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https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope_of_Vpolytope_subsingleton | [478, 1] | [490, 7] | rcases @origin_Hpolytope E _ _ _ _ with ⟨ H_, hH_Fin, hH_ ⟩ | case inr.intro
E : Type
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hStrivial : Set.Subsingleton S
x : E
hx : S = {x}
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case inr.intro.intro.intro
E : Type
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hStrivial : Set.Subsingleton S
x : E
hx : S = {x}
H_ : Set (Halfspace E)
hH_Fin : Set.Finite H_
hH_ : Hpolytope hH_Fin = {0}
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope_of_Vpolytope_subsingleton | [478, 1] | [490, 7] | refine ⟨ Halfspace_translation x '' H_, hH_Fin.image (Halfspace_translation x), ?_ ⟩ | case inr.intro.intro.intro
E : Type
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hStrivial : Set.Subsingleton S
x : E
hx : S = {x}
H_ : Set (Halfspace E)
hH_Fin : Set.Finite H_
hH_ : Hpolytope hH_Fin = {0}
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case inr.intro.intro.intro
E : Type
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hStrivial : Set.Subsingleton S
x : E
hx : S = {x}
H_ : Set (Halfspace E)
hH_Fin : Set.Finite H_
hH_ : Hpolytope hH_Fin = {0}
⊢ Hpolytope ⋯ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope_of_Vpolytope_subsingleton | [478, 1] | [490, 7] | rw [Vpolytope, hx, convexHull_singleton, Hpolytope_translation hH_Fin, hH_, Set.singleton_add_singleton, zero_add] | case inr.intro.intro.intro
E : Type
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hStrivial : Set.Subsingleton S
x : E
hx : S = {x}
H_ : Set (Halfspace E)
hH_Fin : Set.Finite H_
hH_ : Hpolytope hH_Fin = {0}
⊢ Hpolytope ⋯ = Vpolytope hS | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope_of_Vpolytope_0interior | [492, 1] | [503, 7] | rcases DualOfVpolytope_compactHpolytope hS hV0 with ⟨ H_, hH_, hH_eq, hH_cpt ⟩ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hV0 : 0 ∈ interior (Vpolytope hS)
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hV0 : 0 ∈ interior (Vpolytope hS)
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hH_eq : Hpolytope hH_ = polarDual (Vpolytope hS)
hH_cpt : IsCompact (Hpolytope hH_)
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope_of_Vpolytope_0interior | [492, 1] | [503, 7] | rcases Vpolytope_of_Hpolytope hH_ hH_cpt with ⟨ S', hS', hS'eq ⟩ | case intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hV0 : 0 ∈ interior (Vpolytope hS)
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hH_eq : Hpolytope hH_ = polarDual (Vpolytope hS)
hH_cpt : IsCompact (Hpolytope hH_)
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hV0 : 0 ∈ interior (Vpolytope hS)
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hH_eq : Hpolytope hH_ = polarDual (Vpolytope hS)
hH_cpt : IsCompact (Hpolytope hH_)
S' : Set E
hS' : Set.Finite S'
hS'eq : Hpolytope hH_ = Vpolytope hS'
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope_of_Vpolytope_0interior | [492, 1] | [503, 7] | have : 0 ∈ interior (Vpolytope hS') := by
rw [←hS'eq, hH_eq, compact_polarDual_iff (Closed_Vpolytope hS)]
exact Compact_Vpolytope hS | case intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hV0 : 0 ∈ interior (Vpolytope hS)
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hH_eq : Hpolytope hH_ = polarDual (Vpolytope hS)
hH_cpt : IsCompact (Hpolytope hH_)
S' : Set E
hS' : Set.Finite S'
hS'eq : Hpolytope hH_ = Vpolytope hS'
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hV0 : 0 ∈ interior (Vpolytope hS)
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hH_eq : Hpolytope hH_ = polarDual (Vpolytope hS)
hH_cpt : IsCompact (Hpolytope hH_)
S' : Set E
hS' : Set.Finite S'
hS'eq : Hpolytope hH_ = Vpolytope hS'
this : 0 ∈ interior (Vpolytope hS')
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope_of_Vpolytope_0interior | [492, 1] | [503, 7] | rcases DualOfVpolytope_compactHpolytope hS' this with ⟨ H_', hH_', hH_'eq, _ ⟩ | case intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hV0 : 0 ∈ interior (Vpolytope hS)
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hH_eq : Hpolytope hH_ = polarDual (Vpolytope hS)
hH_cpt : IsCompact (Hpolytope hH_)
S' : Set E
hS' : Set.Finite S'
hS'eq : Hpolytope hH_ = Vpolytope hS'
this : 0 ∈ interior (Vpolytope hS')
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case intro.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hV0 : 0 ∈ interior (Vpolytope hS)
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hH_eq : Hpolytope hH_ = polarDual (Vpolytope hS)
hH_cpt : IsCompact (Hpolytope hH_)
S' : Set E
hS' : Set.Finite S'
hS'eq : Hpolytope hH_ = Vpolytope hS'
this : 0 ∈ interior (Vpolytope hS')
H_' : Set (Halfspace E)
hH_' : Set.Finite H_'
hH_'eq : Hpolytope hH_' = polarDual (Vpolytope hS')
right✝ : IsCompact (Hpolytope hH_')
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope_of_Vpolytope_0interior | [492, 1] | [503, 7] | refine ⟨ H_', hH_', ?_ ⟩ | case intro.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hV0 : 0 ∈ interior (Vpolytope hS)
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hH_eq : Hpolytope hH_ = polarDual (Vpolytope hS)
hH_cpt : IsCompact (Hpolytope hH_)
S' : Set E
hS' : Set.Finite S'
hS'eq : Hpolytope hH_ = Vpolytope hS'
this : 0 ∈ interior (Vpolytope hS')
H_' : Set (Halfspace E)
hH_' : Set.Finite H_'
hH_'eq : Hpolytope hH_' = polarDual (Vpolytope hS')
right✝ : IsCompact (Hpolytope hH_')
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case intro.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hV0 : 0 ∈ interior (Vpolytope hS)
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hH_eq : Hpolytope hH_ = polarDual (Vpolytope hS)
hH_cpt : IsCompact (Hpolytope hH_)
S' : Set E
hS' : Set.Finite S'
hS'eq : Hpolytope hH_ = Vpolytope hS'
this : 0 ∈ interior (Vpolytope hS')
H_' : Set (Halfspace E)
hH_' : Set.Finite H_'
hH_'eq : Hpolytope hH_' = polarDual (Vpolytope hS')
right✝ : IsCompact (Hpolytope hH_')
⊢ Hpolytope hH_' = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope_of_Vpolytope_0interior | [492, 1] | [503, 7] | rw [hH_'eq, ←hS'eq, hH_eq, doublePolarDual_self (Closed_Vpolytope hS) (Convex_Vpolytope hS) (interior_subset hV0)] | case intro.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hV0 : 0 ∈ interior (Vpolytope hS)
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hH_eq : Hpolytope hH_ = polarDual (Vpolytope hS)
hH_cpt : IsCompact (Hpolytope hH_)
S' : Set E
hS' : Set.Finite S'
hS'eq : Hpolytope hH_ = Vpolytope hS'
this : 0 ∈ interior (Vpolytope hS')
H_' : Set (Halfspace E)
hH_' : Set.Finite H_'
hH_'eq : Hpolytope hH_' = polarDual (Vpolytope hS')
right✝ : IsCompact (Hpolytope hH_')
⊢ Hpolytope hH_' = Vpolytope hS | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope_of_Vpolytope_0interior | [492, 1] | [503, 7] | rw [←hS'eq, hH_eq, compact_polarDual_iff (Closed_Vpolytope hS)] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hV0 : 0 ∈ interior (Vpolytope hS)
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hH_eq : Hpolytope hH_ = polarDual (Vpolytope hS)
hH_cpt : IsCompact (Hpolytope hH_)
S' : Set E
hS' : Set.Finite S'
hS'eq : Hpolytope hH_ = Vpolytope hS'
⊢ 0 ∈ interior (Vpolytope hS') | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hV0 : 0 ∈ interior (Vpolytope hS)
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hH_eq : Hpolytope hH_ = polarDual (Vpolytope hS)
hH_cpt : IsCompact (Hpolytope hH_)
S' : Set E
hS' : Set.Finite S'
hS'eq : Hpolytope hH_ = Vpolytope hS'
⊢ IsCompact (Vpolytope hS) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope_of_Vpolytope_0interior | [492, 1] | [503, 7] | exact Compact_Vpolytope hS | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hV0 : 0 ∈ interior (Vpolytope hS)
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hH_eq : Hpolytope hH_ = polarDual (Vpolytope hS)
hH_cpt : IsCompact (Hpolytope hH_)
S' : Set E
hS' : Set.Finite S'
hS'eq : Hpolytope hH_ = Vpolytope hS'
⊢ IsCompact (Vpolytope hS) | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | translationHomeo | [505, 1] | [511, 37] | simp | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x y : E
⊢ (fun x_1 => x_1 + -x) ((fun x_1 => x_1 + x) y) = y | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | translationHomeo | [505, 1] | [511, 37] | simp | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x y : E
⊢ (fun x_1 => x_1 + x) ((fun x_1 => x_1 + -x) y) = y | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | translationHomeo | [505, 1] | [511, 37] | continuity | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
⊢ Continuous { toFun := fun x_1 => x_1 + x, invFun := fun x_1 => x_1 + -x, left_inv := ⋯, right_inv := ⋯ }.toFun | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | translationHomeo | [505, 1] | [511, 37] | continuity | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
⊢ Continuous { toFun := fun x_1 => x_1 + x, invFun := fun x_1 => x_1 + -x, left_inv := ⋯, right_inv := ⋯ }.invFun | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | translationHomeo.toFun.def | [513, 1] | [517, 7] | unfold translationHomeo | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
⊢ ⇑(translationHomeo x) = fun x_1 => x_1 + x | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
⊢ ⇑{ toEquiv := { toFun := fun x_1 => x_1 + x, invFun := fun x_1 => x_1 + -x, left_inv := ⋯, right_inv := ⋯ },
continuous_toFun := ⋯, continuous_invFun := ⋯ } =
fun x_1 => x_1 + x |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | translationHomeo.toFun.def | [513, 1] | [517, 7] | simp | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x : E
⊢ ⇑{ toEquiv := { toFun := fun x_1 => x_1 + x, invFun := fun x_1 => x_1 + -x, left_inv := ⋯, right_inv := ⋯ },
continuous_toFun := ⋯, continuous_invFun := ⋯ } =
fun x_1 => x_1 + x | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope_of_Vpolytope_interior | [519, 1] | [542, 7] | let S' := S + {-hVinteriorNonempty.some} | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinteriorNonempty : Set.Nonempty (interior (Vpolytope hS))
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinteriorNonempty : Set.Nonempty (interior (Vpolytope hS))
S' : Set E := S + {-Set.Nonempty.some hVinteriorNonempty}
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope_of_Vpolytope_interior | [519, 1] | [542, 7] | have hS' : S'.Finite := by exact (hS.translation (-hVinteriorNonempty.some)) | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinteriorNonempty : Set.Nonempty (interior (Vpolytope hS))
S' : Set E := S + {-Set.Nonempty.some hVinteriorNonempty}
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinteriorNonempty : Set.Nonempty (interior (Vpolytope hS))
S' : Set E := S + {-Set.Nonempty.some hVinteriorNonempty}
hS' : Set.Finite S'
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope_of_Vpolytope_interior | [519, 1] | [542, 7] | have : 0 ∈ interior (Vpolytope hS') := by
rw [Vpolytope_translation hS, Set.add_singleton, ]
have := @Homeomorph.image_interior _ _ _ _ (translationHomeo (-hVinteriorNonempty.some)) (Vpolytope hS)
rw [translationHomeo.toFun.def] at this
rw [← this]; clear this
rw [← Set.add_singleton, Set.mem_translation, zero_sub, neg_neg]
exact hVinteriorNonempty.some_mem
done | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinteriorNonempty : Set.Nonempty (interior (Vpolytope hS))
S' : Set E := S + {-Set.Nonempty.some hVinteriorNonempty}
hS' : Set.Finite S'
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinteriorNonempty : Set.Nonempty (interior (Vpolytope hS))
S' : Set E := S + {-Set.Nonempty.some hVinteriorNonempty}
hS' : Set.Finite S'
this : 0 ∈ interior (Vpolytope hS')
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope_of_Vpolytope_interior | [519, 1] | [542, 7] | rcases Hpolytope_of_Vpolytope_0interior hS' this with ⟨ H_', hH_', hH_'eq ⟩ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinteriorNonempty : Set.Nonempty (interior (Vpolytope hS))
S' : Set E := S + {-Set.Nonempty.some hVinteriorNonempty}
hS' : Set.Finite S'
this : 0 ∈ interior (Vpolytope hS')
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinteriorNonempty : Set.Nonempty (interior (Vpolytope hS))
S' : Set E := S + {-Set.Nonempty.some hVinteriorNonempty}
hS' : Set.Finite S'
this : 0 ∈ interior (Vpolytope hS')
H_' : Set (Halfspace E)
hH_' : Set.Finite H_'
hH_'eq : Hpolytope hH_' = Vpolytope hS'
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope_of_Vpolytope_interior | [519, 1] | [542, 7] | let H_ := (Halfspace_translation hVinteriorNonempty.some) '' H_' | case intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinteriorNonempty : Set.Nonempty (interior (Vpolytope hS))
S' : Set E := S + {-Set.Nonempty.some hVinteriorNonempty}
hS' : Set.Finite S'
this : 0 ∈ interior (Vpolytope hS')
H_' : Set (Halfspace E)
hH_' : Set.Finite H_'
hH_'eq : Hpolytope hH_' = Vpolytope hS'
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinteriorNonempty : Set.Nonempty (interior (Vpolytope hS))
S' : Set E := S + {-Set.Nonempty.some hVinteriorNonempty}
hS' : Set.Finite S'
this : 0 ∈ interior (Vpolytope hS')
H_' : Set (Halfspace E)
hH_' : Set.Finite H_'
hH_'eq : Hpolytope hH_' = Vpolytope hS'
H_ : Set (Halfspace E) := Halfspace_translation (Set.Nonempty.some hVinteriorNonempty) '' H_'
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope_of_Vpolytope_interior | [519, 1] | [542, 7] | have hH_ : H_.Finite := hH_'.image _ | case intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinteriorNonempty : Set.Nonempty (interior (Vpolytope hS))
S' : Set E := S + {-Set.Nonempty.some hVinteriorNonempty}
hS' : Set.Finite S'
this : 0 ∈ interior (Vpolytope hS')
H_' : Set (Halfspace E)
hH_' : Set.Finite H_'
hH_'eq : Hpolytope hH_' = Vpolytope hS'
H_ : Set (Halfspace E) := Halfspace_translation (Set.Nonempty.some hVinteriorNonempty) '' H_'
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinteriorNonempty : Set.Nonempty (interior (Vpolytope hS))
S' : Set E := S + {-Set.Nonempty.some hVinteriorNonempty}
hS' : Set.Finite S'
this : 0 ∈ interior (Vpolytope hS')
H_' : Set (Halfspace E)
hH_' : Set.Finite H_'
hH_'eq : Hpolytope hH_' = Vpolytope hS'
H_ : Set (Halfspace E) := Halfspace_translation (Set.Nonempty.some hVinteriorNonempty) '' H_'
hH_ : Set.Finite H_
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope_of_Vpolytope_interior | [519, 1] | [542, 7] | refine ⟨ H_, hH_, ?_ ⟩ | case intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinteriorNonempty : Set.Nonempty (interior (Vpolytope hS))
S' : Set E := S + {-Set.Nonempty.some hVinteriorNonempty}
hS' : Set.Finite S'
this : 0 ∈ interior (Vpolytope hS')
H_' : Set (Halfspace E)
hH_' : Set.Finite H_'
hH_'eq : Hpolytope hH_' = Vpolytope hS'
H_ : Set (Halfspace E) := Halfspace_translation (Set.Nonempty.some hVinteriorNonempty) '' H_'
hH_ : Set.Finite H_
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinteriorNonempty : Set.Nonempty (interior (Vpolytope hS))
S' : Set E := S + {-Set.Nonempty.some hVinteriorNonempty}
hS' : Set.Finite S'
this : 0 ∈ interior (Vpolytope hS')
H_' : Set (Halfspace E)
hH_' : Set.Finite H_'
hH_'eq : Hpolytope hH_' = Vpolytope hS'
H_ : Set (Halfspace E) := Halfspace_translation (Set.Nonempty.some hVinteriorNonempty) '' H_'
hH_ : Set.Finite H_
⊢ Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope_of_Vpolytope_interior | [519, 1] | [542, 7] | ext x | case intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinteriorNonempty : Set.Nonempty (interior (Vpolytope hS))
S' : Set E := S + {-Set.Nonempty.some hVinteriorNonempty}
hS' : Set.Finite S'
this : 0 ∈ interior (Vpolytope hS')
H_' : Set (Halfspace E)
hH_' : Set.Finite H_'
hH_'eq : Hpolytope hH_' = Vpolytope hS'
H_ : Set (Halfspace E) := Halfspace_translation (Set.Nonempty.some hVinteriorNonempty) '' H_'
hH_ : Set.Finite H_
⊢ Hpolytope hH_ = Vpolytope hS | case intro.intro.h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinteriorNonempty : Set.Nonempty (interior (Vpolytope hS))
S' : Set E := S + {-Set.Nonempty.some hVinteriorNonempty}
hS' : Set.Finite S'
this : 0 ∈ interior (Vpolytope hS')
H_' : Set (Halfspace E)
hH_' : Set.Finite H_'
hH_'eq : Hpolytope hH_' = Vpolytope hS'
H_ : Set (Halfspace E) := Halfspace_translation (Set.Nonempty.some hVinteriorNonempty) '' H_'
hH_ : Set.Finite H_
x : E
⊢ x ∈ Hpolytope hH_ ↔ x ∈ Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope_of_Vpolytope_interior | [519, 1] | [542, 7] | rw [Hpolytope_translation, hH_'eq, Vpolytope_translation hS, ← Set.sub_eq_neg_add,
Set.neg_add_cancel_right' hVinteriorNonempty.some] | case intro.intro.h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinteriorNonempty : Set.Nonempty (interior (Vpolytope hS))
S' : Set E := S + {-Set.Nonempty.some hVinteriorNonempty}
hS' : Set.Finite S'
this : 0 ∈ interior (Vpolytope hS')
H_' : Set (Halfspace E)
hH_' : Set.Finite H_'
hH_'eq : Hpolytope hH_' = Vpolytope hS'
H_ : Set (Halfspace E) := Halfspace_translation (Set.Nonempty.some hVinteriorNonempty) '' H_'
hH_ : Set.Finite H_
x : E
⊢ x ∈ Hpolytope hH_ ↔ x ∈ Vpolytope hS | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope_of_Vpolytope_interior | [519, 1] | [542, 7] | exact (hS.translation (-hVinteriorNonempty.some)) | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinteriorNonempty : Set.Nonempty (interior (Vpolytope hS))
S' : Set E := S + {-Set.Nonempty.some hVinteriorNonempty}
⊢ Set.Finite S' | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope_of_Vpolytope_interior | [519, 1] | [542, 7] | rw [Vpolytope_translation hS, Set.add_singleton, ] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinteriorNonempty : Set.Nonempty (interior (Vpolytope hS))
S' : Set E := S + {-Set.Nonempty.some hVinteriorNonempty}
hS' : Set.Finite S'
⊢ 0 ∈ interior (Vpolytope hS') | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinteriorNonempty : Set.Nonempty (interior (Vpolytope hS))
S' : Set E := S + {-Set.Nonempty.some hVinteriorNonempty}
hS' : Set.Finite S'
⊢ 0 ∈ interior ((fun x => x + -Set.Nonempty.some hVinteriorNonempty) '' Vpolytope hS) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope_of_Vpolytope_interior | [519, 1] | [542, 7] | have := @Homeomorph.image_interior _ _ _ _ (translationHomeo (-hVinteriorNonempty.some)) (Vpolytope hS) | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinteriorNonempty : Set.Nonempty (interior (Vpolytope hS))
S' : Set E := S + {-Set.Nonempty.some hVinteriorNonempty}
hS' : Set.Finite S'
⊢ 0 ∈ interior ((fun x => x + -Set.Nonempty.some hVinteriorNonempty) '' Vpolytope hS) | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinteriorNonempty : Set.Nonempty (interior (Vpolytope hS))
S' : Set E := S + {-Set.Nonempty.some hVinteriorNonempty}
hS' : Set.Finite S'
this :
⇑(translationHomeo (-Set.Nonempty.some hVinteriorNonempty)) '' interior (Vpolytope hS) =
interior (⇑(translationHomeo (-Set.Nonempty.some hVinteriorNonempty)) '' Vpolytope hS)
⊢ 0 ∈ interior ((fun x => x + -Set.Nonempty.some hVinteriorNonempty) '' Vpolytope hS) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope_of_Vpolytope_interior | [519, 1] | [542, 7] | rw [translationHomeo.toFun.def] at this | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinteriorNonempty : Set.Nonempty (interior (Vpolytope hS))
S' : Set E := S + {-Set.Nonempty.some hVinteriorNonempty}
hS' : Set.Finite S'
this :
⇑(translationHomeo (-Set.Nonempty.some hVinteriorNonempty)) '' interior (Vpolytope hS) =
interior (⇑(translationHomeo (-Set.Nonempty.some hVinteriorNonempty)) '' Vpolytope hS)
⊢ 0 ∈ interior ((fun x => x + -Set.Nonempty.some hVinteriorNonempty) '' Vpolytope hS) | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinteriorNonempty : Set.Nonempty (interior (Vpolytope hS))
S' : Set E := S + {-Set.Nonempty.some hVinteriorNonempty}
hS' : Set.Finite S'
this :
(fun x => x + -Set.Nonempty.some hVinteriorNonempty) '' interior (Vpolytope hS) =
interior ((fun x => x + -Set.Nonempty.some hVinteriorNonempty) '' Vpolytope hS)
⊢ 0 ∈ interior ((fun x => x + -Set.Nonempty.some hVinteriorNonempty) '' Vpolytope hS) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope_of_Vpolytope_interior | [519, 1] | [542, 7] | rw [← this] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinteriorNonempty : Set.Nonempty (interior (Vpolytope hS))
S' : Set E := S + {-Set.Nonempty.some hVinteriorNonempty}
hS' : Set.Finite S'
this :
(fun x => x + -Set.Nonempty.some hVinteriorNonempty) '' interior (Vpolytope hS) =
interior ((fun x => x + -Set.Nonempty.some hVinteriorNonempty) '' Vpolytope hS)
⊢ 0 ∈ interior ((fun x => x + -Set.Nonempty.some hVinteriorNonempty) '' Vpolytope hS) | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinteriorNonempty : Set.Nonempty (interior (Vpolytope hS))
S' : Set E := S + {-Set.Nonempty.some hVinteriorNonempty}
hS' : Set.Finite S'
this :
(fun x => x + -Set.Nonempty.some hVinteriorNonempty) '' interior (Vpolytope hS) =
interior ((fun x => x + -Set.Nonempty.some hVinteriorNonempty) '' Vpolytope hS)
⊢ 0 ∈ (fun x => x + -Set.Nonempty.some hVinteriorNonempty) '' interior (Vpolytope hS) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope_of_Vpolytope_interior | [519, 1] | [542, 7] | clear this | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinteriorNonempty : Set.Nonempty (interior (Vpolytope hS))
S' : Set E := S + {-Set.Nonempty.some hVinteriorNonempty}
hS' : Set.Finite S'
this :
(fun x => x + -Set.Nonempty.some hVinteriorNonempty) '' interior (Vpolytope hS) =
interior ((fun x => x + -Set.Nonempty.some hVinteriorNonempty) '' Vpolytope hS)
⊢ 0 ∈ (fun x => x + -Set.Nonempty.some hVinteriorNonempty) '' interior (Vpolytope hS) | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinteriorNonempty : Set.Nonempty (interior (Vpolytope hS))
S' : Set E := S + {-Set.Nonempty.some hVinteriorNonempty}
hS' : Set.Finite S'
⊢ 0 ∈ (fun x => x + -Set.Nonempty.some hVinteriorNonempty) '' interior (Vpolytope hS) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope_of_Vpolytope_interior | [519, 1] | [542, 7] | rw [← Set.add_singleton, Set.mem_translation, zero_sub, neg_neg] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinteriorNonempty : Set.Nonempty (interior (Vpolytope hS))
S' : Set E := S + {-Set.Nonempty.some hVinteriorNonempty}
hS' : Set.Finite S'
⊢ 0 ∈ (fun x => x + -Set.Nonempty.some hVinteriorNonempty) '' interior (Vpolytope hS) | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinteriorNonempty : Set.Nonempty (interior (Vpolytope hS))
S' : Set E := S + {-Set.Nonempty.some hVinteriorNonempty}
hS' : Set.Finite S'
⊢ Set.Nonempty.some hVinteriorNonempty ∈ interior (Vpolytope hS) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope_of_Vpolytope_interior | [519, 1] | [542, 7] | exact hVinteriorNonempty.some_mem | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinteriorNonempty : Set.Nonempty (interior (Vpolytope hS))
S' : Set E := S + {-Set.Nonempty.some hVinteriorNonempty}
hS' : Set.Finite S'
⊢ Set.Nonempty.some hVinteriorNonempty ∈ interior (Vpolytope hS) | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Vpolytope_inter_cutSpace_fin | [547, 1] | [556, 7] | rcases Hpolytope_of_Vpolytope_interior _ hVinterior with ⟨ H_', hH_', hHV ⟩ | E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : InnerProductSpace ℝ E✝
inst✝⁶ : CompleteSpace E✝
E P : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : PseudoMetricSpace P
inst✝¹ : NormedAddTorsor E P
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinterior : Set.Nonempty (interior (Vpolytope hS))
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
⊢ ∃ S', ∃ (hS' : Set.Finite S'), Vpolytope hS' = Vpolytope hS ∩ Hpolytope hH_ | case intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : InnerProductSpace ℝ E✝
inst✝⁶ : CompleteSpace E✝
E P : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : PseudoMetricSpace P
inst✝¹ : NormedAddTorsor E P
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinterior : Set.Nonempty (interior (Vpolytope hS))
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
H_' : Set (Halfspace E)
hH_' : Set.Finite H_'
hHV : Hpolytope hH_' = Vpolytope hS
⊢ ∃ S', ∃ (hS' : Set.Finite S'), Vpolytope hS' = Vpolytope hS ∩ Hpolytope hH_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Vpolytope_inter_cutSpace_fin | [547, 1] | [556, 7] | have hH_inter := inter_Hpolytope H_' H_ hH_' hH_ | case intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : InnerProductSpace ℝ E✝
inst✝⁶ : CompleteSpace E✝
E P : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : PseudoMetricSpace P
inst✝¹ : NormedAddTorsor E P
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinterior : Set.Nonempty (interior (Vpolytope hS))
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
H_' : Set (Halfspace E)
hH_' : Set.Finite H_'
hHV : Hpolytope hH_' = Vpolytope hS
⊢ ∃ S', ∃ (hS' : Set.Finite S'), Vpolytope hS' = Vpolytope hS ∩ Hpolytope hH_ | case intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : InnerProductSpace ℝ E✝
inst✝⁶ : CompleteSpace E✝
E P : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : PseudoMetricSpace P
inst✝¹ : NormedAddTorsor E P
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinterior : Set.Nonempty (interior (Vpolytope hS))
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
H_' : Set (Halfspace E)
hH_' : Set.Finite H_'
hHV : Hpolytope hH_' = Vpolytope hS
hH_inter : Hpolytope ⋯ = Hpolytope hH_' ∩ Hpolytope hH_
⊢ ∃ S', ∃ (hS' : Set.Finite S'), Vpolytope hS' = Vpolytope hS ∩ Hpolytope hH_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Vpolytope_inter_cutSpace_fin | [547, 1] | [556, 7] | have : IsCompact (Vpolytope hS ∩ Hpolytope hH_) := (Compact_Vpolytope hS).inter_right (Closed_cutSpace H_) | case intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : InnerProductSpace ℝ E✝
inst✝⁶ : CompleteSpace E✝
E P : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : PseudoMetricSpace P
inst✝¹ : NormedAddTorsor E P
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinterior : Set.Nonempty (interior (Vpolytope hS))
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
H_' : Set (Halfspace E)
hH_' : Set.Finite H_'
hHV : Hpolytope hH_' = Vpolytope hS
hH_inter : Hpolytope ⋯ = Hpolytope hH_' ∩ Hpolytope hH_
⊢ ∃ S', ∃ (hS' : Set.Finite S'), Vpolytope hS' = Vpolytope hS ∩ Hpolytope hH_ | case intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : InnerProductSpace ℝ E✝
inst✝⁶ : CompleteSpace E✝
E P : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : PseudoMetricSpace P
inst✝¹ : NormedAddTorsor E P
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinterior : Set.Nonempty (interior (Vpolytope hS))
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
H_' : Set (Halfspace E)
hH_' : Set.Finite H_'
hHV : Hpolytope hH_' = Vpolytope hS
hH_inter : Hpolytope ⋯ = Hpolytope hH_' ∩ Hpolytope hH_
this : IsCompact (Vpolytope hS ∩ Hpolytope hH_)
⊢ ∃ S', ∃ (hS' : Set.Finite S'), Vpolytope hS' = Vpolytope hS ∩ Hpolytope hH_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Vpolytope_inter_cutSpace_fin | [547, 1] | [556, 7] | rw [← hHV, ← hH_inter] at this | case intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : InnerProductSpace ℝ E✝
inst✝⁶ : CompleteSpace E✝
E P : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : PseudoMetricSpace P
inst✝¹ : NormedAddTorsor E P
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinterior : Set.Nonempty (interior (Vpolytope hS))
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
H_' : Set (Halfspace E)
hH_' : Set.Finite H_'
hHV : Hpolytope hH_' = Vpolytope hS
hH_inter : Hpolytope ⋯ = Hpolytope hH_' ∩ Hpolytope hH_
this : IsCompact (Vpolytope hS ∩ Hpolytope hH_)
⊢ ∃ S', ∃ (hS' : Set.Finite S'), Vpolytope hS' = Vpolytope hS ∩ Hpolytope hH_ | case intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : InnerProductSpace ℝ E✝
inst✝⁶ : CompleteSpace E✝
E P : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : PseudoMetricSpace P
inst✝¹ : NormedAddTorsor E P
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinterior : Set.Nonempty (interior (Vpolytope hS))
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
H_' : Set (Halfspace E)
hH_' : Set.Finite H_'
hHV : Hpolytope hH_' = Vpolytope hS
hH_inter : Hpolytope ⋯ = Hpolytope hH_' ∩ Hpolytope hH_
this : IsCompact (Hpolytope ⋯)
⊢ ∃ S', ∃ (hS' : Set.Finite S'), Vpolytope hS' = Vpolytope hS ∩ Hpolytope hH_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Vpolytope_inter_cutSpace_fin | [547, 1] | [556, 7] | rcases Vpolytope_of_Hpolytope (hH_'.union hH_) this with ⟨ S', hS', hSV ⟩ | case intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : InnerProductSpace ℝ E✝
inst✝⁶ : CompleteSpace E✝
E P : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : PseudoMetricSpace P
inst✝¹ : NormedAddTorsor E P
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinterior : Set.Nonempty (interior (Vpolytope hS))
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
H_' : Set (Halfspace E)
hH_' : Set.Finite H_'
hHV : Hpolytope hH_' = Vpolytope hS
hH_inter : Hpolytope ⋯ = Hpolytope hH_' ∩ Hpolytope hH_
this : IsCompact (Hpolytope ⋯)
⊢ ∃ S', ∃ (hS' : Set.Finite S'), Vpolytope hS' = Vpolytope hS ∩ Hpolytope hH_ | case intro.intro.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : InnerProductSpace ℝ E✝
inst✝⁶ : CompleteSpace E✝
E P : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : PseudoMetricSpace P
inst✝¹ : NormedAddTorsor E P
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinterior : Set.Nonempty (interior (Vpolytope hS))
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
H_' : Set (Halfspace E)
hH_' : Set.Finite H_'
hHV : Hpolytope hH_' = Vpolytope hS
hH_inter : Hpolytope ⋯ = Hpolytope hH_' ∩ Hpolytope hH_
this : IsCompact (Hpolytope ⋯)
S' : Set E
hS' : Set.Finite S'
hSV : Hpolytope ⋯ = Vpolytope hS'
⊢ ∃ S', ∃ (hS' : Set.Finite S'), Vpolytope hS' = Vpolytope hS ∩ Hpolytope hH_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Vpolytope_inter_cutSpace_fin | [547, 1] | [556, 7] | exact ⟨ S', hS', by rw [← hSV, ← hHV, ← hH_inter] ⟩ | case intro.intro.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : InnerProductSpace ℝ E✝
inst✝⁶ : CompleteSpace E✝
E P : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : PseudoMetricSpace P
inst✝¹ : NormedAddTorsor E P
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinterior : Set.Nonempty (interior (Vpolytope hS))
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
H_' : Set (Halfspace E)
hH_' : Set.Finite H_'
hHV : Hpolytope hH_' = Vpolytope hS
hH_inter : Hpolytope ⋯ = Hpolytope hH_' ∩ Hpolytope hH_
this : IsCompact (Hpolytope ⋯)
S' : Set E
hS' : Set.Finite S'
hSV : Hpolytope ⋯ = Vpolytope hS'
⊢ ∃ S', ∃ (hS' : Set.Finite S'), Vpolytope hS' = Vpolytope hS ∩ Hpolytope hH_ | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Vpolytope_inter_cutSpace_fin | [547, 1] | [556, 7] | rw [← hSV, ← hHV, ← hH_inter] | E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : InnerProductSpace ℝ E✝
inst✝⁶ : CompleteSpace E✝
E P : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : PseudoMetricSpace P
inst✝¹ : NormedAddTorsor E P
inst✝ : FiniteDimensional ℝ E
S : Set E
hS : Set.Finite S
hVinterior : Set.Nonempty (interior (Vpolytope hS))
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
H_' : Set (Halfspace E)
hH_' : Set.Finite H_'
hHV : Hpolytope hH_' = Vpolytope hS
hH_inter : Hpolytope ⋯ = Hpolytope hH_' ∩ Hpolytope hH_
this : IsCompact (Hpolytope ⋯)
S' : Set E
hS' : Set.Finite S'
hSV : Hpolytope ⋯ = Vpolytope hS'
⊢ Vpolytope hS' = Vpolytope hS ∩ Hpolytope hH_ | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | InDown_eq_DownIn | [558, 1] | [574, 7] | ext y | E✝ : Type
inst✝⁹ : NormedAddCommGroup E✝
inst✝⁸ : InnerProductSpace ℝ E✝
inst✝⁷ : CompleteSpace E✝
E P : Type
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : PseudoMetricSpace P
inst✝² : NormedAddTorsor E P
inst✝¹ : FiniteDimensional ℝ E
p : AffineSubspace ℝ P
inst✝ : Nonempty ↥p
S : Set P
x : ↥p
⊢ ⇑(AffineIsometryEquiv.VSubconst ℝ x) '' (Subtype.val ⁻¹' S) = Subtype.val ⁻¹' (S -ᵥ {↑x}) | case h
E✝ : Type
inst✝⁹ : NormedAddCommGroup E✝
inst✝⁸ : InnerProductSpace ℝ E✝
inst✝⁷ : CompleteSpace E✝
E P : Type
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : PseudoMetricSpace P
inst✝² : NormedAddTorsor E P
inst✝¹ : FiniteDimensional ℝ E
p : AffineSubspace ℝ P
inst✝ : Nonempty ↥p
S : Set P
x : ↥p
y : ↥(AffineSubspace.direction p)
⊢ y ∈ ⇑(AffineIsometryEquiv.VSubconst ℝ x) '' (Subtype.val ⁻¹' S) ↔ y ∈ Subtype.val ⁻¹' (S -ᵥ {↑x}) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | InDown_eq_DownIn | [558, 1] | [574, 7] | simp only [AffineIsometryEquiv.coe_VSubconst, Set.vsub_singleton, Set.mem_image, Set.mem_preimage,
Set.mem_image, Subtype.exists, exists_and_left] | case h
E✝ : Type
inst✝⁹ : NormedAddCommGroup E✝
inst✝⁸ : InnerProductSpace ℝ E✝
inst✝⁷ : CompleteSpace E✝
E P : Type
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : PseudoMetricSpace P
inst✝² : NormedAddTorsor E P
inst✝¹ : FiniteDimensional ℝ E
p : AffineSubspace ℝ P
inst✝ : Nonempty ↥p
S : Set P
x : ↥p
y : ↥(AffineSubspace.direction p)
⊢ y ∈ ⇑(AffineIsometryEquiv.VSubconst ℝ x) '' (Subtype.val ⁻¹' S) ↔ y ∈ Subtype.val ⁻¹' (S -ᵥ {↑x}) | case h
E✝ : Type
inst✝⁹ : NormedAddCommGroup E✝
inst✝⁸ : InnerProductSpace ℝ E✝
inst✝⁷ : CompleteSpace E✝
E P : Type
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : PseudoMetricSpace P
inst✝² : NormedAddTorsor E P
inst✝¹ : FiniteDimensional ℝ E
p : AffineSubspace ℝ P
inst✝ : Nonempty ↥p
S : Set P
x : ↥p
y : ↥(AffineSubspace.direction p)
⊢ (∃ a ∈ S, ∃ (x_1 : a ∈ p), { val := a, property := ⋯ } -ᵥ x = y) ↔ ∃ x_1 ∈ S, x_1 -ᵥ ↑x = ↑y |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | InDown_eq_DownIn | [558, 1] | [574, 7] | constructor | case h
E✝ : Type
inst✝⁹ : NormedAddCommGroup E✝
inst✝⁸ : InnerProductSpace ℝ E✝
inst✝⁷ : CompleteSpace E✝
E P : Type
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : PseudoMetricSpace P
inst✝² : NormedAddTorsor E P
inst✝¹ : FiniteDimensional ℝ E
p : AffineSubspace ℝ P
inst✝ : Nonempty ↥p
S : Set P
x : ↥p
y : ↥(AffineSubspace.direction p)
⊢ (∃ a ∈ S, ∃ (x_1 : a ∈ p), { val := a, property := ⋯ } -ᵥ x = y) ↔ ∃ x_1 ∈ S, x_1 -ᵥ ↑x = ↑y | case h.mp
E✝ : Type
inst✝⁹ : NormedAddCommGroup E✝
inst✝⁸ : InnerProductSpace ℝ E✝
inst✝⁷ : CompleteSpace E✝
E P : Type
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : PseudoMetricSpace P
inst✝² : NormedAddTorsor E P
inst✝¹ : FiniteDimensional ℝ E
p : AffineSubspace ℝ P
inst✝ : Nonempty ↥p
S : Set P
x : ↥p
y : ↥(AffineSubspace.direction p)
⊢ (∃ a ∈ S, ∃ (x_1 : a ∈ p), { val := a, property := ⋯ } -ᵥ x = y) → ∃ x_1 ∈ S, x_1 -ᵥ ↑x = ↑y
case h.mpr
E✝ : Type
inst✝⁹ : NormedAddCommGroup E✝
inst✝⁸ : InnerProductSpace ℝ E✝
inst✝⁷ : CompleteSpace E✝
E P : Type
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : PseudoMetricSpace P
inst✝² : NormedAddTorsor E P
inst✝¹ : FiniteDimensional ℝ E
p : AffineSubspace ℝ P
inst✝ : Nonempty ↥p
S : Set P
x : ↥p
y : ↥(AffineSubspace.direction p)
⊢ (∃ x_1 ∈ S, x_1 -ᵥ ↑x = ↑y) → ∃ a ∈ S, ∃ (x_1 : a ∈ p), { val := a, property := ⋯ } -ᵥ x = y |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | InDown_eq_DownIn | [558, 1] | [574, 7] | rintro ⟨ v, hvmemS, ⟨ hvmemp, rfl ⟩ ⟩ | case h.mp
E✝ : Type
inst✝⁹ : NormedAddCommGroup E✝
inst✝⁸ : InnerProductSpace ℝ E✝
inst✝⁷ : CompleteSpace E✝
E P : Type
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : PseudoMetricSpace P
inst✝² : NormedAddTorsor E P
inst✝¹ : FiniteDimensional ℝ E
p : AffineSubspace ℝ P
inst✝ : Nonempty ↥p
S : Set P
x : ↥p
y : ↥(AffineSubspace.direction p)
⊢ (∃ a ∈ S, ∃ (x_1 : a ∈ p), { val := a, property := ⋯ } -ᵥ x = y) → ∃ x_1 ∈ S, x_1 -ᵥ ↑x = ↑y | case h.mp.intro.intro.intro
E✝ : Type
inst✝⁹ : NormedAddCommGroup E✝
inst✝⁸ : InnerProductSpace ℝ E✝
inst✝⁷ : CompleteSpace E✝
E P : Type
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : PseudoMetricSpace P
inst✝² : NormedAddTorsor E P
inst✝¹ : FiniteDimensional ℝ E
p : AffineSubspace ℝ P
inst✝ : Nonempty ↥p
S : Set P
x : ↥p
v : P
hvmemS : v ∈ S
hvmemp : v ∈ p
⊢ ∃ x_1 ∈ S, x_1 -ᵥ ↑x = ↑({ val := v, property := ⋯ } -ᵥ x) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | InDown_eq_DownIn | [558, 1] | [574, 7] | refine ⟨ v, hvmemS, ?_ ⟩ | case h.mp.intro.intro.intro
E✝ : Type
inst✝⁹ : NormedAddCommGroup E✝
inst✝⁸ : InnerProductSpace ℝ E✝
inst✝⁷ : CompleteSpace E✝
E P : Type
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : PseudoMetricSpace P
inst✝² : NormedAddTorsor E P
inst✝¹ : FiniteDimensional ℝ E
p : AffineSubspace ℝ P
inst✝ : Nonempty ↥p
S : Set P
x : ↥p
v : P
hvmemS : v ∈ S
hvmemp : v ∈ p
⊢ ∃ x_1 ∈ S, x_1 -ᵥ ↑x = ↑({ val := v, property := ⋯ } -ᵥ x) | case h.mp.intro.intro.intro
E✝ : Type
inst✝⁹ : NormedAddCommGroup E✝
inst✝⁸ : InnerProductSpace ℝ E✝
inst✝⁷ : CompleteSpace E✝
E P : Type
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : PseudoMetricSpace P
inst✝² : NormedAddTorsor E P
inst✝¹ : FiniteDimensional ℝ E
p : AffineSubspace ℝ P
inst✝ : Nonempty ↥p
S : Set P
x : ↥p
v : P
hvmemS : v ∈ S
hvmemp : v ∈ p
⊢ v -ᵥ ↑x = ↑({ val := v, property := ⋯ } -ᵥ x) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | InDown_eq_DownIn | [558, 1] | [574, 7] | simp only [hvmemS, AffineSubspace.coe_vsub] | case h.mp.intro.intro.intro
E✝ : Type
inst✝⁹ : NormedAddCommGroup E✝
inst✝⁸ : InnerProductSpace ℝ E✝
inst✝⁷ : CompleteSpace E✝
E P : Type
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : PseudoMetricSpace P
inst✝² : NormedAddTorsor E P
inst✝¹ : FiniteDimensional ℝ E
p : AffineSubspace ℝ P
inst✝ : Nonempty ↥p
S : Set P
x : ↥p
v : P
hvmemS : v ∈ S
hvmemp : v ∈ p
⊢ v -ᵥ ↑x = ↑({ val := v, property := ⋯ } -ᵥ x) | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | InDown_eq_DownIn | [558, 1] | [574, 7] | rintro ⟨ v, hvmemS, h ⟩ | case h.mpr
E✝ : Type
inst✝⁹ : NormedAddCommGroup E✝
inst✝⁸ : InnerProductSpace ℝ E✝
inst✝⁷ : CompleteSpace E✝
E P : Type
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : PseudoMetricSpace P
inst✝² : NormedAddTorsor E P
inst✝¹ : FiniteDimensional ℝ E
p : AffineSubspace ℝ P
inst✝ : Nonempty ↥p
S : Set P
x : ↥p
y : ↥(AffineSubspace.direction p)
⊢ (∃ x_1 ∈ S, x_1 -ᵥ ↑x = ↑y) → ∃ a ∈ S, ∃ (x_1 : a ∈ p), { val := a, property := ⋯ } -ᵥ x = y | case h.mpr.intro.intro
E✝ : Type
inst✝⁹ : NormedAddCommGroup E✝
inst✝⁸ : InnerProductSpace ℝ E✝
inst✝⁷ : CompleteSpace E✝
E P : Type
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : PseudoMetricSpace P
inst✝² : NormedAddTorsor E P
inst✝¹ : FiniteDimensional ℝ E
p : AffineSubspace ℝ P
inst✝ : Nonempty ↥p
S : Set P
x : ↥p
y : ↥(AffineSubspace.direction p)
v : P
hvmemS : v ∈ S
h : v -ᵥ ↑x = ↑y
⊢ ∃ a ∈ S, ∃ (x_1 : a ∈ p), { val := a, property := ⋯ } -ᵥ x = y |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | InDown_eq_DownIn | [558, 1] | [574, 7] | have := y.2 | case h.mpr.intro.intro
E✝ : Type
inst✝⁹ : NormedAddCommGroup E✝
inst✝⁸ : InnerProductSpace ℝ E✝
inst✝⁷ : CompleteSpace E✝
E P : Type
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : PseudoMetricSpace P
inst✝² : NormedAddTorsor E P
inst✝¹ : FiniteDimensional ℝ E
p : AffineSubspace ℝ P
inst✝ : Nonempty ↥p
S : Set P
x : ↥p
y : ↥(AffineSubspace.direction p)
v : P
hvmemS : v ∈ S
h : v -ᵥ ↑x = ↑y
⊢ ∃ a ∈ S, ∃ (x_1 : a ∈ p), { val := a, property := ⋯ } -ᵥ x = y | case h.mpr.intro.intro
E✝ : Type
inst✝⁹ : NormedAddCommGroup E✝
inst✝⁸ : InnerProductSpace ℝ E✝
inst✝⁷ : CompleteSpace E✝
E P : Type
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : PseudoMetricSpace P
inst✝² : NormedAddTorsor E P
inst✝¹ : FiniteDimensional ℝ E
p : AffineSubspace ℝ P
inst✝ : Nonempty ↥p
S : Set P
x : ↥p
y : ↥(AffineSubspace.direction p)
v : P
hvmemS : v ∈ S
h : v -ᵥ ↑x = ↑y
this : ↑y ∈ AffineSubspace.direction p
⊢ ∃ a ∈ S, ∃ (x_1 : a ∈ p), { val := a, property := ⋯ } -ᵥ x = y |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | InDown_eq_DownIn | [558, 1] | [574, 7] | rw [← h, AffineSubspace.vsub_right_mem_direction_iff_mem x.2] at this | case h.mpr.intro.intro
E✝ : Type
inst✝⁹ : NormedAddCommGroup E✝
inst✝⁸ : InnerProductSpace ℝ E✝
inst✝⁷ : CompleteSpace E✝
E P : Type
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : PseudoMetricSpace P
inst✝² : NormedAddTorsor E P
inst✝¹ : FiniteDimensional ℝ E
p : AffineSubspace ℝ P
inst✝ : Nonempty ↥p
S : Set P
x : ↥p
y : ↥(AffineSubspace.direction p)
v : P
hvmemS : v ∈ S
h : v -ᵥ ↑x = ↑y
this : ↑y ∈ AffineSubspace.direction p
⊢ ∃ a ∈ S, ∃ (x_1 : a ∈ p), { val := a, property := ⋯ } -ᵥ x = y | case h.mpr.intro.intro
E✝ : Type
inst✝⁹ : NormedAddCommGroup E✝
inst✝⁸ : InnerProductSpace ℝ E✝
inst✝⁷ : CompleteSpace E✝
E P : Type
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : PseudoMetricSpace P
inst✝² : NormedAddTorsor E P
inst✝¹ : FiniteDimensional ℝ E
p : AffineSubspace ℝ P
inst✝ : Nonempty ↥p
S : Set P
x : ↥p
y : ↥(AffineSubspace.direction p)
v : P
hvmemS : v ∈ S
h : v -ᵥ ↑x = ↑y
this : v ∈ p
⊢ ∃ a ∈ S, ∃ (x_1 : a ∈ p), { val := a, property := ⋯ } -ᵥ x = y |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | InDown_eq_DownIn | [558, 1] | [574, 7] | exact ⟨ v, hvmemS, this, Subtype.val_injective ((AffineSubspace.coe_vsub _ _ x) ▸ h) ⟩ | case h.mpr.intro.intro
E✝ : Type
inst✝⁹ : NormedAddCommGroup E✝
inst✝⁸ : InnerProductSpace ℝ E✝
inst✝⁷ : CompleteSpace E✝
E P : Type
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : PseudoMetricSpace P
inst✝² : NormedAddTorsor E P
inst✝¹ : FiniteDimensional ℝ E
p : AffineSubspace ℝ P
inst✝ : Nonempty ↥p
S : Set P
x : ↥p
y : ↥(AffineSubspace.direction p)
v : P
hvmemS : v ∈ S
h : v -ᵥ ↑x = ↑y
this : v ∈ p
⊢ ∃ a ∈ S, ∃ (x_1 : a ∈ p), { val := a, property := ⋯ } -ᵥ x = y | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Nonempty_iff_Nonempty_interior_in_direction | [577, 1] | [583, 11] | apply subset_affineSpan | E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : InnerProductSpace ℝ E✝
inst✝⁶ : CompleteSpace E✝
E P : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : PseudoMetricSpace P
inst✝¹ : NormedAddTorsor E P
inst✝ : FiniteDimensional ℝ E
S : Set E
s : E
hs : s ∈ S
hS : Nonempty ↑S
⊢ s ∈ affineSpan ℝ S | case a
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : InnerProductSpace ℝ E✝
inst✝⁶ : CompleteSpace E✝
E P : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : PseudoMetricSpace P
inst✝¹ : NormedAddTorsor E P
inst✝ : FiniteDimensional ℝ E
S : Set E
s : E
hs : s ∈ S
hS : Nonempty ↑S
⊢ s ∈ S |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Nonempty_iff_Nonempty_interior_in_direction | [577, 1] | [583, 11] | exact hs | case a
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : InnerProductSpace ℝ E✝
inst✝⁶ : CompleteSpace E✝
E P : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : PseudoMetricSpace P
inst✝¹ : NormedAddTorsor E P
inst✝ : FiniteDimensional ℝ E
S : Set E
s : E
hs : s ∈ S
hS : Nonempty ↑S
⊢ s ∈ S | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Nonempty_iff_Nonempty_interior_in_direction | [577, 1] | [583, 11] | rw [Set.nonempty_coe_sort, ← @convexHull_nonempty_iff ℝ, ← intrinsicInterior_nonempty (convex_convexHull ℝ S),
intrinsicInterior, Set.image_nonempty, affineSpan_convexHull] at hS | E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : InnerProductSpace ℝ E✝
inst✝⁶ : CompleteSpace E✝
E P : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : PseudoMetricSpace P
inst✝¹ : NormedAddTorsor E P
inst✝ : FiniteDimensional ℝ E
S : Set E
s : E
hs : s ∈ S
hS : Nonempty ↑S
⊢ Set.Nonempty
(interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))) | E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : InnerProductSpace ℝ E✝
inst✝⁶ : CompleteSpace E✝
E P : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : PseudoMetricSpace P
inst✝¹ : NormedAddTorsor E P
inst✝ : FiniteDimensional ℝ E
S : Set E
s : E
hs : s ∈ S
hS✝ : Nonempty ↑S
hS : Set.Nonempty (interior (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ Set.Nonempty
(interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Nonempty_iff_Nonempty_interior_in_direction | [577, 1] | [583, 11] | rw [← AffineIsometryEquiv.coe_toHomeomorph, ← Homeomorph.image_interior, Set.image_nonempty] | E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : InnerProductSpace ℝ E✝
inst✝⁶ : CompleteSpace E✝
E P : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : PseudoMetricSpace P
inst✝¹ : NormedAddTorsor E P
inst✝ : FiniteDimensional ℝ E
S : Set E
s : E
hs : s ∈ S
hS✝ : Nonempty ↑S
hS : Set.Nonempty (interior (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ Set.Nonempty
(interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))) | E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : InnerProductSpace ℝ E✝
inst✝⁶ : CompleteSpace E✝
E P : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : PseudoMetricSpace P
inst✝¹ : NormedAddTorsor E P
inst✝ : FiniteDimensional ℝ E
S : Set E
s : E
hs : s ∈ S
hS✝ : Nonempty ↑S
hS : Set.Nonempty (interior (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ Set.Nonempty (interior (Subtype.val ⁻¹' (convexHull ℝ) S)) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Nonempty_iff_Nonempty_interior_in_direction | [577, 1] | [583, 11] | exact hS | E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : InnerProductSpace ℝ E✝
inst✝⁶ : CompleteSpace E✝
E P : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : PseudoMetricSpace P
inst✝¹ : NormedAddTorsor E P
inst✝ : FiniteDimensional ℝ E
S : Set E
s : E
hs : s ∈ S
hS✝ : Nonempty ↑S
hS : Set.Nonempty (interior (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ Set.Nonempty (interior (Subtype.val ⁻¹' (convexHull ℝ) S)) | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | constructor | E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
⊢ (∀ (S : Set E) (hS : Set.Finite S), ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS) ∧
∀ {H_ : Set (Halfspace E)} (hH_ : Set.Finite H_),
IsCompact (Hpolytope hH_) → ∃ S, ∃ (hS : Set.Finite S), Hpolytope hH_ = Vpolytope hS | case left
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
⊢ ∀ (S : Set E) (hS : Set.Finite S), ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS
case right
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
⊢ ∀ {H_ : Set (Halfspace E)} (hH_ : Set.Finite H_),
IsCompact (Hpolytope hH_) → ∃ S, ∃ (hS : Set.Finite S), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | intro S hS | case left
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
⊢ ∀ (S : Set E) (hS : Set.Finite S), ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case left
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | cases' em (S.Nontrivial) with hSnontrivial hStrivial | case left
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case left.inl
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS
case left.inr
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hStrivial : ¬Set.Nontrivial S
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | have := Set.nontrivial_coe_sort.mpr hSnontrivial | case left.inl
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case left.inl
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this : Nontrivial ↑S
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | have hSnonempty := hSnontrivial.nonempty | case left.inl
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this : Nontrivial ↑S
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case left.inl
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this : Nontrivial ↑S
hSnonempty : Set.Nonempty S
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | have := Set.nonempty_coe_sort.mpr hSnonempty | case left.inl
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this : Nontrivial ↑S
hSnonempty : Set.Nonempty S
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case left.inl
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | rcases hSnontrivial.nonempty with ⟨ s, hs ⟩ | case left.inl
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case left.inl.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | have hsaff : s ∈ affineSpan ℝ S := by apply subset_affineSpan; exact hs | case left.inl.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case left.inl.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | let SpanS := affineSpan ℝ S | case left.inl.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case left.inl.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | let s' : SpanS := ⟨ s, hsaff ⟩ | case left.inl.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case left.inl.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | rcases (Nonempty_iff_Nonempty_interior_in_direction hs this) with ⟨ x, hx ⟩ | case left.inl.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case left.inl.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | rcases this with ⟨ S', hS'Fin, hS'eq ⟩ | case left.inl.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
this :
∃ S', Set.Finite S' ∧ (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case left.inl.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
S' : Set ↥(AffineSubspace.direction SpanS)
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | rw [← hS'eq] at hx | case left.inl.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
S' : Set ↥(AffineSubspace.direction SpanS)
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case left.inl.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | have hS' : (interior (Vpolytope hS'Fin)).Nonempty := Set.nonempty_of_mem hx | case left.inl.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case left.inl.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | rcases @Hpolytope_of_Vpolytope_interior SpanS.direction _ _ _ _ _ hS'Fin hS' with ⟨ H_''1, hH''1, hHV ⟩ | case left.inl.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case left.inl.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | let H_'1 : Set (Halfspace E) := (Halfspace.val SpanS.direction) '' H_''1 | case left.inl.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case left.inl.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | have hH_'1 : H_'1.Finite := Set.Finite.image _ hH''1 | case left.inl.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case left.inl.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | rcases Submodule_cutspace SpanS.direction with ⟨ H_'2, hH_'2, hH_'2Span' ⟩ | case left.inl.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case left.inl.intro.intro.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span' : ↑(AffineSubspace.direction SpanS) = cutSpace H_'2
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | have hH_'2Span: Hpolytope hH_'2 = SpanS.direction := hH_'2Span'.symm | case left.inl.intro.intro.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span' : ↑(AffineSubspace.direction SpanS) = cutSpace H_'2
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case left.inl.intro.intro.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span' : ↑(AffineSubspace.direction SpanS) = cutSpace H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | clear hH_'2Span' | case left.inl.intro.intro.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span' : ↑(AffineSubspace.direction SpanS) = cutSpace H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case left.inl.intro.intro.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | let H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2) | case left.inl.intro.intro.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case left.inl.intro.intro.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | have hH_' : H_'.Finite := Set.Finite.image _ (Set.Finite.union hH_'1 hH_'2) | case left.inl.intro.intro.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case left.inl.intro.intro.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | have hH_'12 := inter_Hpolytope H_'1 H_'2 hH_'1 hH_'2 | case left.inl.intro.intro.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case left.inl.intro.intro.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | have : Nontrivial SpanS.direction := by
apply AffineSubspace.direction_nontrivial_of_nontrivial
exact affineSpan_nontrivial ℝ (Set.nontrivial_coe_sort.mpr hSnontrivial) | case left.inl.intro.intro.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case left.inl.intro.intro.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | refine ⟨ H_', hH_', ?_ ⟩ | case left.inl.intro.intro.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case left.inl.intro.intro.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Hpolytope hH_' = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | rw [Hpolytope_translation, hH_'12, hH_'2Span, Hpolytope, ← Set.Nonempty.sInter_inter_comm, Set.image_image,
Set.image_image, @Set.image_congr' _ _ _ _ (H_''1) (Halfspace.val_eq' SpanS.direction),
← Set.image_image, Set.sInter_image, ← Set.Nonempty.image_sInter ?_ (Subtype.val_injective)] | case left.inl.intro.intro.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Hpolytope hH_' = Vpolytope hS | case left.inl.intro.intro.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Subtype.val '' ⋂₀ ((fun x => ↑x) '' H_''1) + {s} = Vpolytope hS
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty ((fun x => ↑x) '' H_''1)
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty (SetLike.coe '' H_'1) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | change Subtype.val '' Hpolytope hH''1 + {s} = Vpolytope hS | case left.inl.intro.intro.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Subtype.val '' ⋂₀ ((fun x => ↑x) '' H_''1) + {s} = Vpolytope hS
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty ((fun x => ↑x) '' H_''1)
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty (SetLike.coe '' H_'1) | case left.inl.intro.intro.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Subtype.val '' Hpolytope hH''1 + {s} = Vpolytope hS
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty ((fun x => ↑x) '' H_''1)
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty (SetLike.coe '' H_'1) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | rw [hHV, Vpolytope, hS'eq] | case left.inl.intro.intro.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Subtype.val '' Hpolytope hH''1 + {s} = Vpolytope hS
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty ((fun x => ↑x) '' H_''1)
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty (SetLike.coe '' H_'1) | case left.inl.intro.intro.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Subtype.val '' (⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)) + {s} = Vpolytope hS
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty ((fun x => ↑x) '' H_''1)
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty (SetLike.coe '' H_'1) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | change Subtype.val '' ((AffineIsometryEquiv.toHomeomorph (AffineIsometryEquiv.VSubconst ℝ s')) '' (Subtype.val ⁻¹' (convexHull ℝ) S)) + {s} = Vpolytope hS | case left.inl.intro.intro.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Subtype.val '' (⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)) + {s} = Vpolytope hS
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty ((fun x => ↑x) '' H_''1)
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty (SetLike.coe '' H_'1) | case left.inl.intro.intro.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Subtype.val ''
(⇑(AffineIsometryEquiv.toHomeomorph (AffineIsometryEquiv.VSubconst ℝ s')) ''
(Subtype.val ⁻¹' (convexHull ℝ) S)) +
{s} =
Vpolytope hS
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty ((fun x => ↑x) '' H_''1)
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty (SetLike.coe '' H_'1) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | rw [AffineIsometryEquiv.coe_toHomeomorph] | case left.inl.intro.intro.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Subtype.val ''
(⇑(AffineIsometryEquiv.toHomeomorph (AffineIsometryEquiv.VSubconst ℝ s')) ''
(Subtype.val ⁻¹' (convexHull ℝ) S)) +
{s} =
Vpolytope hS
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty ((fun x => ↑x) '' H_''1)
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty (SetLike.coe '' H_'1) | case left.inl.intro.intro.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Subtype.val '' (⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)) + {s} = Vpolytope hS
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty ((fun x => ↑x) '' H_''1)
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty (SetLike.coe '' H_'1) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | rw [InDown_eq_DownIn, Set.vsub_eq_sub] | case left.inl.intro.intro.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Subtype.val '' (⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)) + {s} = Vpolytope hS
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty ((fun x => ↑x) '' H_''1)
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty (SetLike.coe '' H_'1) | case left.inl.intro.intro.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Subtype.val '' (Subtype.val ⁻¹' ((convexHull ℝ) S - {↑s'})) + {s} = Vpolytope hS
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty ((fun x => ↑x) '' H_''1)
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty (SetLike.coe '' H_'1) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | change ((↑) : SpanS.direction → E) '' (((↑) : SpanS.direction → E) ⁻¹' ((convexHull ℝ) S - {s})) + {s} = Vpolytope hS | case left.inl.intro.intro.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Subtype.val '' (Subtype.val ⁻¹' ((convexHull ℝ) S - {↑s'})) + {s} = Vpolytope hS
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty ((fun x => ↑x) '' H_''1)
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty (SetLike.coe '' H_'1) | case left.inl.intro.intro.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Subtype.val '' (Subtype.val ⁻¹' ((convexHull ℝ) S - {s})) + {s} = Vpolytope hS
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty ((fun x => ↑x) '' H_''1)
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty (SetLike.coe '' H_'1) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | rw [Subtype.image_preimage_coe, Set.inter_eq_self_of_subset_right ?_, Set.neg_add_cancel_right', Vpolytope] | case left.inl.intro.intro.intro.intro.intro.intro.intro.intro
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Subtype.val '' (Subtype.val ⁻¹' ((convexHull ℝ) S - {s})) + {s} = Vpolytope hS
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty ((fun x => ↑x) '' H_''1)
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty (SetLike.coe '' H_'1) | E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ (convexHull ℝ) S - {s} ⊆ ↑(AffineSubspace.direction SpanS)
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty ((fun x => ↑x) '' H_''1)
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty (SetLike.coe '' H_'1) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | exact AffineSubspace.direction_subset_subset (convexHull_subset_affineSpan S)
(subset_trans (Set.singleton_subset_iff.mpr hs) (subset_affineSpan ℝ S)) | E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ (convexHull ℝ) S - {s} ⊆ ↑(AffineSubspace.direction SpanS)
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty ((fun x => ↑x) '' H_''1)
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty (SetLike.coe '' H_'1) | E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty ((fun x => ↑x) '' H_''1)
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty (SetLike.coe '' H_'1) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | all_goals (apply Set.Nonempty.image) | E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty ((fun x => ↑x) '' H_''1)
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty (SetLike.coe '' H_'1) | case a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty H_''1
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty H_'1 |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | all_goals (try (change Set.Nonempty (Halfspace.val (AffineSubspace.direction SpanS) '' H_''1))) | case a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty H_''1
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty H_'1 | case a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty H_''1
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty (Halfspace.val (AffineSubspace.direction SpanS) '' H_''1) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | all_goals (try apply Set.Nonempty.image) | case a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty H_''1
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty (Halfspace.val (AffineSubspace.direction SpanS) '' H_''1) | case a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty H_''1
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty H_''1 |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | all_goals (by_contra h) | case a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty H_''1
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
⊢ Set.Nonempty H_''1 | case a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
h : ¬Set.Nonempty H_''1
⊢ False
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
h : ¬Set.Nonempty H_''1
⊢ False |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | all_goals (rw [Set.not_nonempty_iff_eq_empty] at h) | case a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
h : ¬Set.Nonempty H_''1
⊢ False
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
h : ¬Set.Nonempty H_''1
⊢ False | case a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
h : H_''1 = ∅
⊢ False
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
h : H_''1 = ∅
⊢ False |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | all_goals (rw [Hpolytope, h, Set.image_empty, Set.sInter_empty] at hHV) | case a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
h : H_''1 = ∅
⊢ False
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Hpolytope hH''1 = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
h : H_''1 = ∅
⊢ False | case a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Set.univ = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
h : H_''1 = ∅
⊢ False
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Set.univ = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
h : H_''1 = ∅
⊢ False |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | all_goals (exact IsCompact.ne_univ (Compact_Vpolytope hS'Fin) hHV.symm) | case a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Set.univ = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
h : H_''1 = ∅
⊢ False
case left.inl.intro.intro.intro.intro.intro.intro.intro.intro.hs.a.a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝¹ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this✝ : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
S' : Set ↥(AffineSubspace.direction SpanS)
hx : x ∈ interior ((convexHull ℝ) S')
hS'Fin : Set.Finite S'
hS'eq : (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S)
hS' : Set.Nonempty (interior (Vpolytope hS'Fin))
H_''1 : Set (Halfspace ↥(AffineSubspace.direction SpanS))
hH''1 : Set.Finite H_''1
hHV : Set.univ = Vpolytope hS'Fin
H_'1 : Set (Halfspace E) := Halfspace.val (AffineSubspace.direction SpanS) '' H_''1
hH_'1 : Set.Finite H_'1
H_'2 : Set (Halfspace E)
hH_'2 : Set.Finite H_'2
hH_'2Span : Hpolytope hH_'2 = ↑(AffineSubspace.direction SpanS)
H_' : Set (Halfspace E) := Halfspace_translation s '' (H_'1 ∪ H_'2)
hH_' : Set.Finite H_'
hH_'12 : Hpolytope ⋯ = Hpolytope hH_'1 ∩ Hpolytope hH_'2
this : Nontrivial ↥(AffineSubspace.direction SpanS)
h : H_''1 = ∅
⊢ False | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | apply subset_affineSpan | E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
⊢ s ∈ affineSpan ℝ S | case a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
⊢ s ∈ S |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | exact hs | case a
E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
⊢ s ∈ S | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | MainTheoremOfPolytopes | [586, 1] | [686, 7] | rw [InDown_eq_DownIn, ← @convexHull_singleton ℝ, Set.vsub_eq_sub, ← convexHull_sub,
← Submodule.coeSubtype] | E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ ∃ S', Set.Finite S' ∧ (convexHull ℝ) S' = ⇑(AffineIsometryEquiv.VSubconst ℝ s') '' (Subtype.val ⁻¹' (convexHull ℝ) S) | E✝ : Type
inst✝¹⁰ : NormedAddCommGroup E✝
inst✝⁹ : InnerProductSpace ℝ E✝
inst✝⁸ : CompleteSpace E✝
E P : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor E P
inst✝² inst✝¹ : FiniteDimensional ℝ E
inst✝ : Nontrivial E
S : Set E
hS : Set.Finite S
hSnontrivial : Set.Nontrivial S
this✝ : Nontrivial ↑S
hSnonempty : Set.Nonempty S
this : Nonempty ↑S
s : E
hs : s ∈ S
hsaff : s ∈ affineSpan ℝ S
SpanS : AffineSubspace ℝ E := affineSpan ℝ S
s' : ↥SpanS := { val := s, property := hsaff }
x : ↥(AffineSubspace.direction (affineSpan ℝ S))
hx : x ∈ interior (⇑(AffineIsometryEquiv.VSubconst ℝ { val := s, property := ⋯ }) '' (Subtype.val ⁻¹' (convexHull ℝ) S))
⊢ ∃ S',
Set.Finite S' ∧
(convexHull ℝ) S' = ⇑(Submodule.subtype (AffineSubspace.direction SpanS)) ⁻¹' (convexHull ℝ) (S - {↑s'}) |
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