
inst✝² : Semiring R
inst✝¹ : IsStrictOrderedRing R
inst✝ : ConvexSpace R M
s t : I → R
hs : ∀ (i : I), 0 ≤ s i
ht : ∀ (i : I), 0 ≤ t i
h : ∀ (i : I), s i + t i = 1
f : StdSimplex R I
m₁ m₂ : I → M
this :
  (iConvexComb f fun i ↦ convexCombPair (s i) (t i) ⋯ ⋯ ⋯ (m₁ i) (m₂ i)) =
    sConvexComb (iConvexComb f fun x ↦ duple (m₁ x) (m₂ x) ⋯ ⋯ ⋯)
⊢ (iConvexComb f fun i ↦ convexCombPair (s i) (t i) ⋯ ⋯ ⋯ (m₁ i) (m₂ i)) =
    sConvexComb (iConvexComb f fun i ↦ duple (m₁ i) (m₂ i) ⋯ ⋯ ⋯)

Tactic:
simpa [convexCombPair, ← convexCombPair_def]

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

Goal state before:
pp: ι : Type u_1
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 X
hs : IsConvexSet R s
w : StdSimplex R ι
f : ι → X
hf : ∀ (i : ι), w.weights i ≠ 0 → f i ∈ s
⊢ iConvexComb w f ∈ s

Tactic:
refine hs ?_

Goal state after:
ι : Type u_1
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 X
hs : IsConvexSet R s
w : StdSimplex R ι
f : ι → X
hf : ∀ (i : ι), w.weights i ≠ 0 → f i ∈ s
⊢ ↑(StdSimplex.map f w).weights.support ⊆ s
Module: Mathlib.Geometry.Convex.ConvexSpace.AffineSpace

Goal state before:
pp: case neg.mp
R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AffineSpace V P
f : StdSimplex R (StdSimplex R P)
b : P
hL :
  (Finset.affineCombination R (StdSimplex.map convexCombination f).weights.support id)
      ⇑(StdSimplex.map convexCombination f).weights =
    ((StdSimplex.map convexCombination f).weights.support.weightedVSubOfPoint id b)
        ⇑(StdSimplex.map convexCombination f).weights +ᵥ
      b
hR :
  (Finset.affineCombination R f.join.weights.support id) ⇑f.join.weights =
    (f.join.weights.support.weightedVSubOfPoint id b) ⇑f.join.weights +ᵥ b
h : P → R → V := ⋯
h_rhs :
  ∑ a ∈ (∑ d ∈ f.weights.support, f.weights d • d.weights).support,
      h a ((∑ d ∈ f.weights.support, f.weights d • d.weights) a) =
    ∑ d ∈ f.weights.support, ∑ a ∈ (f.weights d • d.weights).support, h a ((f.weights d • d.weights) a)
d : StdSimplex R P
hd : ¬f.weights d = 0
p : P
⊢ ¬d.weights p = 0 → ¬(f.weights d • d.weights) p = 0

Tactic:
intro hp

Goal state after:
case neg.mp
R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AffineSpace V P
f : StdSimplex R (StdSimplex R P)
b : P
hL :
  (Finset.affineCombination R (StdSimplex.map convexCombination f).weights.support id)
      ⇑(StdSimplex.map convexCombination f).weights =
    ((StdSimplex.map convexCombination f).weights.support.weightedVSubOfPoint id b)
        ⇑(StdSimplex.map convexCombination f).weights +ᵥ
      b
hR :
  (Finset.affineCombination R f.join.weights.support id) ⇑f.join.weights =
    (f.join.weights.support.weightedVSubOfPoint id b) ⇑f.join.weights +ᵥ b
h : P → R → V := fun x w ↦ w • (x -ᵥ b)
h_rhs :
  ∑ a ∈ (∑ d ∈ f.weights.support, f.weights d • d.weights).support,
      h a ((∑ d ∈ f.weights.support, f.weights d • d.weights) a) =
    ∑ d ∈ f.weights.support, ∑ a ∈ (f.weights d • d.weights).support, h a ((f.weights d • d.weights) a)
d : StdSimplex R P
hd : ¬f.weights d = 0
p : P
hp : ¬d.weights p = 0
⊢ ¬(f.weights d • d.weights) p = 0
Module: Mathlib.Geometry.Convex.ConvexSpace.Defs

Goal state before:
pp: case e'_2
R : Type u_9
M : Type u_10
I : Type u_11
inst✝³ : PartialOrder R
inst✝² : CommSemiring R
inst✝¹ : IsStrictOrderedRing R
inst✝ : ConvexSpace R M
s t : R
hs : 0 ≤ s
ht : 0 ≤ t
h : s + t = 1
f : StdSimplex R I
e₁ e₂ : I → M
i : I
a✝¹ : f.weights i ≠ 0
j : Fin (Nat.succ 0).succ
a✝ : (duple 0 1 hs ht h).weights j ≠ 0
⊢ ![e₁ i, e₂ i] j = ![e₁, e₂] j i

Tactic:
fin_cases j

Goal state after:
case e'_2.«0»
R : Type u_9
M : Type u_10
I : Type u_11
inst✝³ : PartialOrder R
inst✝² : CommSemiring R
inst✝¹ : IsStrictOrderedRing R
inst✝ : ConvexSpace R M
s t : R
hs : 0 ≤ s
ht : 0 ≤ t
h : s + t = 1
f : StdSimplex R I
e₁ e₂ : I → M
i : I
a✝¹ : f.weights i ≠ 0
a✝ : (duple 0 1 hs ht h).weights ((fun i ↦ i) ⟨0, ⋯⟩) ≠ 0
⊢ ![e₁ i, e₂ i] ((fun i ↦ i) ⟨0, ⋯⟩) = ![e₁, e₂] ((fun i ↦ i) ⟨0, ⋯⟩) i

case e'_2.«1»
R : Type u_9
M : Type u_10
I : Type u_11
inst✝³ : PartialOrder R
inst✝² : CommSemiring R
inst✝¹ : IsStrictOrderedRing R
inst✝ : ConvexSpace R M
s t : R
hs : 0 ≤ s
ht : 0 ≤ t
h : s + t = 1
f : StdSimplex R I
e₁ e₂ : I → M
i : I
a✝¹ : f.weights i ≠ 0
a✝ : (duple 0 1 hs ht h).weights ((fun i ↦ i) ⟨1, ⋯⟩) ≠ 0
⊢ ![e₁ i, e₂ i] ((fun i ↦ i) ⟨1, ⋯⟩) = ![e₁, e₂] ((fun i ↦ i) ⟨1, ⋯⟩) i
Module: Mathlib.Geometry.Convex.ConvexSpace.AffineSpace

Goal state before:
pp: case neg.mpr
R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AffineSpace V P
f : StdSimplex R (StdSimplex R P)
b : P
hL :
  (Finset.affineCombination R (StdSimplex.map convexCombination f).weights.support id)
      ⇑(StdSimplex.map convexCombination f).weights =
    ((StdSimplex.map convexCombination f).weights.support.weightedVSubOfPoint id b)
        ⇑(StdSimplex.map convexCombination f).weights +ᵥ
      b
hR :
  (Finset.affineCombination R f.join.weights.support id) ⇑f.join.weights =
    (f.join.weights.support.weightedVSubOfPoint id b) ⇑f.join.weights +ᵥ b
h : P → R → V := ⋯
h_rhs :
  ∑ a ∈ (∑ d ∈ f.weights.support, f.weights d • d.weights).support,
      h a ((∑ d ∈ f.weights.support, f.weights d • d.weights) a) =
    ∑ d ∈ f.weights.support, ∑ a ∈ (f.weights d • d.weights).support, h a ((f.weights d • d.weights) a)
d : StdSimplex R P
hd : ¬f.weights d = 0
p : P
⊢ ¬(f.weights d • d.weights) p = 0 → ¬d.weights p = 0

Tactic:
intro hp

Goal state after:
case neg.mpr
R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AffineSpace V P
f : StdSimplex R (StdSimplex R P)
b : P
hL :
  (Finset.affineCombination R (StdSimplex.map convexCombination f).weights.support id)
      ⇑(StdSimplex.map convexCombination f).weights =
    ((StdSimplex.map convexCombination f).weights.support.weightedVSubOfPoint id b)
        ⇑(StdSimplex.map convexCombination f).weights +ᵥ
      b
hR :
  (Finset.affineCombination R f.join.weights.support id) ⇑f.join.weights =
    (f.join.weights.support.weightedVSubOfPoint id b) ⇑f.join.weights +ᵥ b
h : P → R → V := fun x w ↦ w • (x -ᵥ b)
h_rhs :
  ∑ a ∈ (∑ d ∈ f.weights.support, f.weights d • d.weights).support,
      h a ((∑ d ∈ f.weights.support, f.weights d • d.weights) a) =
    ∑ d ∈ f.weights.support, ∑ a ∈ (f.weights d • d.weights).support, h a ((f.weights d • d.weights) a)
d : StdSimplex R P
hd : ¬f.weights d = 0
p : P
hp : ¬(f.weights d • d.weights) p = 0
⊢ ¬d.weights p = 0
Module: Mathlib.Geometry.Convex.ConvexSpace.Defs

Goal state before:
pp: case e'_3
R : Type u_9
M : Type u_10
I : Type u_11
inst✝³ : PartialOrder R
inst✝² : CommSemiring R
inst✝¹ : IsStrictOrderedRing R
inst✝ : ConvexSpace R M
s t : R
hs : 0 ≤ s
ht : 0 ≤ t
h : s + t = 1
f : StdSimplex R I
e₁ e₂ : I → M
j : Fin (Nat.succ 0).succ
a✝ : (duple 0 1 hs ht h).weights j ≠ 0
⊢ ![iConvexComb f e₁, iConvexComb f e₂] j = iConvexComb f (![e₁, e₂] j)

Tactic:
fin_cases j

Goal state after:
case e'_3.«0»
R : Type u_9
M : Type u_10
I : Type u_11
inst✝³ : PartialOrder R
inst✝² : CommSemiring R
inst✝¹ : IsStrictOrderedRing R
inst✝ : ConvexSpace R M
s t : R
hs : 0 ≤ s
ht : 0 ≤ t
h : s + t = 1
f : StdSimplex R I
e₁ e₂ : I → M
a✝ : (duple 0 1 hs ht h).weights ((fun i ↦ i) ⟨0, ⋯⟩) ≠ 0
⊢ ![iConvexComb f e₁, iConvexComb f e₂] ((fun i ↦ i) ⟨0, ⋯⟩) = iConvexComb f (![e₁, e₂] ((fun i ↦ i) ⟨0, ⋯⟩))

case e'_3.«1»
R : Type u_9
M : Type u_10
I : Type u_11
inst✝³ : PartialOrder R
inst✝² : CommSemiring R
inst✝¹ : IsStrictOrderedRing R
inst✝ : ConvexSpace R M
s t : R
hs : 0 ≤ s
ht : 0 ≤ t
h : s + t = 1
f : StdSimplex R I
e₁ e₂ : I → M
a✝ : (duple 0 1 hs ht h).weights ((fun i ↦ i) ⟨1, ⋯⟩) ≠ 0
⊢ ![iConvexComb f e₁, iConvexComb f e₂] ((fun i ↦ i) ⟨1, ⋯⟩) = iConvexComb f (![e₁, e₂] ((fun i ↦ i) ⟨1, ⋯⟩))
Module: Mathlib.Geometry.Convex.Set

Goal state before:
pp: ι : Type u_1
I : Type u_2
R : Type u_3
K : Type u_4
X : Type u_5
Y : Type u_6
inst✝⁴ : Semiring R
inst✝³ : PartialOrder R
inst✝² : IsStrictOrderedRing R
inst✝¹ : ConvexSpace R X
inst✝ : ConvexSpace R Y
f : X → Y
w✝ : StdSimplex R X
s✝ t : Set X
x y : X
s : Set X
hs : IsConvexSet R s
w : StdSimplex R (StdSimplex R ↑s)
⊢ ⟨iConvexComb (StdSimplex.map (fun w ↦ ⟨iConvexComb w Subtype.val, ⋯⟩) w) Subtype.val, ⋯⟩ =
    ⟨iConvexComb (sConvexComb w) Subtype.val, ⋯⟩

Tactic:
by ext; simp [iConvexComb_assoc]

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 : SMul 𝕜 E
s : Set E
f g : E → β
inst✝ : SemilatticeSup β
hf : QuasiconvexOn 𝕜 s f
hg : QuasiconvexOn 𝕜 s g
r : β
⊢ Convex 𝕜 {x | x ∈ s ∧ (f ⊔ g) x ≤ r}

Tactic:
[

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 : SMul 𝕜 E
s : Set E
f g : E → β
inst✝ : SemilatticeSup β
hf : QuasiconvexOn 𝕜 s f
hg : QuasiconvexOn 𝕜 s g
r : β
⊢ Convex 𝕜 {x | x ∈ s ∧ (f ⊔ g) x ≤ r}
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
⊢ ((∀ ⦃x : E⦄,
        x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ max (f x) (f y)) ∧
      ∀ ⦃x : E⦄,
        x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → min (f x) (f y) ≤ f (a • x + b • y)) ↔
    ∀ ⦃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:
[

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
⊢ ((∀ ⦃x : E⦄,
        x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ max (f x) (f y)) ∧
      ∀ ⦃x : E⦄,
        x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → min (f x) (f y) ≤ f (a • x + b • y)) ↔
    ∀ ⦃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)
Module: Mathlib.Analysis.Convex.Piecewise

Goal state before:
pp: case inr.inl.inr
𝕜 : Type u_1
E : Type u_2
β : Type u_3
inst✝¹¹ : Semiring 𝕜
inst✝¹⁰ : PartialOrder 𝕜
inst✝⁹ : AddCommMonoid E
inst✝⁸ : LinearOrder E
inst✝⁷ : IsOrderedAddMonoid E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : PosSMulMono 𝕜 E
inst✝⁴ : AddCommGroup β
inst✝³ : PartialOrder β
inst✝² : IsOrderedAddMonoid β
inst✝¹ : Module 𝕜 β
inst✝ : PosSMulMono 𝕜 β
e : E
f g : E → β
hf : ConvexOn 𝕜 (Set.Iic e) f
hg : ConvexOn 𝕜 (Set.Ici e) g
h_anti : AntitoneOn f (Set.Iic e)
h_mono : MonotoneOn g (Set.Ici e)
h_eq : f e = g e
x : E
x✝¹ : x ∈ Set.univ
y : E
x✝ : y ∈ Set.univ
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
hx : e < x
hy : y ≤ e
hc : e < a • x + b • y
⊢ g (a • x + b • y) ≤ a • g x + b • f y

Tactic:
have hc' : a • x + b • y ≤ a • x + b • e := by gcongr

Goal state after:
case inr.inl.inr
𝕜 : Type u_1
E : Type u_2
β : Type u_3
inst✝¹¹ : Semiring 𝕜
inst✝¹⁰ : PartialOrder 𝕜
inst✝⁹ : AddCommMonoid E
inst✝⁸ : LinearOrder E
inst✝⁷ : IsOrderedAddMonoid E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : PosSMulMono 𝕜 E
inst✝⁴ : AddCommGroup β
inst✝³ : PartialOrder β
inst✝² : IsOrderedAddMonoid β
inst✝¹ : Module 𝕜 β
inst✝ : PosSMulMono 𝕜 β
e : E
f g : E → β
hf : ConvexOn 𝕜 (Set.Iic e) f
hg : ConvexOn 𝕜 (Set.Ici e) g
h_anti : AntitoneOn f (Set.Iic e)
h_mono : MonotoneOn g (Set.Ici e)
h_eq : f e = g e
x : E
x✝¹ : x ∈ Set.univ
y : E
x✝ : y ∈ Set.univ
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
hx : e < x
hy : y ≤ e
hc : e < a • x + b • y
hc' : a • x + b • y ≤ a • x + b • e
⊢ g (a • x + b • y) ≤ a • g x + b • f y
Module: Mathlib.Analysis.Convex.Quasiconvex

Goal state before:
pp: 𝕜 : Type u_1
β : Type u_3
inst✝³ : Field 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → β
inst✝ : LinearOrder β
hf : QuasilinearOn 𝕜 s f
⊢ MonotoneOn f s ∨ AntitoneOn f s

Tactic:
[

Goal state after:
𝕜 : Type u_1
β : Type u_3
inst✝³ : Field 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → β
inst✝ : LinearOrder β
hf : QuasilinearOn 𝕜 s f
⊢ MonotoneOn f s ∨ AntitoneOn f s
Module: Mathlib.Analysis.Convex.MetricSpace

Goal state before:
pp: case ha
X : Type u_2
inst✝² : ConvexSpace ℝ X
inst✝¹ : MetricSpace X
inst✝ : IsConvexDist X
i : ↑(Set.Icc 0 1)
x y : X × X
⊢ ↑i * max (dist x.1 y.1) (dist x.2 y.2) + (1 - ↑i) * max (dist x.1 y.1) (dist x.2 y.2) ≤
    max (dist x.1 y.1) (dist x.2 y.2)

Tactic:
rw [← add_mul, add_sub_cancel, one_mul]

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

Goal state before:
pp: case refine_2
𝕜 : Type u_1
E : Type u_2
inst✝³ : Ring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
s : Finset E
hs✝ : s ∈ K.faces
t : Finset E
ht : t ∈ K.faces ∧ s ⊂ t
hs : s ∈ K.faces
hs' : ∀ ⦃t : Finset E⦄, t ∈ K.faces → s ⊆ t → s = t
this : s = t
⊢ False

Tactic:
this

Goal state after:
case refine_2
𝕜 : Type u_1
E : Type u_2
inst✝³ : Ring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
s : Finset E
hs✝ : s ∈ K.faces
t : Finset E
ht : t ∈ K.faces ∧ t ⊂ t
hs : s ∈ K.faces
hs' : ∀ ⦃t : Finset E⦄, t ∈ K.faces → s ⊆ t → s = t
this : s = t
⊢ False
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁸ : CommRing R
inst✝⁷ : PartialOrder R
inst✝⁶ : IsStrictOrderedRing R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : AddTorsor V P
inst✝² : AddCommGroup V'
inst✝¹ : Module R V'
inst✝ : AddTorsor V' P'
s : AffineSubspace R P
x y : P
f : P →ᵃ[R] P'
p₁ : P
hp₁ : p₁ ∈ s
p₂ : P
hp₂ : p₂ ∈ s
h : SameRay R (x -ᵥ p₁) (y -ᵥ p₂)
⊢ SameRay R (f x -ᵥ f p₁) (f y -ᵥ f p₂)

Tactic:
[

Goal state after:
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁸ : CommRing R
inst✝⁷ : PartialOrder R
inst✝⁶ : IsStrictOrderedRing R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : AddTorsor V P
inst✝² : AddCommGroup V'
inst✝¹ : Module R V'
inst✝ : AddTorsor V' P'
s : AffineSubspace R P
x y : P
f : P →ᵃ[R] P'
p₁ : P
hp₁ : p₁ ∈ s
p₂ : P
hp₂ : p₂ ∈ s
h : SameRay R (x -ᵥ p₁) (y -ᵥ p₂)
⊢ SameRay R (f x -ᵥ f p₁) (f y -ᵥ f p₂)
Module: Mathlib.Combinatorics.SimpleGraph.Basic

Goal state before:
pp: V : Type u
G H : SimpleGraph V
h : Disjoint G.support H.support
⊢ Disjoint G H

Tactic:
[

Goal state after:
V : Type u
G H : SimpleGraph V
h : Disjoint G.support H.support
⊢ Disjoint G H
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁸ : CommRing R
inst✝⁷ : PartialOrder R
inst✝⁶ : IsStrictOrderedRing R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : AddTorsor V P
inst✝² : AddCommGroup V'
inst✝¹ : Module R V'
inst✝ : AddTorsor V' P'
s : AffineSubspace R P
x y : P
f : P →ᵃ[R] P'
hf : Function.Injective ⇑f
p₁ : P
hp₁ : p₁ ∈ s
p₂ : P
hp₂ : p₂ ∈ s
h : SameRay R (f x -ᵥ f p₁) (f y -ᵥ f p₂)
⊢ SameRay R (x -ᵥ p₁) (y -ᵥ p₂)

Tactic:
[

Goal state after:
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁸ : CommRing R
inst✝⁷ : PartialOrder R
inst✝⁶ : IsStrictOrderedRing R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : AddTorsor V P
inst✝² : AddCommGroup V'
inst✝¹ : Module R V'
inst✝ : AddTorsor V' P'
s : AffineSubspace R P
x y : P
f : P →ᵃ[R] P'
hf : Function.Injective ⇑f
p₁ : P
hp₁ : p₁ ∈ s
p₂ : P
hp₂ : p₂ ∈ s
h : SameRay R (f x -ᵥ f p₁) (f y -ᵥ f p₂)
⊢ SameRay R (x -ᵥ p₁) (y -ᵥ p₂)
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁸ : CommRing R
inst✝⁷ : PartialOrder R
inst✝⁶ : IsStrictOrderedRing R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : AddTorsor V P
inst✝² : AddCommGroup V'
inst✝¹ : Module R V'
inst✝ : AddTorsor V' P'
s : AffineSubspace R P
x y : P
f : P →ᵃ[R] P'
hf : Function.Injective ⇑f
⊢ (map f s).SSameSide (f x) (f y) ↔ s.SSameSide x y

Tactic:
[

Goal state after:
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁸ : CommRing R
inst✝⁷ : PartialOrder R
inst✝⁶ : IsStrictOrderedRing R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : AddTorsor V P
inst✝² : AddCommGroup V'
inst✝¹ : Module R V'
inst✝ : AddTorsor V' P'
s : AffineSubspace R P
x y : P
f : P →ᵃ[R] P'
hf : Function.Injective ⇑f
⊢ (map f s).SSameSide (f x) (f y) ↔ s.SSameSide x y
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁸ : CommRing R
inst✝⁷ : PartialOrder R
inst✝⁶ : IsStrictOrderedRing R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : AddTorsor V P
inst✝² : AddCommGroup V'
inst✝¹ : Module R V'
inst✝ : AddTorsor V' P'
s : AffineSubspace R P
x y : P
f : P →ᵃ[R] P'
p₁ : P
hp₁ : p₁ ∈ s
p₂ : P
hp₂ : p₂ ∈ s
h : SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)
⊢ SameRay R (f x -ᵥ f p₁) (f p₂ -ᵥ f y)

Tactic:
[

Goal state after:
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁸ : CommRing R
inst✝⁷ : PartialOrder R
inst✝⁶ : IsStrictOrderedRing R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : AddTorsor V P
inst✝² : AddCommGroup V'
inst✝¹ : Module R V'
inst✝ : AddTorsor V' P'
s : AffineSubspace R P
x y : P
f : P →ᵃ[R] P'
p₁ : P
hp₁ : p₁ ∈ s
p₂ : P
hp₂ : p₂ ∈ s
h : SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)
⊢ SameRay R (f x -ᵥ f p₁) (f p₂ -ᵥ f y)
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁸ : CommRing R
inst✝⁷ : PartialOrder R
inst✝⁶ : IsStrictOrderedRing R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : AddTorsor V P
inst✝² : AddCommGroup V'
inst✝¹ : Module R V'
inst✝ : AddTorsor V' P'
s : AffineSubspace R P
x y : P
f : P →ᵃ[R] P'
hf : Function.Injective ⇑f
p₁ : P
hp₁ : p₁ ∈ s
p₂ : P
hp₂ : p₂ ∈ s
h : SameRay R (f x -ᵥ f p₁) (f p₂ -ᵥ f y)
⊢ SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)

Tactic:
[

Goal state after:
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁸ : CommRing R
inst✝⁷ : PartialOrder R
inst✝⁶ : IsStrictOrderedRing R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : AddTorsor V P
inst✝² : AddCommGroup V'
inst✝¹ : Module R V'
inst✝ : AddTorsor V' P'
s : AffineSubspace R P
x y : P
f : P →ᵃ[R] P'
hf : Function.Injective ⇑f
p₁ : P
hp₁ : p₁ ∈ s
p₂ : P
hp₂ : p₂ ∈ s
h : SameRay R (f x -ᵥ f p₁) (f p₂ -ᵥ f y)
⊢ SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁸ : CommRing R
inst✝⁷ : PartialOrder R
inst✝⁶ : IsStrictOrderedRing R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : AddTorsor V P
inst✝² : AddCommGroup V'
inst✝¹ : Module R V'
inst✝ : AddTorsor V' P'
s : AffineSubspace R P
x y : P
f : P →ᵃ[R] P'
hf : Function.Injective ⇑f
⊢ (map f s).SOppSide (f x) (f y) ↔ s.SOppSide x y

Tactic:
[

Goal state after:
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁸ : CommRing R
inst✝⁷ : PartialOrder R
inst✝⁶ : IsStrictOrderedRing R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : AddTorsor V P
inst✝² : AddCommGroup V'
inst✝¹ : Module R V'
inst✝ : AddTorsor V' P'
s : AffineSubspace R P
x y : P
f : P →ᵃ[R] P'
hf : Function.Injective ⇑f
⊢ (map f s).SOppSide (f x) (f y) ↔ s.SOppSide x y
Module: Mathlib.Analysis.Convex.SpecificFunctions.Pow

Goal state before:
pp: this : ⇑sqrt = fun x ↦ x ^ (1 / 2)
⊢ StrictConcaveOn ℝ≥0 univ ⇑sqrt

Tactic:
this

Goal state after:
this : ⇑sqrt = fun x ↦ x ^ (1 / 2)
⊢ StrictConcaveOn ℝ≥0 univ fun x ↦ x ^ (1 / 2)
Module: Mathlib.Analysis.Convex.Radon

Goal state before:
pp: case a
𝕜 : Type u_2
E : Type u_3
inst✝⁷ : Field 𝕜
inst✝⁶ : LinearOrder 𝕜
inst✝⁵ : IsStrictOrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : T2Space E
F : Set (Set E)
h_card : ↑(finrank 𝕜 E) + 1 ≤ F.encard
h_convex : ∀ X ∈ F, Convex 𝕜 X
h_compact : ∀ X ∈ F, IsCompact X
h_inter : ∀ (G : Finset (Set E)), ↑G ⊆ F → #G = finrank 𝕜 E + 1 → (⋂₀ ↑G).Nonempty
I : Finset (Set E)
hI_ss : ↑I ⊆ F
hI_card : #I ≤ finrank 𝕜 E + 1
J : Set (Set E)
left✝ : ↑I ⊆ J
hJ_ss : J ⊆ F
hJ_fin : Fintype ↑J
hJ_card : ↑(#J.toFinset) = ↑(finrank 𝕜 E) + 1
J' : Finset (Set E) := J.toFinset
⊢ #J' = finrank 𝕜 E + 1

Tactic:
← ENat.natCast_one,

Goal state after:
case a
𝕜 : Type u_2
E : Type u_3
inst✝⁷ : Field 𝕜
inst✝⁶ : LinearOrder 𝕜
inst✝⁵ : IsStrictOrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : T2Space E
F : Set (Set E)
h_card : ↑(finrank 𝕜 E) + 1 ≤ F.encard
h_convex : ∀ X ∈ F, Convex 𝕜 X
h_compact : ∀ X ∈ F, IsCompact X
h_inter : ∀ (G : Finset (Set E)), ↑G ⊆ F → #G = finrank 𝕜 E + 1 → (⋂₀ ↑G).Nonempty
I : Finset (Set E)
hI_ss : ↑I ⊆ F
hI_card : #I ≤ finrank 𝕜 E + 1
J : Set (Set E)
left✝ : ↑I ⊆ J
hJ_ss : J ⊆ F
hJ_fin : Fintype ↑J
hJ_card : ↑(#J.toFinset) = ↑(finrank 𝕜 E) + ↑1
J' : Finset (Set E) := J.toFinset
⊢ #J' = finrank 𝕜 E + 1