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

Goal state before:
pp: case mp
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
x z : P
ty : R
hty : ty ∈ Set.Icc 0 1
tx : R
htx : tx ∈ Set.Icc 0 1
hx : ((tx • (z -ᵥ (ty • (z -ᵥ x) +ᵥ x)) + ty • (z -ᵥ x)) +ᵥ x) -ᵥ x = 0
⊢ x = (lineMap x z) ty

Tactic:
vadd_vsub,

Goal state after:
case mp
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
x z : P
ty : R
hty : ty ∈ Set.Icc 0 1
tx : R
htx : tx ∈ Set.Icc 0 1
hx : tx • (z -ᵥ (ty • (z -ᵥ x) +ᵥ x)) + ty • (z -ᵥ x) = 0
⊢ x = (lineMap x z) ty
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
x z : P
ty : R
hty : ty ∈ Set.Icc 0 1
tx : R
htx : tx ∈ Set.Icc 0 1
h : -(tx * (1 - ty)) = ty
⊢ ty = 0

Tactic:
refine le_antisymm ?_ hty.1

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
x z : P
ty : R
hty : ty ∈ Set.Icc 0 1
tx : R
htx : tx ∈ Set.Icc 0 1
h : -(tx * (1 - ty)) = ty
⊢ ty ≤ 0
Module: Mathlib.Analysis.Convex.Between

Goal state before:
pp: case mp.inr
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
x z : P
ty : R
hty : ty ∈ Set.Icc 0 1
tx : R
htx : tx ∈ Set.Icc 0 1
h : z -ᵥ x = 0
⊢ x = (lineMap x z) ty

Tactic:
rw [vsub_eq_zero_iff_eq] at h
      rw [h, lineMap_same_apply]

Goal state after:
Module: Mathlib.CategoryTheory.CofilteredSystem

Goal state before:
pp: case refine_1
J : Type u
inst✝¹ : Category.{v_1, u} J
F : J ⥤ Type v
i : J
s : Set (F.obj i)
inst✝ : IsCofilteredOrEmpty J
h : F.IsMittagLeffler
j j₁ : J
g₁ : j₁ ⟶ i
f₁ : j₁ ⟶ j
j₂ : J
f₂ : j₂ ⟶ j₁
h₂ : F.eventualRange j₁ = range ⇑(ConcreteCategory.hom (F.map f₂))
j₃ : J
f₃ : j₃ ⟶ j₂
x : F.obj j₂
hx : x ∈ ⋂ f, ⇑(ConcreteCategory.hom (F.map f)) ⁻¹' s
this : (ConcreteCategory.hom (F.map f₂)) x ∈ F.eventualRange j₁
y : F.obj j₃
hy : y ∈ F.eventualRange j₃
h₃ : (ConcreteCategory.hom (F.map (f₃ ≫ f₂))) y = (ConcreteCategory.hom (F.map f₂)) x
g₂ : j₃ ⟶ i
j₄ : J
f₄ : j₄ ⟶ j₃
h₄ : f₄ ≫ g₂ = f₄ ≫ (f₃ ≫ f₂) ≫ g₁
⊢ y ∈ ⇑(ConcreteCategory.hom (F.map g₂)) ⁻¹' s

Tactic:
obtain ⟨y, rfl⟩ := F.mem_eventualRange_iff.1 hy f₄

Goal state after:
case refine_1
J : Type u
inst✝¹ : Category.{v_1, u} J
F : J ⥤ Type v
i : J
s : Set (F.obj i)
inst✝ : IsCofilteredOrEmpty J
h : F.IsMittagLeffler
j j₁ : J
g₁ : j₁ ⟶ i
f₁ : j₁ ⟶ j
j₂ : J
f₂ : j₂ ⟶ j₁
h₂ : F.eventualRange j₁ = range ⇑(ConcreteCategory.hom (F.map f₂))
j₃ : J
f₃ : j₃ ⟶ j₂
x : F.obj j₂
hx : x ∈ ⋂ f, ⇑(ConcreteCategory.hom (F.map f)) ⁻¹' s
this : (ConcreteCategory.hom (F.map f₂)) x ∈ F.eventualRange j₁
g₂ : j₃ ⟶ i
j₄ : J
f₄ : j₄ ⟶ j₃
h₄ : f₄ ≫ g₂ = f₄ ≫ (f₃ ≫ f₂) ≫ g₁
y : F.obj j₄
hy : (ConcreteCategory.hom (F.map f₄)) y ∈ F.eventualRange j₃
h₃ :
  (ConcreteCategory.hom (F.map (f₃ ≫ f₂))) ((ConcreteCategory.hom (F.map f₄)) y) = (ConcreteCategory.hom (F.map f₂)) x
⊢ (ConcreteCategory.hom (F.map f₄)) y ∈ ⇑(ConcreteCategory.hom (F.map g₂)) ⁻¹' s
Module: Mathlib.CategoryTheory.CofilteredSystem

Goal state before:
pp: J : Type u
inst✝³ : Category.{v_1, u} J
F : J ⥤ Type v
inst✝² : IsCofilteredOrEmpty J
inst✝¹ : ∀ (j : J), Nonempty (F.obj j)
inst✝ : ∀ (j : J), Finite (F.obj j)
Fsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective ⇑(ConcreteCategory.hom (F.map f))
i : J
x : F.obj i
s : Set (F.obj i) := {x}
this : ∀ (j : J), Nonempty ((F.toPreimages s).obj j)
sec : (j : J) → (F.toPreimages s).obj j
h : sec ∈ (F.toPreimages s).sections
j✝ j'✝ : J
jk : j✝ ⟶ j'✝
⊢ (ConcreteCategory.hom (F.map jk)) ((fun j ↦ ↑(sec j)) j✝) = (fun j ↦ ↑(sec j)) j'✝

Tactic:
simpa [Subtype.ext_iff] using! h jk

Goal state after:
Module: Mathlib.Data.PEquiv

Goal state before:
pp: α : Type u
β : Type v
γ : Type w
δ : Type x
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f g h : α ≃. β
fg : f ≤ g
gh : f ≤ h
a : α
b : β
H : b ∈ f a
⊢ b ∈
    { toFun := fun a ↦ if g a = h a then g a else none,
        invFun := fun b ↦ if g.symm b = h.symm b then g.symm b else none, inv := ⋯ }
      a

Tactic:
have hf := fg a b H

Goal state after:
α : Type u
β : Type v
γ : Type w
δ : Type x
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f g h : α ≃. β
fg : f ≤ g
gh : f ≤ h
a : α
b : β
H : b ∈ f a
hf : b ∈ g a
⊢ b ∈
    { toFun := fun a ↦ if g a = h a then g a else none,
        invFun := fun b ↦ if g.symm b = h.symm b then g.symm b else none, inv := ⋯ }
      a
Module: Mathlib.Data.Matrix.PEquiv

Goal state before:
pp: l : Type u_2
m : Type u_3
n : Type u_4
α : Type u_5
inst✝² : Fintype m
inst✝¹ : DecidableEq m
inst✝ : NonAssocSemiring α
f : l ≃ m
M : Matrix m n α
i : l
j : n
⊢ (f.toPEquiv.toMatrix * M) i j = M.submatrix (⇑f) id i j

Tactic:
rw [toMatrix_mul_apply, Equiv.toPEquiv_apply, submatrix_apply, id]

Goal state after:
Module: Mathlib.Data.Matrix.PEquiv

Goal state before:
pp: case h.none.none
m : Type u_3
n : Type u_4
α : Type u_5
inst✝² : DecidableEq n
inst✝¹ : MulZeroOneClass α
inst✝ : Nontrivial α
f g : m ≃. n
i : m
hi : ¬f i = g i
hf : f i = none
hg : g i = none
⊢ ∃ x, ¬(if x ∈ none then 1 else 0) = if x ∈ none then 1 else 0

Tactic:
rw [hf, hg] at hi

Goal state after:
case h.none.none
m : Type u_3
n : Type u_4
α : Type u_5
inst✝² : DecidableEq n
inst✝¹ : MulZeroOneClass α
inst✝ : Nontrivial α
f g : m ≃. n
i : m
hi : ¬none = none
hf : f i = none
hg : g i = none
⊢ ∃ x, ¬(if x ∈ none then 1 else 0) = if x ∈ none then 1 else 0
Module: Mathlib.Analysis.Convex.Birkhoff

Goal state before:
pp: case inr
R : Type u_1
n : Type u_2
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : Field R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
h✝ : Nonempty n
d : ℕ
ih :
  ∀ m < d,
    ∀ (M : Matrix n n R) (s : R),
      0 ≤ s →
        (∃ M' ∈ doublyStochastic R n, M = s • M') →
          #{i | M i.1 i.2 ≠ 0} = m → ∃ w, (∀ (σ : Equiv.Perm n), 0 ≤ w σ) ∧ ∑ σ, w σ • Equiv.Perm.permMatrix R σ = M
M : Matrix n n R
s : R
hs : 0 ≤ s
hM : (∀ (i j : n), 0 ≤ M i j) ∧ (∀ (i : n), ∑ j, M i j = s) ∧ ∀ (j : n), ∑ i, M i j = s
hd : #{i | M i.1 i.2 ≠ 0} = d
hs' : 0 < s
σ : Equiv.Perm n
hσ : ∀ (i j : n), M i j = 0 → Equiv.Perm.permMatrix R σ i j = 0
i : n
hi : i ∈ univ
hi' : ∀ x' ∈ univ, M i (σ i) ≤ M x' (σ x')
N : Matrix n n R := M - M i (σ i) • Equiv.Perm.permMatrix R σ
hMi' : 0 < M i (σ i)
⊢ ∃ w, (∀ (σ : Equiv.Perm n), 0 ≤ w σ) ∧ ∑ σ, w σ • Equiv.Perm.permMatrix R σ = M

Tactic:
let s' : R := s - M i (σ i)

Goal state after:
case inr
R : Type u_1
n : Type u_2
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : Field R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
h✝ : Nonempty n
d : ℕ
ih :
  ∀ m < d,
    ∀ (M : Matrix n n R) (s : R),
      0 ≤ s →
        (∃ M' ∈ doublyStochastic R n, M = s • M') →
          #{i | M i.1 i.2 ≠ 0} = m → ∃ w, (∀ (σ : Equiv.Perm n), 0 ≤ w σ) ∧ ∑ σ, w σ • Equiv.Perm.permMatrix R σ = M
M : Matrix n n R
s : R
hs : 0 ≤ s
hM : (∀ (i j : n), 0 ≤ M i j) ∧ (∀ (i : n), ∑ j, M i j = s) ∧ ∀ (j : n), ∑ i, M i j = s
hd : #{i | M i.1 i.2 ≠ 0} = d
hs' : 0 < s
σ : Equiv.Perm n
hσ : ∀ (i j : n), M i j = 0 → Equiv.Perm.permMatrix R σ i j = 0
i : n
hi : i ∈ univ
hi' : ∀ x' ∈ univ, M i (σ i) ≤ M x' (σ x')
N : Matrix n n R := M - M i (σ i) • Equiv.Perm.permMatrix R σ
hMi' : 0 < M i (σ i)
s' : R := s - M i (σ i)
⊢ ∃ w, (∀ (σ : Equiv.Perm n), 0 ≤ w σ) ∧ ∑ σ, w σ • Equiv.Perm.permMatrix R σ = M
Module: Mathlib.Analysis.Convex.Caratheodory

Goal state before:
pp: case refine_4
𝕜 : Type u_1
E : Type u
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
x : E
t : Finset E
ht₁ : ↑t ⊆ s
ht₂ : AffineIndependent 𝕜 Subtype.val
w : E → 𝕜
hw₁ : ∀ y ∈ t, 0 ≤ w y
hw₂ : ∑ y ∈ t, w y = 1
hw₃ : t.centerMass w id = x
t' : Finset E := {i ∈ t | w i ≠ 0}
⊢ ∑ i ∈ t'.attach, w ↑i = 1

Tactic:
Finset.sum_attach,

Goal state after:
case refine_4
𝕜 : Type u_1
E : Type u
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
x : E
t : Finset E
ht₁ : ↑t ⊆ s
ht₂ : AffineIndependent 𝕜 Subtype.val
w : E → 𝕜
hw₁ : ∀ y ∈ t, 0 ≤ w y
hw₂ : ∑ y ∈ t, w y = 1
hw₃ : t.centerMass w id = x
t' : Finset E := {i ∈ t | w i ≠ 0}
⊢ ∑ x ∈ t', w x = 1
Module: Mathlib.Analysis.Convex.Between

Goal state before:
pp: case refine_2
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
h_ne : x ≠ y
t₁ : R
ht₁ : t₁ ∈ Set.Icc 0 1
hx : (lineMap w y) t₁ = x
t₂ : R
ht₂ : t₂ ∈ Set.Icc 0 1
hy : (lineMap x z) t₂ = y
⊢ (lineMap w z) (t₁ * t₂ / (1 - t₁ + t₁ * t₂)) = x

Tactic:
have h_denom : 1 - t₁ + t₁ * t₂ ≠ 0 := by
    contrapose h_ne
    have h1 : t₁ = 1 := by nlinarith [ht₁.1, ht₁.2, ht₂.1, ht₂.2]
    rw [← hx, h1, lineMap_apply_one]

Goal state after:
case refine_2
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
h_ne : x ≠ y
t₁ : R
ht₁ : t₁ ∈ Set.Icc 0 1
hx : (lineMap w y) t₁ = x
t₂ : R
ht₂ : t₂ ∈ Set.Icc 0 1
hy : (lineMap x z) t₂ = y
h_denom : 1 - t₁ + t₁ * t₂ ≠ 0
⊢ (lineMap w z) (t₁ * t₂ / (1 - t₁ + t₁ * t₂)) = x
Module: Mathlib.Analysis.Convex.Between

Goal state before:
pp: case refine_2
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
h_ne : x ≠ y
t₁ : R
ht₁ : t₁ ∈ Set.Icc 0 1
t₂ : R
hx : x -ᵥ w = t₁ • ((t₂ • (z -ᵥ x) +ᵥ x) -ᵥ w)
ht₂ : t₂ ∈ Set.Icc 0 1
hy : (lineMap x z) t₂ = y
h_denom : 1 - t₁ + t₁ * t₂ ≠ 0
⊢ t₁ • ((t₂ • (z -ᵥ x) +ᵥ x) -ᵥ x) + t₁ • t₂ • (x -ᵥ w) = t₁ • t₂ • (z -ᵥ w)

Tactic:
vadd_vsub,

Goal state after:
case refine_2
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
h_ne : x ≠ y
t₁ : R
ht₁ : t₁ ∈ Set.Icc 0 1
t₂ : R
hx : x -ᵥ w = t₁ • ((t₂ • (z -ᵥ x) +ᵥ x) -ᵥ w)
ht₂ : t₂ ∈ Set.Icc 0 1
hy : (lineMap x z) t₂ = y
h_denom : 1 - t₁ + t₁ * t₂ ≠ 0
⊢ t₁ • t₂ • (z -ᵥ x) + t₁ • t₂ • (x -ᵥ w) = t₁ • t₂ • (z -ᵥ w)
Module: Mathlib.Analysis.Convex.Cone.InnerDual

Goal state before:
pp: E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
hx : x ∈ innerDual univ
⊢ x ∈ ⊥

Tactic:
by simpa using hx (mem_univ (-x))

Goal state after:
Module: Mathlib.Analysis.Convex.Cone.Dual

Goal state before:
pp: E : Type u_1
inst✝⁵ : TopologicalSpace E
inst✝⁴ : AddCommGroup E
inst✝³ : IsTopologicalAddGroup E
inst✝² : Module ℝ E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : LocallyConvexSpace ℝ E
K : Set E
C : ProperCone ℝ E
hKconv : Convex ℝ K
hKcomp : IsCompact K
hKC : Disjoint K ↑C
⊢ ∃ f, (∀ x ∈ C, 0 ≤ f x) ∧ ∀ x ∈ K, f x < 0

Tactic:
obtain rfl | ⟨x₀, hx₀⟩ := K.eq_empty_or_nonempty

Goal state after:
case inl
E : Type u_1
inst✝⁵ : TopologicalSpace E
inst✝⁴ : AddCommGroup E
inst✝³ : IsTopologicalAddGroup E
inst✝² : Module ℝ E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : LocallyConvexSpace ℝ E
C : ProperCone ℝ E
hKconv : Convex ℝ ∅
hKcomp : IsCompact ∅
hKC : Disjoint ∅ ↑C
⊢ ∃ f, (∀ x ∈ C, 0 ≤ f x) ∧ ∀ x ∈ ∅, f x < 0

case inr
E : Type u_1
inst✝⁵ : TopologicalSpace E
inst✝⁴ : AddCommGroup E
inst✝³ : IsTopologicalAddGroup E
inst✝² : Module ℝ E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : LocallyConvexSpace ℝ E
K : Set E
C : ProperCone ℝ E
hKconv : Convex ℝ K
hKcomp : IsCompact K
hKC : Disjoint K ↑C
x₀ : E
hx₀ : x₀ ∈ K
⊢ ∃ f, (∀ x ∈ C, 0 ≤ f x) ∧ ∀ x ∈ K, f x < 0
Module: Mathlib.Analysis.Convex.Cone.Dual

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁹ : TopologicalSpace E
inst✝⁸ : AddCommGroup E
inst✝⁷ : IsTopologicalAddGroup E
inst✝⁶ : TopologicalSpace F
inst✝⁵ : AddCommGroup F
inst✝⁴ : Module ℝ E
inst✝³ : ContinuousSMul ℝ E
inst✝² : LocallyConvexSpace ℝ E
inst✝¹ : Module ℝ F
p : E →ₗ[ℝ] F →ₗ[ℝ] ℝ
inst✝ : p.IsContPerfPair
C : ProperCone ℝ E
x : E
⊢ (∀ ⦃x_1 : F⦄, (∀ ⦃x : E⦄, x ∈ C → 0 ≤ (p x) x_1) → 0 ≤ (p.flip x_1) x) → x ∈ C

Tactic:
contrapose!

Goal state after:
E : Type u_1
F : Type u_2
inst✝⁹ : TopologicalSpace E
inst✝⁸ : AddCommGroup E
inst✝⁷ : IsTopologicalAddGroup E
inst✝⁶ : TopologicalSpace F
inst✝⁵ : AddCommGroup F
inst✝⁴ : Module ℝ E
inst✝³ : ContinuousSMul ℝ E
inst✝² : LocallyConvexSpace ℝ E
inst✝¹ : Module ℝ F
p : E →ₗ[ℝ] F →ₗ[ℝ] ℝ
inst✝ : p.IsContPerfPair
C : ProperCone ℝ E
x : E
⊢ x ∉ C → ∃ x_1, (∀ ⦃x : E⦄, x ∈ C → 0 ≤ (p x) x_1) ∧ (p.flip x_1) x < 0
Module: Mathlib.Analysis.Convex.GaugeRescale

Goal state before:
pp: case inr
E : Type u_1
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
inst✝¹ : TopologicalSpace E
inst✝ : T1Space E
s : Set E
hsa : Absorbent ℝ s
hsb : IsVonNBounded ℝ s
x : E
hx : x ≠ 0
⊢ gauge s x ≠ 0

Tactic:
exact ((gauge_pos hsa hsb).2 hx).ne'

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

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
P : Type u_5
inst✝⁴ : Ring 𝕜
inst✝³ : AddCommGroup V
inst✝² : Module 𝕜 V
inst✝¹ : TopologicalSpace P
inst✝ : AddTorsor V P
s : Set P
hs : IsClosed[instTopologicalSpaceSubtype] (Subtype.val ⁻¹' s)
⊢ Subtype.val '' Subtype.val ⁻¹' s = s

Tactic:
image_preimage_eq_of_subset

Goal state after:
𝕜 : Type u_1
V : Type u_2
P : Type u_5
inst✝⁴ : Ring 𝕜
inst✝³ : AddCommGroup V
inst✝² : Module 𝕜 V
inst✝¹ : TopologicalSpace P
inst✝ : AddTorsor V P
s : Set P
hs : IsClosed[instTopologicalSpaceSubtype] (Subtype.val ⁻¹' s)
⊢ s = s

𝕜 : Type u_1
V : Type u_2
P : Type u_5
inst✝⁴ : Ring 𝕜
inst✝³ : AddCommGroup V
inst✝² : Module 𝕜 V
inst✝¹ : TopologicalSpace P
inst✝ : AddTorsor V P
s : Set P
hs : IsClosed[instTopologicalSpaceSubtype] (Subtype.val ⁻¹' s)
⊢ s ⊆ range Subtype.val
Module: Mathlib.Analysis.Convex.Intrinsic

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
W : Type u_3
Q : Type u_4
P : Type u_5
inst✝⁸ : Ring 𝕜
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module 𝕜 V
inst✝⁵ : TopologicalSpace P
inst✝⁴ : AddTorsor V P
inst✝³ : AddCommGroup W
inst✝² : Module 𝕜 W
inst✝¹ : TopologicalSpace Q
inst✝ : AddTorsor W Q
s : Set P
t : Set Q
e : ↥(affineSpan 𝕜 (s ×ˢ t)) ≃ₜ ↥(affineSpan 𝕜 s) × ↥(affineSpan 𝕜 t) :=
  (Homeomorph.setCongr ⋯).trans (Homeomorph.Set.prod ↑(affineSpan 𝕜 s) ↑(affineSpan 𝕜 t))
this : Subtype.val ∘ ⇑e.symm = fun p ↦ (↑p.1, ↑p.2)
h : Subtype.val ⁻¹' s ×ˢ t = ⇑e ⁻¹' (Subtype.val ⁻¹' s) ×ˢ (Subtype.val ⁻¹' t)
⊢ intrinsicInterior 𝕜 (s ×ˢ t) = intrinsicInterior 𝕜 s ×ˢ intrinsicInterior 𝕜 t

Tactic:
simp_rw [intrinsicInterior, h, ← e.preimage_interior, interior_prod_eq, ← e.image_symm,
    ← image_comp, prod_image_image_eq, this]

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

Goal state before:
pp: case inr
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
C : Set E
f : E → ℝ
inst✝ : FiniteDimensional ℝ E
hC : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] C
hf : ConvexOn ℝ C f
x₀ : E
hx₀ : x₀ ∈ C
b : AffineBasis (Fin (Module.finrank ℝ E + 1)) ℝ E
hx₀b : x₀ ∈ interior ((convexHull ℝ) (Set.range ⇑b))
hbC : (convexHull ℝ) (Set.range ⇑b) ⊆ C
⊢ BddAbove (f '' interior ((convexHull ℝ) (Set.range ⇑b)))

Tactic:
exact (hf.bddAbove_convexHull ((subset_convexHull ..).trans hbC)
      ((finite_range _).image _).bddAbove).mono (by gcongr; exact interior_subset)

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

Goal state before:
pp: case f
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 : StdSimplex R P
| f.weights d • ((((d.weights.support.weightedVSubOfPoint id b) fun i ↦ (fun x r ↦ r) i (d.weights i)) +ᵥ b) -ᵥ b)

Tactic:
vadd_vsub,

Goal state after:
case f
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 : StdSimplex R P
| f.weights d • (d.weights.support.weightedVSubOfPoint id b) fun i ↦ (fun x r ↦ r) i (d.weights i)
Module: Mathlib.Analysis.Convex.Radon

Goal state before:
pp: case h_inter
𝕜 : Type u_2
E : Type u_3
inst✝⁵ : Field 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
n k : ℕ
h_card : finrank 𝕜 E + 1 ≤ k
hk :
  ∀ {ι : Type u_1} {F : ι → Set E} {s : Finset ι},
    (∀ i ∈ s, Convex 𝕜 (F i)) →
      (∀ I ⊆ s, #I ≤ finrank 𝕜 E + 1 → (⋂ i ∈ I, F i).Nonempty) → #s = k → (⋂ i ∈ s, F i).Nonempty
ι : Type u_1
F : ι → Set E
s : Finset ι
h_convex : ∀ i ∈ s, Convex 𝕜 (F i)
h_inter : ∀ I ⊆ s, #I ≤ finrank 𝕜 E + 1 → (⋂ i ∈ I, F i).Nonempty
hn : #s = k + 1
i : ↥s
⊢ ∀ I ⊆ s.erase ↑i, #I ≤ finrank 𝕜 E + 1 → (⋂ i ∈ I, F i).Nonempty

pp: case hn
𝕜 : Type u_2
E : Type u_3
inst✝⁵ : Field 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
n k : ℕ
h_card : finrank 𝕜 E + 1 ≤ k
hk :
  ∀ {ι : Type u_1} {F : ι → Set E} {s : Finset ι},
    (∀ i ∈ s, Convex 𝕜 (F i)) →
      (∀ I ⊆ s, #I ≤ finrank 𝕜 E + 1 → (⋂ i ∈ I, F i).Nonempty) → #s = k → (⋂ i ∈ s, F i).Nonempty
ι : Type u_1
F : ι → Set E
s : Finset ι
h_convex : ∀ i ∈ s, Convex 𝕜 (F i)
h_inter : ∀ I ⊆ s, #I ≤ finrank 𝕜 E + 1 → (⋂ i ∈ I, F i).Nonempty
hn : #s = k + 1
i : ↥s
⊢ #(s.erase ↑i) = k

Tactic:
· intro J hJ_ss hJ_card
      exact h_inter J (subset_trans hJ_ss (erase_subset i.val s)) hJ_card

Goal state after:
case hn
𝕜 : Type u_2
E : Type u_3
inst✝⁵ : Field 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
n k : ℕ
h_card : finrank 𝕜 E + 1 ≤ k
hk :
  ∀ {ι : Type u_1} {F : ι → Set E} {s : Finset ι},
    (∀ i ∈ s, Convex 𝕜 (F i)) →
      (∀ I ⊆ s, #I ≤ finrank 𝕜 E + 1 → (⋂ i ∈ I, F i).Nonempty) → #s = k → (⋂ i ∈ s, F i).Nonempty
ι : Type u_1
F : ι → Set E
s : Finset ι
h_convex : ∀ i ∈ s, Convex 𝕜 (F i)
h_inter : ∀ I ⊆ s, #I ≤ finrank 𝕜 E + 1 → (⋂ i ∈ I, F i).Nonempty
hn : #s = k + 1
i : ↥s
⊢ #(s.erase ↑i) = k
Module: Mathlib.Analysis.Convex.SimplicialComplex.AffineIndependentUnion

Goal state before:
pp: ι : Type u_1
inst✝ : DecidableEq ι
G : SimpleGraph ι
v : ι
⊢ (∃ v_1, {v} = {v_1}) ∨ ∃ x ∈ G.edgeSet, x.toFinset = {v}

Tactic:
exact Or.inl ⟨v, rfl⟩

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

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
inst✝⁵ : Field 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
F : Finset (Set E)
h_card : finrank 𝕜 E + 1 ≤ #F
h_convex : ∀ X ∈ F, Convex 𝕜 X
h_inter : ∀ G ⊆ F, #G = finrank 𝕜 E + 1 → (⋂₀ ↑G).Nonempty
⊢ (⋂₀ ↑F).Nonempty

Tactic:
by
  apply helly_theorem_set' h_convex
  intro I hI_ss hI_card
  obtain ⟨J, _, hJ_ss, hJ_card⟩ := exists_subsuperset_card_eq hI_ss hI_card h_card
  have : ⋂₀ (J : Set (Set E)) ⊆ ⋂₀ I := sInter_mono (by simpa [hI_ss])
  apply Set.Nonempty.mono this
  exact h_inter J hJ_ss (by lia)

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

Goal state before:
pp: case neg
ι : Type u_1
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 : ι → Set E
h_card : ↑(finrank 𝕜 E) + 1 ≤ ENat.card ι
h_convex : ∀ (i : ι), Convex 𝕜 (F i)
h_compact : ∀ (i : ι), IsCompact (F i)
h_inter : ∀ (I : Finset ι), #I = finrank 𝕜 E + 1 → (⋂ i ∈ I, F i).Nonempty
I : Finset ι
hI_card : #I ≤ finrank 𝕜 E + 1
h : ¬Infinite ι
this✝ : Finite ι
this : Fintype ι
⊢ ∃ J, I ⊆ J ∧ #J = finrank 𝕜 E + 1

Tactic:
apply exists_superset_card_eq hI_card

Goal state after:
case neg
ι : Type u_1
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 : ι → Set E
h_card : ↑(finrank 𝕜 E) + 1 ≤ ENat.card ι
h_convex : ∀ (i : ι), Convex 𝕜 (F i)
h_compact : ∀ (i : ι), IsCompact (F i)
h_inter : ∀ (I : Finset ι), #I = finrank 𝕜 E + 1 → (⋂ i ∈ I, F i).Nonempty
I : Finset ι
hI_card : #I ≤ finrank 𝕜 E + 1
h : ¬Infinite ι
this✝ : Finite ι
this : Fintype ι
⊢ finrank 𝕜 E + 1 ≤ Fintype.card ι
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : CommRing R
inst✝⁴ : PartialOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
p₁ p₂ x : P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
t : R
ht : t ≤ 0
⊢ SameRay R ((t • (x -ᵥ p₁) +ᵥ p₂) -ᵥ p₂) (p₁ -ᵥ x)

Tactic:
vadd_vsub,

Goal state after:
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : CommRing R
inst✝⁴ : PartialOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
p₁ p₂ x : P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
t : R
ht : t ≤ 0
⊢ SameRay R (t • (x -ᵥ p₁)) (p₁ -ᵥ x)
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : CommRing 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 p₂ : P
m : V
c₁ c₂ : R
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
h₁ : x -ᵥ p₁ = c₁ • m
h₂ : y -ᵥ p₂ = c₂ • m
hc : c₁ * c₂ ≤ 0
⊢ p₂ -ᵥ y = -c₂ • m

Tactic:
rw [← neg_vsub_eq_vsub_rev, h₂, neg_smul]

Goal state after: