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https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rw [openSegment_eq_image', Set.mem_image] at hx | case mpr.intro.intro.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
hx : x ∈ openSegment ℝ x1 x2
Hi_ : Halfspace E
hfxα : ↑Hi_.f x = Hi_.α
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
hlt : ↑Hi_.f x1 < Hi_.α
⊢ False | case mpr.intro.intro.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
hx : ∃ x_1 ∈ Set.Ioo 0 1, x1 + x_1 • (x2 - x1) = x
Hi_ : Halfspace E
hfxα : ↑Hi_.f x = Hi_.α
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
hlt : ↑Hi_.f x1 < Hi_.α
⊢ False |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rcases hx with ⟨ t, ht, rfl ⟩ | case mpr.intro.intro.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
hx : ∃ x_1 ∈ Set.Ioo 0 1, x1 + x_1 • (x2 - x1) = x
Hi_ : Halfspace E
hfxα : ↑Hi_.f x = Hi_.α
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
hlt : ↑Hi_.f x1 < Hi_.α
⊢ False | case mpr.intro.intro.a.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x1 x2 : E
Hi_ : Halfspace E
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
hlt : ↑Hi_.f x1 < Hi_.α
t : ℝ
ht : t ∈ Set.Ioo 0 1
hfxα : ↑Hi_.f (x1 + t • (x2 - x1)) = Hi_.α
⊢ False |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rw [Hi_.f.1.map_add, Hi_.f.1.map_smul] at hfxα | case mpr.intro.intro.a.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x1 x2 : E
Hi_ : Halfspace E
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
hlt : ↑Hi_.f x1 < Hi_.α
t : ℝ
ht : t ∈ Set.Ioo 0 1
hfxα : ↑Hi_.f (x1 + t • (x2 - x1)) = Hi_.α
⊢ False | case mpr.intro.intro.a.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x1 x2 : E
Hi_ : Halfspace E
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
hlt : ↑Hi_.f x1 < Hi_.α
t : ℝ
ht : t ∈ Set.Ioo 0 1
hfxα : ↑Hi_.f x1 + t • ↑Hi_.f (x2 - x1) = Hi_.α
⊢ False |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | have : Hi_.f.1 x1 + t • Hi_.f.1 (x2 - x1) + (1-t) • Hi_.f.1 (x2 - x1) > Hi_.α := by
rw [hfxα, gt_iff_lt]
exact lt_add_of_le_of_pos (by linarith) <|
(smul_pos_iff_of_pos_left (by linarith [ht.2])).mpr <|
(smul_pos_iff_of_pos_left ht.1).mp <| pos_of_lt_add_right <| hfxα.symm ▸ hlt | case mpr.intro.intro.a.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x1 x2 : E
Hi_ : Halfspace E
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
hlt : ↑Hi_.f x1 < Hi_.α
t : ℝ
ht : t ∈ Set.Ioo 0 1
hfxα : ↑Hi_.f x1 + t • ↑Hi_.f (x2 - x1) = Hi_.α
⊢ False | case mpr.intro.intro.a.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x1 x2 : E
Hi_ : Halfspace E
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
hlt : ↑Hi_.f x1 < Hi_.α
t : ℝ
ht : t ∈ Set.Ioo 0 1
hfxα : ↑Hi_.f x1 + t • ↑Hi_.f (x2 - x1) = Hi_.α
this : ↑Hi_.f x1 + t • ↑Hi_.f (x2 - x1) + (1 - t) • ↑Hi_.f (x2 - x1) > Hi_.α
⊢ False |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rw [add_assoc, ← add_smul, add_sub, add_comm t 1, add_sub_cancel, one_smul, ← Hi_.f.1.map_add, add_comm, sub_add_cancel] at this | case mpr.intro.intro.a.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x1 x2 : E
Hi_ : Halfspace E
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
hlt : ↑Hi_.f x1 < Hi_.α
t : ℝ
ht : t ∈ Set.Ioo 0 1
hfxα : ↑Hi_.f x1 + t • ↑Hi_.f (x2 - x1) = Hi_.α
this : ↑Hi_.f x1 + t • ↑Hi_.f (x2 - x1) + (1 - t) • ↑Hi_.f (x2 - x1) > Hi_.α
⊢ False | case mpr.intro.intro.a.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x1 x2 : E
Hi_ : Halfspace E
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
hlt : ↑Hi_.f x1 < Hi_.α
t : ℝ
ht : t ∈ Set.Ioo 0 1
hfxα : ↑Hi_.f x1 + t • ↑Hi_.f (x2 - x1) = Hi_.α
this : ↑Hi_.f x2 > Hi_.α
⊢ False |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | linarith | case mpr.intro.intro.a.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x1 x2 : E
Hi_ : Halfspace E
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
hlt : ↑Hi_.f x1 < Hi_.α
t : ℝ
ht : t ∈ Set.Ioo 0 1
hfxα : ↑Hi_.f x1 + t • ↑Hi_.f (x2 - x1) = Hi_.α
this : ↑Hi_.f x2 > Hi_.α
⊢ False | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rw [hfxα, gt_iff_lt] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x1 x2 : E
Hi_ : Halfspace E
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
hlt : ↑Hi_.f x1 < Hi_.α
t : ℝ
ht : t ∈ Set.Ioo 0 1
hfxα : ↑Hi_.f x1 + t • ↑Hi_.f (x2 - x1) = Hi_.α
⊢ ↑Hi_.f x1 + t • ↑Hi_.f (x2 - x1) + (1 - t) • ↑Hi_.f (x2 - x1) > Hi_.α | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x1 x2 : E
Hi_ : Halfspace E
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
hlt : ↑Hi_.f x1 < Hi_.α
t : ℝ
ht : t ∈ Set.Ioo 0 1
hfxα : ↑Hi_.f x1 + t • ↑Hi_.f (x2 - x1) = Hi_.α
⊢ Hi_.α < Hi_.α + (1 - t) • ↑Hi_.f (x2 - x1) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | exact lt_add_of_le_of_pos (by linarith) <|
(smul_pos_iff_of_pos_left (by linarith [ht.2])).mpr <|
(smul_pos_iff_of_pos_left ht.1).mp <| pos_of_lt_add_right <| hfxα.symm ▸ hlt | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x1 x2 : E
Hi_ : Halfspace E
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
hlt : ↑Hi_.f x1 < Hi_.α
t : ℝ
ht : t ∈ Set.Ioo 0 1
hfxα : ↑Hi_.f x1 + t • ↑Hi_.f (x2 - x1) = Hi_.α
⊢ Hi_.α < Hi_.α + (1 - t) • ↑Hi_.f (x2 - x1) | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | linarith | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x1 x2 : E
Hi_ : Halfspace E
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
hlt : ↑Hi_.f x1 < Hi_.α
t : ℝ
ht : t ∈ Set.Ioo 0 1
hfxα : ↑Hi_.f x1 + t • ↑Hi_.f (x2 - x1) = Hi_.α
⊢ Hi_.α ≤ Hi_.α | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | linarith [ht.2] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x1 x2 : E
Hi_ : Halfspace E
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
hlt : ↑Hi_.f x1 < Hi_.α
t : ℝ
ht : t ∈ Set.Ioo 0 1
hfxα : ↑Hi_.f x1 + t • ↑Hi_.f (x2 - x1) = Hi_.α
⊢ 0 < 1 - t | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rw [openSegment_symm, openSegment_eq_image', Set.mem_image] at hx | case mpr.intro.intro.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
hx : x ∈ openSegment ℝ x1 x2
Hi_ : Halfspace E
hfxα : ↑Hi_.f x = Hi_.α
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
hlt : ↑Hi_.f x2 < Hi_.α
⊢ False | case mpr.intro.intro.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
hx : ∃ x_1 ∈ Set.Ioo 0 1, x2 + x_1 • (x1 - x2) = x
Hi_ : Halfspace E
hfxα : ↑Hi_.f x = Hi_.α
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
hlt : ↑Hi_.f x2 < Hi_.α
⊢ False |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rcases hx with ⟨ t, ht, rfl ⟩ | case mpr.intro.intro.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x x1 x2 : E
hx : ∃ x_1 ∈ Set.Ioo 0 1, x2 + x_1 • (x1 - x2) = x
Hi_ : Halfspace E
hfxα : ↑Hi_.f x = Hi_.α
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
hlt : ↑Hi_.f x2 < Hi_.α
⊢ False | case mpr.intro.intro.a.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x1 x2 : E
Hi_ : Halfspace E
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
hlt : ↑Hi_.f x2 < Hi_.α
t : ℝ
ht : t ∈ Set.Ioo 0 1
hfxα : ↑Hi_.f (x2 + t • (x1 - x2)) = Hi_.α
⊢ False |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rw [Hi_.f.1.map_add, Hi_.f.1.map_smul] at hfxα | case mpr.intro.intro.a.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x1 x2 : E
Hi_ : Halfspace E
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
hlt : ↑Hi_.f x2 < Hi_.α
t : ℝ
ht : t ∈ Set.Ioo 0 1
hfxα : ↑Hi_.f (x2 + t • (x1 - x2)) = Hi_.α
⊢ False | case mpr.intro.intro.a.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x1 x2 : E
Hi_ : Halfspace E
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
hlt : ↑Hi_.f x2 < Hi_.α
t : ℝ
ht : t ∈ Set.Ioo 0 1
hfxα : ↑Hi_.f x2 + t • ↑Hi_.f (x1 - x2) = Hi_.α
⊢ False |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | have : Hi_.f.1 x2 + t • Hi_.f.1 (x1 - x2) + (1-t) • Hi_.f.1 (x1 - x2) > Hi_.α := by
rw [hfxα, gt_iff_lt]
exact lt_add_of_le_of_pos (by linarith) <|
(smul_pos_iff_of_pos_left (by linarith [ht.2])).mpr <|
(smul_pos_iff_of_pos_left ht.1).mp <| pos_of_lt_add_right <| hfxα.symm ▸ hlt | case mpr.intro.intro.a.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x1 x2 : E
Hi_ : Halfspace E
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
hlt : ↑Hi_.f x2 < Hi_.α
t : ℝ
ht : t ∈ Set.Ioo 0 1
hfxα : ↑Hi_.f x2 + t • ↑Hi_.f (x1 - x2) = Hi_.α
⊢ False | case mpr.intro.intro.a.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x1 x2 : E
Hi_ : Halfspace E
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
hlt : ↑Hi_.f x2 < Hi_.α
t : ℝ
ht : t ∈ Set.Ioo 0 1
hfxα : ↑Hi_.f x2 + t • ↑Hi_.f (x1 - x2) = Hi_.α
this : ↑Hi_.f x2 + t • ↑Hi_.f (x1 - x2) + (1 - t) • ↑Hi_.f (x1 - x2) > Hi_.α
⊢ False |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rw [add_assoc, ← add_smul, add_sub, add_comm t 1, add_sub_cancel, one_smul,
← Hi_.f.1.map_add, add_comm, sub_add_cancel] at this | case mpr.intro.intro.a.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x1 x2 : E
Hi_ : Halfspace E
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
hlt : ↑Hi_.f x2 < Hi_.α
t : ℝ
ht : t ∈ Set.Ioo 0 1
hfxα : ↑Hi_.f x2 + t • ↑Hi_.f (x1 - x2) = Hi_.α
this : ↑Hi_.f x2 + t • ↑Hi_.f (x1 - x2) + (1 - t) • ↑Hi_.f (x1 - x2) > Hi_.α
⊢ False | case mpr.intro.intro.a.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x1 x2 : E
Hi_ : Halfspace E
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
hlt : ↑Hi_.f x2 < Hi_.α
t : ℝ
ht : t ∈ Set.Ioo 0 1
hfxα : ↑Hi_.f x2 + t • ↑Hi_.f (x1 - x2) = Hi_.α
this : ↑Hi_.f x1 > Hi_.α
⊢ False |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | linarith | case mpr.intro.intro.a.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x1 x2 : E
Hi_ : Halfspace E
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
hlt : ↑Hi_.f x2 < Hi_.α
t : ℝ
ht : t ∈ Set.Ioo 0 1
hfxα : ↑Hi_.f x2 + t • ↑Hi_.f (x1 - x2) = Hi_.α
this : ↑Hi_.f x1 > Hi_.α
⊢ False | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rw [hfxα, gt_iff_lt] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x1 x2 : E
Hi_ : Halfspace E
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
hlt : ↑Hi_.f x2 < Hi_.α
t : ℝ
ht : t ∈ Set.Ioo 0 1
hfxα : ↑Hi_.f x2 + t • ↑Hi_.f (x1 - x2) = Hi_.α
⊢ ↑Hi_.f x2 + t • ↑Hi_.f (x1 - x2) + (1 - t) • ↑Hi_.f (x1 - x2) > Hi_.α | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x1 x2 : E
Hi_ : Halfspace E
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
hlt : ↑Hi_.f x2 < Hi_.α
t : ℝ
ht : t ∈ Set.Ioo 0 1
hfxα : ↑Hi_.f x2 + t • ↑Hi_.f (x1 - x2) = Hi_.α
⊢ Hi_.α < Hi_.α + (1 - t) • ↑Hi_.f (x1 - x2) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | exact lt_add_of_le_of_pos (by linarith) <|
(smul_pos_iff_of_pos_left (by linarith [ht.2])).mpr <|
(smul_pos_iff_of_pos_left ht.1).mp <| pos_of_lt_add_right <| hfxα.symm ▸ hlt | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x1 x2 : E
Hi_ : Halfspace E
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
hlt : ↑Hi_.f x2 < Hi_.α
t : ℝ
ht : t ∈ Set.Ioo 0 1
hfxα : ↑Hi_.f x2 + t • ↑Hi_.f (x1 - x2) = Hi_.α
⊢ Hi_.α < Hi_.α + (1 - t) • ↑Hi_.f (x1 - x2) | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | linarith | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x1 x2 : E
Hi_ : Halfspace E
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
hlt : ↑Hi_.f x2 < Hi_.α
t : ℝ
ht : t ∈ Set.Ioo 0 1
hfxα : ↑Hi_.f x2 + t • ↑Hi_.f (x1 - x2) = Hi_.α
⊢ Hi_.α ≤ Hi_.α | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | linarith [ht.2] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x1 x2 : E
Hi_ : Halfspace E
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
hlt : ↑Hi_.f x2 < Hi_.α
t : ℝ
ht : t ∈ Set.Ioo 0 1
hfxα : ↑Hi_.f x2 + t • ↑Hi_.f (x1 - x2) = Hi_.α
⊢ 0 < 1 - t | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | suffices hHeqVdual : ∃ (H_ : Set (Halfspace E)) (hH_ : H_.Finite),
Hpolytope hH_ = polarDual (Vpolytope hS) from by
rcases hHeqVdual with ⟨H_, hH_, hHeqVdual⟩
refine ⟨ H_, hH_, hHeqVdual, ?_ ⟩
exact hHeqVdual ▸ (polarDual_compact_if (Closed_Vpolytope hS) (Convex_Vpolytope hS) hS0) | 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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = polarDual (Vpolytope hS) ∧ IsCompact (Hpolytope hH_) | 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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = polarDual (Vpolytope hS) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | use pointDual '' (Subtype.val ⁻¹' (S \ {0})) | 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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = polarDual (Vpolytope hS) | case 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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ ∃ (hH_ : Set.Finite (pointDual '' (Subtype.val ⁻¹' (S \ {0})))), Hpolytope hH_ = polarDual (Vpolytope hS) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | use (by
apply Set.Finite.image
apply Set.Finite.preimage _ _
apply Set.injOn_of_injective
exact Subtype.val_injective
exact Set.Finite.diff hS {0}
done) | case 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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ ∃ (hH_ : Set.Finite (pointDual '' (Subtype.val ⁻¹' (S \ {0})))), Hpolytope hH_ = polarDual (Vpolytope hS) | case 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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ Hpolytope ⋯ = polarDual (Vpolytope hS) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | apply subset_antisymm | case 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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ Hpolytope ⋯ = polarDual (Vpolytope hS) | case h.a
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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ Hpolytope ⋯ ⊆ polarDual (Vpolytope hS)
case h.a
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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ polarDual (Vpolytope hS) ⊆ Hpolytope ⋯ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | rcases hHeqVdual with ⟨H_, hH_, hHeqVdual⟩ | 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
hS0 : 0 ∈ interior (Vpolytope hS)
hHeqVdual : ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = polarDual (Vpolytope hS)
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = polarDual (Vpolytope hS) ∧ IsCompact (Hpolytope 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
hS0 : 0 ∈ interior (Vpolytope hS)
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHeqVdual : Hpolytope hH_ = polarDual (Vpolytope hS)
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = polarDual (Vpolytope hS) ∧ IsCompact (Hpolytope hH_) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | refine ⟨ H_, hH_, hHeqVdual, ?_ ⟩ | 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
hS0 : 0 ∈ interior (Vpolytope hS)
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHeqVdual : Hpolytope hH_ = polarDual (Vpolytope hS)
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = polarDual (Vpolytope hS) ∧ IsCompact (Hpolytope 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
hS0 : 0 ∈ interior (Vpolytope hS)
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHeqVdual : Hpolytope hH_ = polarDual (Vpolytope hS)
⊢ IsCompact (Hpolytope hH_) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | exact hHeqVdual ▸ (polarDual_compact_if (Closed_Vpolytope hS) (Convex_Vpolytope hS) hS0) | 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
hS0 : 0 ∈ interior (Vpolytope hS)
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHeqVdual : Hpolytope hH_ = polarDual (Vpolytope hS)
⊢ IsCompact (Hpolytope hH_) | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | apply Set.Finite.image | 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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ Set.Finite (pointDual '' (Subtype.val ⁻¹' (S \ {0}))) | case 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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ Set.Finite (Subtype.val ⁻¹' (S \ {0})) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | apply Set.Finite.preimage _ _ | case 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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ Set.Finite (Subtype.val ⁻¹' (S \ {0})) | 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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ Set.InjOn Subtype.val (Subtype.val ⁻¹' (S \ {0}))
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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ Set.Finite (S \ {0}) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | apply Set.injOn_of_injective | 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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ Set.InjOn Subtype.val (Subtype.val ⁻¹' (S \ {0}))
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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ Set.Finite (S \ {0}) | case 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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ Function.Injective Subtype.val
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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ Set.Finite (S \ {0}) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | exact Subtype.val_injective | case 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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ Function.Injective Subtype.val
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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ Set.Finite (S \ {0}) | 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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ Set.Finite (S \ {0}) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | exact Set.Finite.diff hS {0} | 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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ Set.Finite (S \ {0}) | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | intro x hx | case h.a
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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ Hpolytope ⋯ ⊆ polarDual (Vpolytope hS) | case h.a
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : x ∈ Hpolytope ⋯
⊢ x ∈ polarDual (Vpolytope hS) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | cases' (em (x = 0)) with h h | case h.a
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : x ∈ Hpolytope ⋯
⊢ x ∈ polarDual (Vpolytope hS) | case h.a.inl
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : x ∈ Hpolytope ⋯
h : x = 0
⊢ x ∈ polarDual (Vpolytope hS)
case h.a.inr
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : x ∈ Hpolytope ⋯
h : ¬x = 0
⊢ x ∈ polarDual (Vpolytope hS) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | rw [mem_Hpolytope] at hx | case h.a.inr
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : x ∈ Hpolytope ⋯
h : ¬x = 0
⊢ x ∈ polarDual (Vpolytope hS) | case h.a.inr
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : ∀ Hi ∈ pointDual '' (Subtype.val ⁻¹' (S \ {0})), ↑Hi.f x ≤ Hi.α
h : ¬x = 0
⊢ x ∈ polarDual (Vpolytope hS) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | rw [mem_polarDual] | case h.a.inr
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : ∀ Hi ∈ pointDual '' (Subtype.val ⁻¹' (S \ {0})), ↑Hi.f x ≤ Hi.α
h : ¬x = 0
⊢ x ∈ polarDual (Vpolytope hS) | case h.a.inr
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : ∀ Hi ∈ pointDual '' (Subtype.val ⁻¹' (S \ {0})), ↑Hi.f x ≤ Hi.α
h : ¬x = 0
⊢ ∀ x_1 ∈ Vpolytope hS, ⟪x_1, x⟫_ℝ ≤ 1 |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | intro p hp | case h.a.inr
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : ∀ Hi ∈ pointDual '' (Subtype.val ⁻¹' (S \ {0})), ↑Hi.f x ≤ Hi.α
h : ¬x = 0
⊢ ∀ x_1 ∈ Vpolytope hS, ⟪x_1, x⟫_ℝ ≤ 1 | case h.a.inr
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : ∀ Hi ∈ pointDual '' (Subtype.val ⁻¹' (S \ {0})), ↑Hi.f x ≤ Hi.α
h : ¬x = 0
p : E
hp : p ∈ Vpolytope hS
⊢ ⟪p, x⟫_ℝ ≤ 1 |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | let x' := (⟨ x, h ⟩ : { p : E // p ≠ 0 }) | case h.a.inr
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : ∀ Hi ∈ pointDual '' (Subtype.val ⁻¹' (S \ {0})), ↑Hi.f x ≤ Hi.α
h : ¬x = 0
p : E
hp : p ∈ Vpolytope hS
⊢ ⟪p, x⟫_ℝ ≤ 1 | case h.a.inr
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : ∀ Hi ∈ pointDual '' (Subtype.val ⁻¹' (S \ {0})), ↑Hi.f x ≤ Hi.α
h : ¬x = 0
p : E
hp : p ∈ Vpolytope hS
x' : { p // p ≠ 0 } := { val := x, property := h }
⊢ ⟪p, x⟫_ℝ ≤ 1 |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | have hx' : ↑x' = x := rfl | case h.a.inr
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : ∀ Hi ∈ pointDual '' (Subtype.val ⁻¹' (S \ {0})), ↑Hi.f x ≤ Hi.α
h : ¬x = 0
p : E
hp : p ∈ Vpolytope hS
x' : { p // p ≠ 0 } := { val := x, property := h }
⊢ ⟪p, x⟫_ℝ ≤ 1 | case h.a.inr
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : ∀ Hi ∈ pointDual '' (Subtype.val ⁻¹' (S \ {0})), ↑Hi.f x ≤ Hi.α
h : ¬x = 0
p : E
hp : p ∈ Vpolytope hS
x' : { p // p ≠ 0 } := { val := x, property := h }
hx' : ↑x' = x
⊢ ⟪p, x⟫_ℝ ≤ 1 |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | rw [← hx', real_inner_comm, ←mem_pointDual] | case h.a.inr
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : ∀ Hi ∈ pointDual '' (Subtype.val ⁻¹' (S \ {0})), ↑Hi.f x ≤ Hi.α
h : ¬x = 0
p : E
hp : p ∈ Vpolytope hS
x' : { p // p ≠ 0 } := { val := x, property := h }
hx' : ↑x' = x
⊢ ⟪p, x⟫_ℝ ≤ 1 | case h.a.inr
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : ∀ Hi ∈ pointDual '' (Subtype.val ⁻¹' (S \ {0})), ↑Hi.f x ≤ Hi.α
h : ¬x = 0
p : E
hp : p ∈ Vpolytope hS
x' : { p // p ≠ 0 } := { val := x, property := h }
hx' : ↑x' = x
⊢ p ∈ ↑(pointDual x') |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | suffices h : S ⊆ SetLike.coe (pointDual x') from by
apply convexHull_min h <| Halfspace_convex (pointDual x')
exact hp | case h.a.inr
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : ∀ Hi ∈ pointDual '' (Subtype.val ⁻¹' (S \ {0})), ↑Hi.f x ≤ Hi.α
h : ¬x = 0
p : E
hp : p ∈ Vpolytope hS
x' : { p // p ≠ 0 } := { val := x, property := h }
hx' : ↑x' = x
⊢ p ∈ ↑(pointDual x') | case h.a.inr
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : ∀ Hi ∈ pointDual '' (Subtype.val ⁻¹' (S \ {0})), ↑Hi.f x ≤ Hi.α
h : ¬x = 0
p : E
hp : p ∈ Vpolytope hS
x' : { p // p ≠ 0 } := { val := x, property := h }
hx' : ↑x' = x
⊢ S ⊆ ↑(pointDual x') |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | intro s hs | case h.a.inr
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : ∀ Hi ∈ pointDual '' (Subtype.val ⁻¹' (S \ {0})), ↑Hi.f x ≤ Hi.α
h : ¬x = 0
p : E
hp : p ∈ Vpolytope hS
x' : { p // p ≠ 0 } := { val := x, property := h }
hx' : ↑x' = x
⊢ S ⊆ ↑(pointDual x') | case h.a.inr
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : ∀ Hi ∈ pointDual '' (Subtype.val ⁻¹' (S \ {0})), ↑Hi.f x ≤ Hi.α
h : ¬x = 0
p : E
hp : p ∈ Vpolytope hS
x' : { p // p ≠ 0 } := { val := x, property := h }
hx' : ↑x' = x
s : E
hs : s ∈ S
⊢ s ∈ ↑(pointDual x') |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | cases' (em (s = 0)) with h h | case h.a.inr
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : ∀ Hi ∈ pointDual '' (Subtype.val ⁻¹' (S \ {0})), ↑Hi.f x ≤ Hi.α
h : ¬x = 0
p : E
hp : p ∈ Vpolytope hS
x' : { p // p ≠ 0 } := { val := x, property := h }
hx' : ↑x' = x
s : E
hs : s ∈ S
⊢ s ∈ ↑(pointDual x') | case h.a.inr.inl
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : ∀ Hi ∈ pointDual '' (Subtype.val ⁻¹' (S \ {0})), ↑Hi.f x ≤ Hi.α
h✝ : ¬x = 0
p : E
hp : p ∈ Vpolytope hS
x' : { p // p ≠ 0 } := { val := x, property := h✝ }
hx' : ↑x' = x
s : E
hs : s ∈ S
h : s = 0
⊢ s ∈ ↑(pointDual x')
case h.a.inr.inr
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : ∀ Hi ∈ pointDual '' (Subtype.val ⁻¹' (S \ {0})), ↑Hi.f x ≤ Hi.α
h✝ : ¬x = 0
p : E
hp : p ∈ Vpolytope hS
x' : { p // p ≠ 0 } := { val := x, property := h✝ }
hx' : ↑x' = x
s : E
hs : s ∈ S
h : ¬s = 0
⊢ s ∈ ↑(pointDual x') |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | specialize hx (pointDual ⟨ s, h ⟩) (Set.mem_image_of_mem _ ?_) | case h.a.inr.inr
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : ∀ Hi ∈ pointDual '' (Subtype.val ⁻¹' (S \ {0})), ↑Hi.f x ≤ Hi.α
h✝ : ¬x = 0
p : E
hp : p ∈ Vpolytope hS
x' : { p // p ≠ 0 } := { val := x, property := h✝ }
hx' : ↑x' = x
s : E
hs : s ∈ S
h : ¬s = 0
⊢ s ∈ ↑(pointDual x') | case h.a.inr.inr
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : ∀ Hi ∈ pointDual '' (Subtype.val ⁻¹' (S \ {0})), ↑Hi.f x ≤ Hi.α
h✝ : ¬x = 0
p : E
hp : p ∈ Vpolytope hS
x' : { p // p ≠ 0 } := { val := x, property := h✝ }
hx' : ↑x' = x
s : E
hs : s ∈ S
h : ¬s = 0
⊢ { val := s, property := h } ∈ Subtype.val ⁻¹' (S \ {0})
case h.a.inr.inr
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
h✝ : ¬x = 0
p : E
hp : p ∈ Vpolytope hS
x' : { p // p ≠ 0 } := { val := x, property := h✝ }
hx' : ↑x' = x
s : E
hs : s ∈ S
h : ¬s = 0
hx : ↑(pointDual { val := s, property := h }).f x ≤ (pointDual { val := s, property := h }).α
⊢ s ∈ ↑(pointDual x') |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | rw [← Halfspace_mem, mem_pointDual, Subtype.coe_mk] at hx | case h.a.inr.inr
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
h✝ : ¬x = 0
p : E
hp : p ∈ Vpolytope hS
x' : { p // p ≠ 0 } := { val := x, property := h✝ }
hx' : ↑x' = x
s : E
hs : s ∈ S
h : ¬s = 0
hx : ↑(pointDual { val := s, property := h }).f x ≤ (pointDual { val := s, property := h }).α
⊢ s ∈ ↑(pointDual x') | case h.a.inr.inr
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
h✝ : ¬x = 0
p : E
hp : p ∈ Vpolytope hS
x' : { p // p ≠ 0 } := { val := x, property := h✝ }
hx' : ↑x' = x
s : E
hs : s ∈ S
h : ¬s = 0
hx : ⟪s, x⟫_ℝ ≤ 1
⊢ s ∈ ↑(pointDual x') |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | rw [mem_pointDual, Subtype.coe_mk, real_inner_comm] | case h.a.inr.inr
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
h✝ : ¬x = 0
p : E
hp : p ∈ Vpolytope hS
x' : { p // p ≠ 0 } := { val := x, property := h✝ }
hx' : ↑x' = x
s : E
hs : s ∈ S
h : ¬s = 0
hx : ⟪s, x⟫_ℝ ≤ 1
⊢ s ∈ ↑(pointDual x') | case h.a.inr.inr
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
h✝ : ¬x = 0
p : E
hp : p ∈ Vpolytope hS
x' : { p // p ≠ 0 } := { val := x, property := h✝ }
hx' : ↑x' = x
s : E
hs : s ∈ S
h : ¬s = 0
hx : ⟪s, x⟫_ℝ ≤ 1
⊢ ⟪s, x⟫_ℝ ≤ 1 |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | exact hx | case h.a.inr.inr
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
h✝ : ¬x = 0
p : E
hp : p ∈ Vpolytope hS
x' : { p // p ≠ 0 } := { val := x, property := h✝ }
hx' : ↑x' = x
s : E
hs : s ∈ S
h : ¬s = 0
hx : ⟪s, x⟫_ℝ ≤ 1
⊢ ⟪s, x⟫_ℝ ≤ 1 | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | rw [h] | case h.a.inl
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : x ∈ Hpolytope ⋯
h : x = 0
⊢ x ∈ polarDual (Vpolytope hS) | case h.a.inl
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : x ∈ Hpolytope ⋯
h : x = 0
⊢ 0 ∈ polarDual (Vpolytope hS) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | exact polarDual_origin (Vpolytope hS) | case h.a.inl
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : x ∈ Hpolytope ⋯
h : x = 0
⊢ 0 ∈ polarDual (Vpolytope hS) | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | apply convexHull_min h <| Halfspace_convex (pointDual x') | 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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : ∀ Hi ∈ pointDual '' (Subtype.val ⁻¹' (S \ {0})), ↑Hi.f x ≤ Hi.α
h✝ : ¬x = 0
p : E
hp : p ∈ Vpolytope hS
x' : { p // p ≠ 0 } := { val := x, property := h✝ }
hx' : ↑x' = x
h : S ⊆ ↑(pointDual x')
⊢ p ∈ ↑(pointDual x') | case a
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : ∀ Hi ∈ pointDual '' (Subtype.val ⁻¹' (S \ {0})), ↑Hi.f x ≤ Hi.α
h✝ : ¬x = 0
p : E
hp : p ∈ Vpolytope hS
x' : { p // p ≠ 0 } := { val := x, property := h✝ }
hx' : ↑x' = x
h : S ⊆ ↑(pointDual x')
⊢ p ∈ (convexHull ℝ) S |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | exact hp | case a
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : ∀ Hi ∈ pointDual '' (Subtype.val ⁻¹' (S \ {0})), ↑Hi.f x ≤ Hi.α
h✝ : ¬x = 0
p : E
hp : p ∈ Vpolytope hS
x' : { p // p ≠ 0 } := { val := x, property := h✝ }
hx' : ↑x' = x
h : S ⊆ ↑(pointDual x')
⊢ p ∈ (convexHull ℝ) S | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | exact h ▸ pointDual_origin x' | case h.a.inr.inl
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : ∀ Hi ∈ pointDual '' (Subtype.val ⁻¹' (S \ {0})), ↑Hi.f x ≤ Hi.α
h✝ : ¬x = 0
p : E
hp : p ∈ Vpolytope hS
x' : { p // p ≠ 0 } := { val := x, property := h✝ }
hx' : ↑x' = x
s : E
hs : s ∈ S
h : s = 0
⊢ s ∈ ↑(pointDual x') | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | rw [Set.mem_preimage, Subtype.coe_mk, Set.mem_diff] | case h.a.inr.inr
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : ∀ Hi ∈ pointDual '' (Subtype.val ⁻¹' (S \ {0})), ↑Hi.f x ≤ Hi.α
h✝ : ¬x = 0
p : E
hp : p ∈ Vpolytope hS
x' : { p // p ≠ 0 } := { val := x, property := h✝ }
hx' : ↑x' = x
s : E
hs : s ∈ S
h : ¬s = 0
⊢ { val := s, property := h } ∈ Subtype.val ⁻¹' (S \ {0}) | case h.a.inr.inr
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : ∀ Hi ∈ pointDual '' (Subtype.val ⁻¹' (S \ {0})), ↑Hi.f x ≤ Hi.α
h✝ : ¬x = 0
p : E
hp : p ∈ Vpolytope hS
x' : { p // p ≠ 0 } := { val := x, property := h✝ }
hx' : ↑x' = x
s : E
hs : s ∈ S
h : ¬s = 0
⊢ s ∈ S ∧ s ∉ {0} |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | exact ⟨ hs, h ⟩ | case h.a.inr.inr
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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : ∀ Hi ∈ pointDual '' (Subtype.val ⁻¹' (S \ {0})), ↑Hi.f x ≤ Hi.α
h✝ : ¬x = 0
p : E
hp : p ∈ Vpolytope hS
x' : { p // p ≠ 0 } := { val := x, property := h✝ }
hx' : ↑x' = x
s : E
hs : s ∈ S
h : ¬s = 0
⊢ s ∈ S ∧ s ∉ {0} | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | apply Set.sInter_subset_sInter | case h.a
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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ polarDual (Vpolytope hS) ⊆ Hpolytope ⋯ | case h.a.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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' (S \ {0}))) ⊆
SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' Vpolytope hS)) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | apply Set.image_subset | case h.a.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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' (S \ {0}))) ⊆
SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' Vpolytope hS)) | case h.a.h.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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ pointDual '' (Subtype.val ⁻¹' (S \ {0})) ⊆ pointDual '' (Subtype.val ⁻¹' Vpolytope hS) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | apply Set.image_subset | case h.a.h.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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ pointDual '' (Subtype.val ⁻¹' (S \ {0})) ⊆ pointDual '' (Subtype.val ⁻¹' Vpolytope hS) | case h.a.h.h.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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ Subtype.val ⁻¹' (S \ {0}) ⊆ Subtype.val ⁻¹' Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | rw [Set.preimage_subset_preimage_iff] | case h.a.h.h.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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ Subtype.val ⁻¹' (S \ {0}) ⊆ Subtype.val ⁻¹' Vpolytope hS | case h.a.h.h.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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ S \ {0} ⊆ Vpolytope hS
case h.a.h.h.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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ S \ {0} ⊆ Set.range Subtype.val |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | apply subset_trans (by simp) <| subset_convexHull _ _ | case h.a.h.h.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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ S \ {0} ⊆ Vpolytope hS
case h.a.h.h.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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ S \ {0} ⊆ Set.range Subtype.val | case h.a.h.h.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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ S \ {0} ⊆ Set.range Subtype.val |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | rw [Subtype.range_coe_subtype] | case h.a.h.h.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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ S \ {0} ⊆ Set.range Subtype.val | case h.a.h.h.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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ S \ {0} ⊆ {x | x ≠ 0} |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | intro x hx | case h.a.h.h.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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ S \ {0} ⊆ {x | x ≠ 0} | case h.a.h.h.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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : x ∈ S \ {0}
⊢ x ∈ {x | x ≠ 0} |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | rw [Set.mem_diff, Set.mem_singleton_iff] at hx | case h.a.h.h.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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : x ∈ S \ {0}
⊢ x ∈ {x | x ≠ 0} | case h.a.h.h.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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : x ∈ S ∧ ¬x = 0
⊢ x ∈ {x | x ≠ 0} |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | rw [Set.mem_setOf] | case h.a.h.h.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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : x ∈ S ∧ ¬x = 0
⊢ x ∈ {x | x ≠ 0} | case h.a.h.h.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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : x ∈ S ∧ ¬x = 0
⊢ x ≠ 0 |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | exact hx.2 | case h.a.h.h.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
hS0 : 0 ∈ interior (Vpolytope hS)
x : E
hx : x ∈ S ∧ ¬x = 0
⊢ x ≠ 0 | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | DualOfVpolytope_compactHpolytope | [341, 1] | [417, 7] | simp | 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
hS0 : 0 ∈ interior (Vpolytope hS)
⊢ S \ {0} ⊆ S | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Hpolytope | [419, 1] | [476, 7] | intro H_ hH_ hHcpt | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
⊢ ∀ {H_ : Set (Halfspace E)} (hH_ : Set.Finite H_),
IsCompact (Hpolytope hH_) → ∃ S, ∃ (hS : Set.Finite S), Hpolytope hH_ = Vpolytope hS | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : 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 | Vpolytope_of_Hpolytope | [419, 1] | [476, 7] | have : closure (convexHull ℝ ((Hpolytope hH_).extremePoints ℝ)) = Hpolytope hH_ :=
closure_convexHull_extremePoints hHcpt (Convex_Hpolytope hH_) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
hExHFinite : Set.Finite (Set.extremePoints ℝ (Hpolytope hH_))
⊢ ∃ S, ∃ (hS : Set.Finite S), Hpolytope hH_ = Vpolytope hS | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
hExHFinite : Set.Finite (Set.extremePoints ℝ (Hpolytope hH_))
this : closure ((convexHull ℝ) (Set.extremePoints ℝ (Hpolytope hH_))) = Hpolytope hH_
⊢ ∃ S, ∃ (hS : Set.Finite S), Hpolytope hH_ = Vpolytope hS |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Hpolytope | [419, 1] | [476, 7] | rw [← this, IsClosed.closure_eq] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
hExHFinite : Set.Finite (Set.extremePoints ℝ (Hpolytope hH_))
this : closure ((convexHull ℝ) (Set.extremePoints ℝ (Hpolytope hH_))) = Hpolytope hH_
⊢ ∃ S, ∃ (hS : Set.Finite S), Hpolytope hH_ = Vpolytope hS | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
hExHFinite : Set.Finite (Set.extremePoints ℝ (Hpolytope hH_))
this : closure ((convexHull ℝ) (Set.extremePoints ℝ (Hpolytope hH_))) = Hpolytope hH_
⊢ ∃ S, ∃ (hS : Set.Finite S), (convexHull ℝ) (Set.extremePoints ℝ (Hpolytope hH_)) = Vpolytope hS
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
hExHFinite : Set.Finite (Set.extremePoints ℝ (Hpolytope hH_))
this : closure ((convexHull ℝ) (Set.extremePoints ℝ (Hpolytope hH_))) = Hpolytope hH_
⊢ IsClosed ((convexHull ℝ) (Set.extremePoints ℝ (Hpolytope hH_))) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Hpolytope | [419, 1] | [476, 7] | use (Hpolytope hH_).extremePoints ℝ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
hExHFinite : Set.Finite (Set.extremePoints ℝ (Hpolytope hH_))
this : closure ((convexHull ℝ) (Set.extremePoints ℝ (Hpolytope hH_))) = Hpolytope hH_
⊢ ∃ S, ∃ (hS : Set.Finite S), (convexHull ℝ) (Set.extremePoints ℝ (Hpolytope hH_)) = Vpolytope hS
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
hExHFinite : Set.Finite (Set.extremePoints ℝ (Hpolytope hH_))
this : closure ((convexHull ℝ) (Set.extremePoints ℝ (Hpolytope hH_))) = Hpolytope hH_
⊢ IsClosed ((convexHull ℝ) (Set.extremePoints ℝ (Hpolytope hH_))) | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
hExHFinite : Set.Finite (Set.extremePoints ℝ (Hpolytope hH_))
this : closure ((convexHull ℝ) (Set.extremePoints ℝ (Hpolytope hH_))) = Hpolytope hH_
⊢ ∃ (hS : Set.Finite (Set.extremePoints ℝ (Hpolytope hH_))),
(convexHull ℝ) (Set.extremePoints ℝ (Hpolytope hH_)) = Vpolytope hS
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
hExHFinite : Set.Finite (Set.extremePoints ℝ (Hpolytope hH_))
this : closure ((convexHull ℝ) (Set.extremePoints ℝ (Hpolytope hH_))) = Hpolytope hH_
⊢ IsClosed ((convexHull ℝ) (Set.extremePoints ℝ (Hpolytope hH_))) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Hpolytope | [419, 1] | [476, 7] | use hExHFinite | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
hExHFinite : Set.Finite (Set.extremePoints ℝ (Hpolytope hH_))
this : closure ((convexHull ℝ) (Set.extremePoints ℝ (Hpolytope hH_))) = Hpolytope hH_
⊢ ∃ (hS : Set.Finite (Set.extremePoints ℝ (Hpolytope hH_))),
(convexHull ℝ) (Set.extremePoints ℝ (Hpolytope hH_)) = Vpolytope hS
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
hExHFinite : Set.Finite (Set.extremePoints ℝ (Hpolytope hH_))
this : closure ((convexHull ℝ) (Set.extremePoints ℝ (Hpolytope hH_))) = Hpolytope hH_
⊢ IsClosed ((convexHull ℝ) (Set.extremePoints ℝ (Hpolytope hH_))) | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
hExHFinite : Set.Finite (Set.extremePoints ℝ (Hpolytope hH_))
this : closure ((convexHull ℝ) (Set.extremePoints ℝ (Hpolytope hH_))) = Hpolytope hH_
⊢ (convexHull ℝ) (Set.extremePoints ℝ (Hpolytope hH_)) = Vpolytope hExHFinite
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
hExHFinite : Set.Finite (Set.extremePoints ℝ (Hpolytope hH_))
this : closure ((convexHull ℝ) (Set.extremePoints ℝ (Hpolytope hH_))) = Hpolytope hH_
⊢ IsClosed ((convexHull ℝ) (Set.extremePoints ℝ (Hpolytope hH_))) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Hpolytope | [419, 1] | [476, 7] | rfl | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
hExHFinite : Set.Finite (Set.extremePoints ℝ (Hpolytope hH_))
this : closure ((convexHull ℝ) (Set.extremePoints ℝ (Hpolytope hH_))) = Hpolytope hH_
⊢ (convexHull ℝ) (Set.extremePoints ℝ (Hpolytope hH_)) = Vpolytope hExHFinite
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
hExHFinite : Set.Finite (Set.extremePoints ℝ (Hpolytope hH_))
this : closure ((convexHull ℝ) (Set.extremePoints ℝ (Hpolytope hH_))) = Hpolytope hH_
⊢ IsClosed ((convexHull ℝ) (Set.extremePoints ℝ (Hpolytope hH_))) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
hExHFinite : Set.Finite (Set.extremePoints ℝ (Hpolytope hH_))
this : closure ((convexHull ℝ) (Set.extremePoints ℝ (Hpolytope hH_))) = Hpolytope hH_
⊢ IsClosed ((convexHull ℝ) (Set.extremePoints ℝ (Hpolytope hH_))) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Hpolytope | [419, 1] | [476, 7] | exact Closed_Vpolytope hExHFinite | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
hExHFinite : Set.Finite (Set.extremePoints ℝ (Hpolytope hH_))
this : closure ((convexHull ℝ) (Set.extremePoints ℝ (Hpolytope hH_))) = Hpolytope hH_
⊢ IsClosed ((convexHull ℝ) (Set.extremePoints ℝ (Hpolytope hH_))) | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Hpolytope | [419, 1] | [476, 7] | have := ExtremePointsofHpolytope hH_ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
⊢ Set.Finite (Set.extremePoints ℝ (Hpolytope hH_)) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
⊢ Set.Finite (Set.extremePoints ℝ (Hpolytope hH_)) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Hpolytope | [419, 1] | [476, 7] | let g : E ↪ Set E :=
⟨ fun x : E => Set.singleton x, Set.singleton_injective ⟩ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
⊢ Set.Finite (Set.extremePoints ℝ (Hpolytope hH_)) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
⊢ Set.Finite (Set.extremePoints ℝ (Hpolytope hH_)) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Hpolytope | [419, 1] | [476, 7] | rcases Set.Finite.exists_finset_coe hH_ with ⟨ Hfin, hHfin ⟩ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
⊢ Set.Finite (Set.extremePoints ℝ (Hpolytope hH_)) | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
⊢ Set.Finite (Set.extremePoints ℝ (Hpolytope hH_)) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Hpolytope | [419, 1] | [476, 7] | let PHfin := Hfin.powerset | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
⊢ Set.Finite (Set.extremePoints ℝ (Hpolytope hH_)) | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
⊢ Set.Finite (Set.extremePoints ℝ (Hpolytope hH_)) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Hpolytope | [419, 1] | [476, 7] | let PH := Finset.toSet '' PHfin.toSet | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
⊢ Set.Finite (Set.extremePoints ℝ (Hpolytope hH_)) | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
⊢ Set.Finite (Set.extremePoints ℝ (Hpolytope hH_)) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Hpolytope | [419, 1] | [476, 7] | have hPH : PH.Finite := PHfin.finite_toSet.image _ | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
⊢ Set.Finite (Set.extremePoints ℝ (Hpolytope hH_)) | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
⊢ Set.Finite (Set.extremePoints ℝ (Hpolytope hH_)) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Hpolytope | [419, 1] | [476, 7] | have hfPH : (f '' PH).Finite := hPH.image f | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
⊢ Set.Finite (Set.extremePoints ℝ (Hpolytope hH_)) | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
⊢ Set.Finite (Set.extremePoints ℝ (Hpolytope hH_)) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Hpolytope | [419, 1] | [476, 7] | have hgExFin : Set.Finite <| g '' (Set.extremePoints ℝ (Hpolytope hH_)) := Set.Finite.subset hfPH hgfPH | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
hgfPH : ⇑g '' Set.extremePoints ℝ (Hpolytope hH_) ⊆ f '' PH
⊢ Set.Finite (Set.extremePoints ℝ (Hpolytope hH_)) | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
hgfPH : ⇑g '' Set.extremePoints ℝ (Hpolytope hH_) ⊆ f '' PH
hgExFin : Set.Finite (⇑g '' Set.extremePoints ℝ (Hpolytope hH_))
⊢ Set.Finite (Set.extremePoints ℝ (Hpolytope hH_)) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Hpolytope | [419, 1] | [476, 7] | have := hgExFin.preimage_embedding g | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
hgfPH : ⇑g '' Set.extremePoints ℝ (Hpolytope hH_) ⊆ f '' PH
hgExFin : Set.Finite (⇑g '' Set.extremePoints ℝ (Hpolytope hH_))
⊢ Set.Finite (Set.extremePoints ℝ (Hpolytope hH_)) | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this✝ :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
hgfPH : ⇑g '' Set.extremePoints ℝ (Hpolytope hH_) ⊆ f '' PH
hgExFin : Set.Finite (⇑g '' Set.extremePoints ℝ (Hpolytope hH_))
this : Set.Finite (⇑g ⁻¹' (⇑g '' Set.extremePoints ℝ (Hpolytope hH_)))
⊢ Set.Finite (Set.extremePoints ℝ (Hpolytope hH_)) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Hpolytope | [419, 1] | [476, 7] | rw [Function.Injective.preimage_image] at this | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this✝ :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
hgfPH : ⇑g '' Set.extremePoints ℝ (Hpolytope hH_) ⊆ f '' PH
hgExFin : Set.Finite (⇑g '' Set.extremePoints ℝ (Hpolytope hH_))
this : Set.Finite (⇑g ⁻¹' (⇑g '' Set.extremePoints ℝ (Hpolytope hH_)))
⊢ Set.Finite (Set.extremePoints ℝ (Hpolytope hH_)) | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this✝ :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
hgfPH : ⇑g '' Set.extremePoints ℝ (Hpolytope hH_) ⊆ f '' PH
hgExFin : Set.Finite (⇑g '' Set.extremePoints ℝ (Hpolytope hH_))
this : Set.Finite (Set.extremePoints ℝ (Hpolytope hH_))
⊢ Set.Finite (Set.extremePoints ℝ (Hpolytope hH_))
case intro.hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this✝ :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
hgfPH : ⇑g '' Set.extremePoints ℝ (Hpolytope hH_) ⊆ f '' PH
hgExFin : Set.Finite (⇑g '' Set.extremePoints ℝ (Hpolytope hH_))
this : Set.Finite (⇑g ⁻¹' (⇑g '' Set.extremePoints ℝ (Hpolytope hH_)))
⊢ Function.Injective ⇑g |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Hpolytope | [419, 1] | [476, 7] | exact this | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this✝ :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
hgfPH : ⇑g '' Set.extremePoints ℝ (Hpolytope hH_) ⊆ f '' PH
hgExFin : Set.Finite (⇑g '' Set.extremePoints ℝ (Hpolytope hH_))
this : Set.Finite (Set.extremePoints ℝ (Hpolytope hH_))
⊢ Set.Finite (Set.extremePoints ℝ (Hpolytope hH_))
case intro.hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this✝ :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
hgfPH : ⇑g '' Set.extremePoints ℝ (Hpolytope hH_) ⊆ f '' PH
hgExFin : Set.Finite (⇑g '' Set.extremePoints ℝ (Hpolytope hH_))
this : Set.Finite (⇑g ⁻¹' (⇑g '' Set.extremePoints ℝ (Hpolytope hH_)))
⊢ Function.Injective ⇑g | case intro.hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this✝ :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
hgfPH : ⇑g '' Set.extremePoints ℝ (Hpolytope hH_) ⊆ f '' PH
hgExFin : Set.Finite (⇑g '' Set.extremePoints ℝ (Hpolytope hH_))
this : Set.Finite (⇑g ⁻¹' (⇑g '' Set.extremePoints ℝ (Hpolytope hH_)))
⊢ Function.Injective ⇑g |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Hpolytope | [419, 1] | [476, 7] | exact g.2 | case intro.hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this✝ :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
hgfPH : ⇑g '' Set.extremePoints ℝ (Hpolytope hH_) ⊆ f '' PH
hgExFin : Set.Finite (⇑g '' Set.extremePoints ℝ (Hpolytope hH_))
this : Set.Finite (⇑g ⁻¹' (⇑g '' Set.extremePoints ℝ (Hpolytope hH_)))
⊢ Function.Injective ⇑g | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Hpolytope | [419, 1] | [476, 7] | intro Sx hSx | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
⊢ ⇑g '' Set.extremePoints ℝ (Hpolytope hH_) ⊆ f '' PH | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
Sx : Set E
hSx : Sx ∈ ⇑g '' Set.extremePoints ℝ (Hpolytope hH_)
⊢ Sx ∈ f '' PH |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Hpolytope | [419, 1] | [476, 7] | rcases hSx with ⟨ x, hx, rfl ⟩ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
Sx : Set E
hSx : Sx ∈ ⇑g '' Set.extremePoints ℝ (Hpolytope hH_)
⊢ Sx ∈ f '' PH | case intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
x : E
hx : x ∈ Set.extremePoints ℝ (Hpolytope hH_)
⊢ g x ∈ f '' PH |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Hpolytope | [419, 1] | [476, 7] | change {x} ∈ f '' PH | case intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
x : E
hx : x ∈ Set.extremePoints ℝ (Hpolytope hH_)
⊢ g x ∈ f '' PH | case intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
x : E
hx : x ∈ Set.extremePoints ℝ (Hpolytope hH_)
⊢ {x} ∈ f '' PH |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Hpolytope | [419, 1] | [476, 7] | rw [PH.mem_image] | case intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
x : E
hx : x ∈ Set.extremePoints ℝ (Hpolytope hH_)
⊢ {x} ∈ f '' PH | case intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
x : E
hx : x ∈ Set.extremePoints ℝ (Hpolytope hH_)
⊢ ∃ x_1 ∈ PH, f x_1 = {x} |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Hpolytope | [419, 1] | [476, 7] | refine ⟨ Hpolytope.I H_ x, ?_, ?_ ⟩ | case intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
x : E
hx : x ∈ Set.extremePoints ℝ (Hpolytope hH_)
⊢ ∃ x_1 ∈ PH, f x_1 = {x} | case intro.intro.refine_1
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
x : E
hx : x ∈ Set.extremePoints ℝ (Hpolytope hH_)
⊢ Hpolytope.I H_ x ∈ PH
case intro.intro.refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
x : E
hx : x ∈ Set.extremePoints ℝ (Hpolytope hH_)
⊢ f (Hpolytope.I H_ x) = {x} |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Hpolytope | [419, 1] | [476, 7] | rw [Set.mem_image] | case intro.intro.refine_1
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
x : E
hx : x ∈ Set.extremePoints ℝ (Hpolytope hH_)
⊢ Hpolytope.I H_ x ∈ PH | case intro.intro.refine_1
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
x : E
hx : x ∈ Set.extremePoints ℝ (Hpolytope hH_)
⊢ ∃ x_1 ∈ ↑PHfin, ↑x_1 = Hpolytope.I H_ x |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Hpolytope | [419, 1] | [476, 7] | rcases (hH_.subset (Hpolytope.I_sub x)).exists_finset_coe with ⟨ Ifin, hIfin ⟩ | case intro.intro.refine_1
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
x : E
hx : x ∈ Set.extremePoints ℝ (Hpolytope hH_)
⊢ ∃ x_1 ∈ ↑PHfin, ↑x_1 = Hpolytope.I H_ x | case intro.intro.refine_1.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
x : E
hx : x ∈ Set.extremePoints ℝ (Hpolytope hH_)
Ifin : Finset (Halfspace E)
hIfin : ↑Ifin = Hpolytope.I H_ x
⊢ ∃ x_1 ∈ ↑PHfin, ↑x_1 = Hpolytope.I H_ x |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Hpolytope | [419, 1] | [476, 7] | refine ⟨ Ifin, ?_, hIfin ⟩ | case intro.intro.refine_1.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
x : E
hx : x ∈ Set.extremePoints ℝ (Hpolytope hH_)
Ifin : Finset (Halfspace E)
hIfin : ↑Ifin = Hpolytope.I H_ x
⊢ ∃ x_1 ∈ ↑PHfin, ↑x_1 = Hpolytope.I H_ x | case intro.intro.refine_1.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
x : E
hx : x ∈ Set.extremePoints ℝ (Hpolytope hH_)
Ifin : Finset (Halfspace E)
hIfin : ↑Ifin = Hpolytope.I H_ x
⊢ Ifin ∈ ↑PHfin |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Hpolytope | [419, 1] | [476, 7] | rw [Finset.mem_coe, Finset.mem_powerset, ← Finset.coe_subset, hHfin, hIfin] | case intro.intro.refine_1.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
x : E
hx : x ∈ Set.extremePoints ℝ (Hpolytope hH_)
Ifin : Finset (Halfspace E)
hIfin : ↑Ifin = Hpolytope.I H_ x
⊢ Ifin ∈ ↑PHfin | case intro.intro.refine_1.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
x : E
hx : x ∈ Set.extremePoints ℝ (Hpolytope hH_)
Ifin : Finset (Halfspace E)
hIfin : ↑Ifin = Hpolytope.I H_ x
⊢ Hpolytope.I H_ x ⊆ H_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Hpolytope | [419, 1] | [476, 7] | exact Hpolytope.I_sub x | case intro.intro.refine_1.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
x : E
hx : x ∈ Set.extremePoints ℝ (Hpolytope hH_)
Ifin : Finset (Halfspace E)
hIfin : ↑Ifin = Hpolytope.I H_ x
⊢ Hpolytope.I H_ x ⊆ H_ | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Hpolytope | [419, 1] | [476, 7] | rw [← ExtremePointsofHpolytope hH_ x (extremePoints_subset hx)] | case intro.intro.refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
x : E
hx : x ∈ Set.extremePoints ℝ (Hpolytope hH_)
⊢ f (Hpolytope.I H_ x) = {x} | case intro.intro.refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
x : E
hx : x ∈ Set.extremePoints ℝ (Hpolytope hH_)
⊢ x ∈ Set.extremePoints ℝ (Hpolytope hH_) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Vpolytope_of_Hpolytope | [419, 1] | [476, 7] | exact hx | case intro.intro.refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
hHcpt : IsCompact (Hpolytope hH_)
this :
∀ x ∈ Hpolytope hH_, x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
f : Set (Halfspace E) → Set E := fun T => ⋂₀ ((fun x => frontier ↑x) '' T)
g : E ↪ Set E := { toFun := fun x => Set.singleton x, inj' := ⋯ }
Hfin : Finset (Halfspace E)
hHfin : ↑Hfin = H_
PHfin : Finset (Finset (Halfspace E)) := Finset.powerset Hfin
PH : Set (Set (Halfspace E)) := Finset.toSet '' ↑PHfin
hPH : Set.Finite PH
hfPH : Set.Finite (f '' PH)
x : E
hx : x ∈ Set.extremePoints ℝ (Hpolytope hH_)
⊢ x ∈ Set.extremePoints ℝ (Hpolytope hH_) | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope_of_Vpolytope_subsingleton | [478, 1] | [490, 7] | cases' hStrivial.eq_empty_or_singleton with hSempty hSsingleton | 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
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case inl
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
hSempty : S = ∅
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS
case inr
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
hSsingleton : ∃ x, S = {x}
⊢ ∃ 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] | rw [Vpolytope, hSempty, convexHull_empty] | case inl
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
hSempty : S = ∅
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | case inl
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
hSempty : S = ∅
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = ∅ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope_of_Vpolytope_subsingleton | [478, 1] | [490, 7] | exact empty_Hpolytope | case inl
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
hSempty : S = ∅
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = ∅ | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | Hpolytope_of_Vpolytope_subsingleton | [478, 1] | [490, 7] | rcases hSsingleton with ⟨ x, hx ⟩ | case inr
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
hSsingleton : ∃ x, S = {x}
⊢ ∃ H_, ∃ (hH_ : Set.Finite H_), Hpolytope hH_ = Vpolytope hS | 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 |
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