url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
|
|---|---|---|---|---|---|---|---|---|
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
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.