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https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | constructor | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
⊢ x ∈ Set.extremePoints ℝ (Hpolytope hH_) ↔ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x} | case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
⊢ x ∈ Set.extremePoints ℝ (Hpolytope hH_) → ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
⊢ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x} → x ∈ Set.extremePoints ℝ (Hpolytope hH_) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | intro hxEx | case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
⊢ x ∈ Set.extremePoints ℝ (Hpolytope hH_) → ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x} | case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hxEx : x ∈ Set.extremePoints ℝ (Hpolytope hH_)
⊢ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x} |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rw [Set.eq_singleton_iff_unique_mem] | case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hxEx : x ∈ Set.extremePoints ℝ (Hpolytope hH_)
⊢ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x} | case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hxEx : x ∈ Set.extremePoints ℝ (Hpolytope hH_)
⊢ x ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) ∧ ∀ x_1 ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x), x_1 = x |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | refine ⟨ fun HiS ⟨ Hi_, hHi_, h ⟩ => h ▸ hHi_.2, ?_ ⟩ | case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hxEx : x ∈ Set.extremePoints ℝ (Hpolytope hH_)
⊢ x ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) ∧ ∀ x_1 ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x), x_1 = x | case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hxEx : x ∈ Set.extremePoints ℝ (Hpolytope hH_)
⊢ ∀ x_1 ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x), x_1 = x |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | contrapose! hxEx | case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hxEx : x ∈ Set.extremePoints ℝ (Hpolytope hH_)
⊢ ∀ x_1 ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x), x_1 = x | case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hxEx : ∃ x_1 ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x), x_1 ≠ x
⊢ x ∉ Set.extremePoints ℝ (Hpolytope hH_) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rcases hxEx with ⟨ y, hy, hyx ⟩ | case mp
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hxEx : ∃ x_1 ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x), x_1 ≠ x
⊢ x ∉ Set.extremePoints ℝ (Hpolytope hH_) | case mp.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
⊢ x ∉ Set.extremePoints ℝ (Hpolytope hH_) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | have hxyy : x ∈ openSegment ℝ ((2:ℝ) • x - y) y := by
clear hyx hy hxH hH_
rw [openSegment_eq_image, Set.mem_image]
refine ⟨ 1/2, by norm_num, ?_ ⟩
rw [(by norm_num : (1:ℝ) - 1 / 2 = 1 / 2), smul_sub, sub_add_cancel, smul_smul,
div_mul_cancel _ (by linarith), one_smul]
done | case mp.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
⊢ x ∉ Set.extremePoints ℝ (Hpolytope hH_) | case mp.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
⊢ x ∉ Set.extremePoints ℝ (Hpolytope hH_) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | have hmemsegmemI : ∀ v, v ∈ segment ℝ ((2:ℝ) • x - y) y →
∀ Hi_, Hi_ ∈ Hpolytope.I H_ x → v ∈ SetLike.coe Hi_ := by
rintro v hv Hi_ hHi_
rw [Set.mem_sInter] at hy
specialize hy (frontier <| SetLike.coe Hi_) ⟨ Hi_, hHi_, rfl ⟩
have hHi_2 := hHi_.2
rw [frontierHalfspace_Hyperplane] at hy hHi_2
apply IsClosed.frontier_subset <| Halfspace_closed Hi_
rw [frontierHalfspace_Hyperplane]
apply Set.mem_of_mem_of_subset hv
apply (convex_iff_segment_subset.mp <| Hyperplane_convex Hi_) _ hy
have h21 : Finset.sum Finset.univ ![(2:ℝ), -1] = 1 := by
rw [Fin.sum_univ_two, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.head_cons]
linarith
done
have h2x_y := Hyperplane_affineClosed Hi_ ![x, y] (by
rw [Matrix.range_cons, Matrix.range_cons, Matrix.range_empty, Set.union_empty];
exact Set.union_subset (Set.singleton_subset_iff.mpr hHi_2) (Set.singleton_subset_iff.mpr hy))
![2, -1] h21
rw [Finset.affineCombination_eq_linear_combination _ _ _ h21, Fin.sum_univ_two, Matrix.cons_val_zero,
Matrix.cons_val_one, Matrix.head_cons, Matrix.cons_val_zero, Matrix.cons_val_one,
Matrix.head_cons, neg_one_smul, ← sub_eq_add_neg] at h2x_y
exact h2x_y | case mp.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
⊢ x ∉ Set.extremePoints ℝ (Hpolytope hH_) | case mp.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
⊢ x ∉ Set.extremePoints ℝ (Hpolytope hH_) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rw [mem_extremePoints] | case mp.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
⊢ x ∉ Set.extremePoints ℝ (Hpolytope hH_) | case mp.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
⊢ ¬(x ∈ Hpolytope hH_ ∧ ∀ x₁ ∈ Hpolytope hH_, ∀ x₂ ∈ Hpolytope hH_, x ∈ openSegment ℝ x₁ x₂ → x₁ = x ∧ x₂ = x) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | push_neg | case mp.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
⊢ ¬(x ∈ Hpolytope hH_ ∧ ∀ x₁ ∈ Hpolytope hH_, ∀ x₂ ∈ Hpolytope hH_, x ∈ openSegment ℝ x₁ x₂ → x₁ = x ∧ x₂ = x) | case mp.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
⊢ x ∈ Hpolytope hH_ → ∃ x₁ ∈ Hpolytope hH_, ∃ x₂ ∈ Hpolytope hH_, x ∈ openSegment ℝ x₁ x₂ ∧ (x₁ = x → x₂ ≠ x) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rintro hxH' | case mp.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
⊢ x ∈ Hpolytope hH_ → ∃ x₁ ∈ Hpolytope hH_, ∃ x₂ ∈ Hpolytope hH_, x ∈ openSegment ℝ x₁ x₂ ∧ (x₁ = x → x₂ ≠ x) | case mp.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
hxH' : x ∈ Hpolytope hH_
⊢ ∃ x₁ ∈ Hpolytope hH_, ∃ x₂ ∈ Hpolytope hH_, x ∈ openSegment ℝ x₁ x₂ ∧ (x₁ = x → x₂ ≠ x) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rcases hmemballmemIc with ⟨ ε, hε, hmemballmemIc ⟩ | case mp.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
hxH' : x ∈ Hpolytope hH_
⊢ ∃ x₁ ∈ Hpolytope hH_, ∃ x₂ ∈ Hpolytope hH_, x ∈ openSegment ℝ x₁ x₂ ∧ (x₁ = x → x₂ ≠ x) | case mp.intro.intro.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
hxH' : x ∈ Hpolytope hH_
ε : ℝ
hε : ε > 0
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
⊢ ∃ x₁ ∈ Hpolytope hH_, ∃ x₂ ∈ Hpolytope hH_, x ∈ openSegment ℝ x₁ x₂ ∧ (x₁ = x → x₂ ≠ x) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rcases hxSegBallInterSeg ((2:ℝ) • x - y) y ε ⟨ hxyy, fun h => hyx h.2 ⟩ hε with
⟨ x1, x2, hmem, hsub, hne ⟩ | case mp.intro.intro.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
hxH' : x ∈ Hpolytope hH_
ε : ℝ
hε : ε > 0
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
⊢ ∃ x₁ ∈ Hpolytope hH_, ∃ x₂ ∈ Hpolytope hH_, x ∈ openSegment ℝ x₁ x₂ ∧ (x₁ = x → x₂ ≠ x) | case mp.intro.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
hxH' : x ∈ Hpolytope hH_
ε : ℝ
hε : ε > 0
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
hmem : x ∈ openSegment ℝ x1 x2
hsub : segment ℝ x1 x2 ⊆ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hne : ¬(x1 = x ∧ x2 = x)
⊢ ∃ x₁ ∈ Hpolytope hH_, ∃ x₂ ∈ Hpolytope hH_, x ∈ openSegment ℝ x₁ x₂ ∧ (x₁ = x → x₂ ≠ x) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | push_neg at hne | case mp.intro.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
hxH' : x ∈ Hpolytope hH_
ε : ℝ
hε : ε > 0
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
hmem : x ∈ openSegment ℝ x1 x2
hsub : segment ℝ x1 x2 ⊆ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hne : ¬(x1 = x ∧ x2 = x)
⊢ ∃ x₁ ∈ Hpolytope hH_, ∃ x₂ ∈ Hpolytope hH_, x ∈ openSegment ℝ x₁ x₂ ∧ (x₁ = x → x₂ ≠ x) | case mp.intro.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
hxH' : x ∈ Hpolytope hH_
ε : ℝ
hε : ε > 0
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
hmem : x ∈ openSegment ℝ x1 x2
hsub : segment ℝ x1 x2 ⊆ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hne : x1 = x → x2 ≠ x
⊢ ∃ x₁ ∈ Hpolytope hH_, ∃ x₂ ∈ Hpolytope hH_, x ∈ openSegment ℝ x₁ x₂ ∧ (x₁ = x → x₂ ≠ x) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | clear hxH' hε hyx hy hxH hxyy | case mp.intro.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
hxH' : x ∈ Hpolytope hH_
ε : ℝ
hε : ε > 0
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
hmem : x ∈ openSegment ℝ x1 x2
hsub : segment ℝ x1 x2 ⊆ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hne : x1 = x → x2 ≠ x
⊢ ∃ x₁ ∈ Hpolytope hH_, ∃ x₂ ∈ Hpolytope hH_, x ∈ openSegment ℝ x₁ x₂ ∧ (x₁ = x → x₂ ≠ x) | case mp.intro.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
hmem : x ∈ openSegment ℝ x1 x2
hsub : segment ℝ x1 x2 ⊆ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hne : x1 = x → x2 ≠ x
⊢ ∃ x₁ ∈ Hpolytope hH_, ∃ x₂ ∈ Hpolytope hH_, x ∈ openSegment ℝ x₁ x₂ ∧ (x₁ = x → x₂ ≠ x) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | unfold Hpolytope | case mp.intro.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
hmem : x ∈ openSegment ℝ x1 x2
hsub : segment ℝ x1 x2 ⊆ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hne : x1 = x → x2 ≠ x
⊢ ∃ x₁ ∈ Hpolytope hH_, ∃ x₂ ∈ Hpolytope hH_, x ∈ openSegment ℝ x₁ x₂ ∧ (x₁ = x → x₂ ≠ x) | case mp.intro.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
hmem : x ∈ openSegment ℝ x1 x2
hsub : segment ℝ x1 x2 ⊆ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hne : x1 = x → x2 ≠ x
⊢ ∃ x₁ ∈ ⋂₀ (SetLike.coe '' H_), ∃ x₂ ∈ ⋂₀ (SetLike.coe '' H_), x ∈ openSegment ℝ x₁ x₂ ∧ (x₁ = x → x₂ ≠ x) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | refine ⟨ x1, ?_, x2, ?_, ⟨ hmem, hne ⟩ ⟩ <;> clear hmem hne <;>
rw [Set.mem_sInter] <;>
intro Hi_s hHi_s <;>
rw [Set.mem_image] at hHi_s <;>
rcases hHi_s with ⟨ Hi_, hHi_, rfl ⟩ | case mp.intro.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
hmem : x ∈ openSegment ℝ x1 x2
hsub : segment ℝ x1 x2 ⊆ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hne : x1 = x → x2 ≠ x
⊢ ∃ x₁ ∈ ⋂₀ (SetLike.coe '' H_), ∃ x₂ ∈ ⋂₀ (SetLike.coe '' H_), x ∈ openSegment ℝ x₁ x₂ ∧ (x₁ = x → x₂ ≠ x) | case mp.intro.intro.intro.intro.intro.intro.intro.intro.refine_1.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
hsub : segment ℝ x1 x2 ⊆ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
⊢ x1 ∈ ↑Hi_
case mp.intro.intro.intro.intro.intro.intro.intro.intro.refine_2.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
hsub : segment ℝ x1 x2 ⊆ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
⊢ x2 ∈ ↑Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | clear hyx hy hxH hH_ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
⊢ x ∈ openSegment ℝ (2 • x - y) y | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
x y : E
⊢ x ∈ openSegment ℝ (2 • x - y) y |
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] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
x y : E
⊢ x ∈ openSegment ℝ (2 • x - y) y | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
x y : E
⊢ ∃ x_1 ∈ Set.Ioo 0 1, (1 - x_1) • (2 • x - y) + x_1 • y = x |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | refine ⟨ 1/2, by norm_num, ?_ ⟩ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
x y : E
⊢ ∃ x_1 ∈ Set.Ioo 0 1, (1 - x_1) • (2 • x - y) + x_1 • y = x | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
x y : E
⊢ (1 - 1 / 2) • (2 • x - y) + (1 / 2) • y = x |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rw [(by norm_num : (1:ℝ) - 1 / 2 = 1 / 2), smul_sub, sub_add_cancel, smul_smul,
div_mul_cancel _ (by linarith), one_smul] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
x y : E
⊢ (1 - 1 / 2) • (2 • x - y) + (1 / 2) • y = x | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | norm_num | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
x y : E
⊢ 1 / 2 ∈ Set.Ioo 0 1 | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | norm_num | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
x y : E
⊢ 1 - 1 / 2 = 1 / 2 | 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
H_ : Set (Halfspace E)
x y : E
⊢ 2 ≠ 0 | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rcases hball with ⟨ ε, hε, hball ⟩ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hball : ∃ ε > 0, Metric.ball x ε ⊆ ⋂₀ ((fun x => interior ↑x) '' (H_ \ Hpolytope.I H_ x))
⊢ ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_ | 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_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
ε : ℝ
hε : ε > 0
hball : Metric.ball x ε ⊆ ⋂₀ ((fun x => interior ↑x) '' (H_ \ Hpolytope.I H_ x))
⊢ ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | refine ⟨ ε, hε, fun v hv Hi_ hHi_ => ?_ ⟩ | 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_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
ε : ℝ
hε : ε > 0
hball : Metric.ball x ε ⊆ ⋂₀ ((fun x => interior ↑x) '' (H_ \ Hpolytope.I H_ x))
⊢ ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_ | 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_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
ε : ℝ
hε : ε > 0
hball : Metric.ball x ε ⊆ ⋂₀ ((fun x => interior ↑x) '' (H_ \ Hpolytope.I H_ x))
v : E
hv : v ∈ Metric.ball x ε
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_ \ Hpolytope.I H_ x
⊢ v ∈ ↑Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | apply interior_subset | 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_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
ε : ℝ
hε : ε > 0
hball : Metric.ball x ε ⊆ ⋂₀ ((fun x => interior ↑x) '' (H_ \ Hpolytope.I H_ x))
v : E
hv : v ∈ Metric.ball x ε
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_ \ Hpolytope.I H_ x
⊢ v ∈ ↑Hi_ | case intro.intro.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
ε : ℝ
hε : ε > 0
hball : Metric.ball x ε ⊆ ⋂₀ ((fun x => interior ↑x) '' (H_ \ Hpolytope.I H_ x))
v : E
hv : v ∈ Metric.ball x ε
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_ \ Hpolytope.I H_ x
⊢ v ∈ interior ↑Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | exact (Set.mem_sInter.mp <| hball hv) (interior <| SetLike.coe Hi_) ⟨ Hi_, hHi_, rfl ⟩ | case intro.intro.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
ε : ℝ
hε : ε > 0
hball : Metric.ball x ε ⊆ ⋂₀ ((fun x => interior ↑x) '' (H_ \ Hpolytope.I H_ x))
v : E
hv : v ∈ Metric.ball x ε
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_ \ Hpolytope.I H_ x
⊢ v ∈ interior ↑Hi_ | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | unfold Hpolytope at hxH | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
⊢ ∃ ε > 0, Metric.ball x ε ⊆ ⋂₀ ((fun x => interior ↑x) '' (H_ \ Hpolytope.I H_ x)) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ ⋂₀ (SetLike.coe '' H_)
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
⊢ ∃ ε > 0, Metric.ball x ε ⊆ ⋂₀ ((fun x => interior ↑x) '' (H_ \ Hpolytope.I H_ x)) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rw [Metric.isOpen_iff] at hIcinteriorOpen | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ ⋂₀ (SetLike.coe '' H_)
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hxIcinterior : x ∈ ⋂₀ ((fun x => interior ↑x) '' (H_ \ Hpolytope.I H_ x))
hIcinteriorOpen : IsOpen (⋂₀ ((fun x => interior ↑x) '' (H_ \ Hpolytope.I H_ x)))
⊢ ∃ ε > 0, Metric.ball x ε ⊆ ⋂₀ ((fun x => interior ↑x) '' (H_ \ Hpolytope.I H_ x)) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ ⋂₀ (SetLike.coe '' H_)
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hxIcinterior : x ∈ ⋂₀ ((fun x => interior ↑x) '' (H_ \ Hpolytope.I H_ x))
hIcinteriorOpen :
∀ x_1 ∈ ⋂₀ ((fun x => interior ↑x) '' (H_ \ Hpolytope.I H_ x)),
∃ ε > 0, Metric.ball x_1 ε ⊆ ⋂₀ ((fun x => interior ↑x) '' (H_ \ Hpolytope.I H_ x))
⊢ ∃ ε > 0, Metric.ball x ε ⊆ ⋂₀ ((fun x => interior ↑x) '' (H_ \ Hpolytope.I H_ x)) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | exact hIcinteriorOpen x hxIcinterior | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ ⋂₀ (SetLike.coe '' H_)
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hxIcinterior : x ∈ ⋂₀ ((fun x => interior ↑x) '' (H_ \ Hpolytope.I H_ x))
hIcinteriorOpen :
∀ x_1 ∈ ⋂₀ ((fun x => interior ↑x) '' (H_ \ Hpolytope.I H_ x)),
∃ ε > 0, Metric.ball x_1 ε ⊆ ⋂₀ ((fun x => interior ↑x) '' (H_ \ Hpolytope.I H_ x))
⊢ ∃ ε > 0, Metric.ball x ε ⊆ ⋂₀ ((fun x => interior ↑x) '' (H_ \ Hpolytope.I H_ x)) | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rintro HiS ⟨ Hi_, hHi_, rfl ⟩ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ ⋂₀ (SetLike.coe '' H_)
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
⊢ x ∈ ⋂₀ ((fun x => interior ↑x) '' (H_ \ 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_
x : E
hxH : x ∈ ⋂₀ (SetLike.coe '' H_)
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_ \ Hpolytope.I H_ x
⊢ x ∈ (fun x => interior ↑x) Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rw [Set.mem_diff, Hpolytope.I_mem, IsClosed.frontier_eq <| Halfspace_closed Hi_,
Set.mem_diff] at hHi_ | 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_
x : E
hxH : x ∈ ⋂₀ (SetLike.coe '' H_)
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_ \ Hpolytope.I H_ x
⊢ x ∈ (fun x => interior ↑x) Hi_ | 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_
x : E
hxH : x ∈ ⋂₀ (SetLike.coe '' H_)
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_ ∧ ¬(Hi_ ∈ H_ ∧ x ∈ ↑Hi_ ∧ x ∉ interior ↑Hi_)
⊢ x ∈ (fun x => interior ↑x) Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | push_neg at hHi_ | 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_
x : E
hxH : x ∈ ⋂₀ (SetLike.coe '' H_)
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_ ∧ ¬(Hi_ ∈ H_ ∧ x ∈ ↑Hi_ ∧ x ∉ interior ↑Hi_)
⊢ x ∈ (fun x => interior ↑x) Hi_ | 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_
x : E
hxH : x ∈ ⋂₀ (SetLike.coe '' H_)
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_ ∧ (Hi_ ∈ H_ → x ∈ ↑Hi_ → x ∈ interior ↑Hi_)
⊢ x ∈ (fun x => interior ↑x) Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | exact hHi_.2 hHi_.1 <| hxH Hi_ ⟨ Hi_, hHi_.1, rfl ⟩ | 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_
x : E
hxH : x ∈ ⋂₀ (SetLike.coe '' H_)
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_ ∧ (Hi_ ∈ H_ → x ∈ ↑Hi_ → x ∈ interior ↑Hi_)
⊢ x ∈ (fun x => interior ↑x) Hi_ | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | apply Set.Finite.isOpen_sInter (Set.Finite.image _ (Set.Finite.diff hH_ _)) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ ⋂₀ (SetLike.coe '' H_)
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hxIcinterior : x ∈ ⋂₀ ((fun x => interior ↑x) '' (H_ \ Hpolytope.I H_ x))
⊢ IsOpen (⋂₀ ((fun x => interior ↑x) '' (H_ \ Hpolytope.I H_ x))) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ ⋂₀ (SetLike.coe '' H_)
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hxIcinterior : x ∈ ⋂₀ ((fun x => interior ↑x) '' (H_ \ Hpolytope.I H_ x))
⊢ ∀ t ∈ (fun x => interior ↑x) '' (H_ \ Hpolytope.I H_ x), IsOpen t |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | exact fun _ ⟨ Hi_, _, h ⟩ => h ▸ isOpen_interior | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ ⋂₀ (SetLike.coe '' H_)
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hxIcinterior : x ∈ ⋂₀ ((fun x => interior ↑x) '' (H_ \ Hpolytope.I H_ x))
⊢ ∀ t ∈ (fun x => interior ↑x) '' (H_ \ Hpolytope.I H_ x), IsOpen t | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rintro v hv Hi_ hHi_ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
⊢ ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
v : E
hv : v ∈ segment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
⊢ v ∈ ↑Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rw [Set.mem_sInter] at hy | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hy : y ∈ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x)
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
v : E
hv : v ∈ segment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
⊢ v ∈ ↑Hi_ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hy : ∀ t ∈ (fun x => frontier ↑x) '' Hpolytope.I H_ x, y ∈ t
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
v : E
hv : v ∈ segment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
⊢ v ∈ ↑Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | specialize hy (frontier <| SetLike.coe Hi_) ⟨ Hi_, hHi_, rfl ⟩ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hy : ∀ t ∈ (fun x => frontier ↑x) '' Hpolytope.I H_ x, y ∈ t
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
v : E
hv : v ∈ segment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
⊢ v ∈ ↑Hi_ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
v : E
hv : v ∈ segment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hy : y ∈ frontier ↑Hi_
⊢ v ∈ ↑Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | have hHi_2 := hHi_.2 | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
v : E
hv : v ∈ segment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hy : y ∈ frontier ↑Hi_
⊢ v ∈ ↑Hi_ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
v : E
hv : v ∈ segment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hy : y ∈ frontier ↑Hi_
hHi_2 : x ∈ frontier ↑Hi_
⊢ v ∈ ↑Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rw [frontierHalfspace_Hyperplane] at hy hHi_2 | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
v : E
hv : v ∈ segment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hy : y ∈ frontier ↑Hi_
hHi_2 : x ∈ frontier ↑Hi_
⊢ v ∈ ↑Hi_ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
v : E
hv : v ∈ segment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hy : y ∈ {x | ↑Hi_.f x = Hi_.α}
hHi_2 : x ∈ {x | ↑Hi_.f x = Hi_.α}
⊢ v ∈ ↑Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | apply IsClosed.frontier_subset <| Halfspace_closed Hi_ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
v : E
hv : v ∈ segment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hy : y ∈ {x | ↑Hi_.f x = Hi_.α}
hHi_2 : x ∈ {x | ↑Hi_.f x = Hi_.α}
⊢ v ∈ ↑Hi_ | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
v : E
hv : v ∈ segment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hy : y ∈ {x | ↑Hi_.f x = Hi_.α}
hHi_2 : x ∈ {x | ↑Hi_.f x = Hi_.α}
⊢ v ∈ frontier ↑Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rw [frontierHalfspace_Hyperplane] | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
v : E
hv : v ∈ segment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hy : y ∈ {x | ↑Hi_.f x = Hi_.α}
hHi_2 : x ∈ {x | ↑Hi_.f x = Hi_.α}
⊢ v ∈ frontier ↑Hi_ | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
v : E
hv : v ∈ segment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hy : y ∈ {x | ↑Hi_.f x = Hi_.α}
hHi_2 : x ∈ {x | ↑Hi_.f x = Hi_.α}
⊢ v ∈ {x | ↑Hi_.f x = Hi_.α} |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | apply Set.mem_of_mem_of_subset hv | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
v : E
hv : v ∈ segment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hy : y ∈ {x | ↑Hi_.f x = Hi_.α}
hHi_2 : x ∈ {x | ↑Hi_.f x = Hi_.α}
⊢ v ∈ {x | ↑Hi_.f x = Hi_.α} | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
v : E
hv : v ∈ segment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hy : y ∈ {x | ↑Hi_.f x = Hi_.α}
hHi_2 : x ∈ {x | ↑Hi_.f x = Hi_.α}
⊢ segment ℝ (2 • x - y) y ⊆ {x | ↑Hi_.f x = Hi_.α} |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | apply (convex_iff_segment_subset.mp <| Hyperplane_convex Hi_) _ hy | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
v : E
hv : v ∈ segment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hy : y ∈ {x | ↑Hi_.f x = Hi_.α}
hHi_2 : x ∈ {x | ↑Hi_.f x = Hi_.α}
⊢ segment ℝ (2 • x - y) y ⊆ {x | ↑Hi_.f x = Hi_.α} | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
v : E
hv : v ∈ segment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hy : y ∈ {x | ↑Hi_.f x = Hi_.α}
hHi_2 : x ∈ {x | ↑Hi_.f x = Hi_.α}
⊢ 2 • x - y ∈ {x | ↑Hi_.f x = Hi_.α} |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | have h21 : Finset.sum Finset.univ ![(2:ℝ), -1] = 1 := by
rw [Fin.sum_univ_two, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.head_cons]
linarith
done | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
v : E
hv : v ∈ segment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hy : y ∈ {x | ↑Hi_.f x = Hi_.α}
hHi_2 : x ∈ {x | ↑Hi_.f x = Hi_.α}
⊢ 2 • x - y ∈ {x | ↑Hi_.f x = Hi_.α} | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
v : E
hv : v ∈ segment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hy : y ∈ {x | ↑Hi_.f x = Hi_.α}
hHi_2 : x ∈ {x | ↑Hi_.f x = Hi_.α}
h21 : Finset.sum Finset.univ ![2, -1] = 1
⊢ 2 • x - y ∈ {x | ↑Hi_.f x = Hi_.α} |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | have h2x_y := Hyperplane_affineClosed Hi_ ![x, y] (by
rw [Matrix.range_cons, Matrix.range_cons, Matrix.range_empty, Set.union_empty];
exact Set.union_subset (Set.singleton_subset_iff.mpr hHi_2) (Set.singleton_subset_iff.mpr hy))
![2, -1] h21 | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
v : E
hv : v ∈ segment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hy : y ∈ {x | ↑Hi_.f x = Hi_.α}
hHi_2 : x ∈ {x | ↑Hi_.f x = Hi_.α}
h21 : Finset.sum Finset.univ ![2, -1] = 1
⊢ 2 • x - y ∈ {x | ↑Hi_.f x = Hi_.α} | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
v : E
hv : v ∈ segment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hy : y ∈ {x | ↑Hi_.f x = Hi_.α}
hHi_2 : x ∈ {x | ↑Hi_.f x = Hi_.α}
h21 : Finset.sum Finset.univ ![2, -1] = 1
h2x_y : (Finset.affineCombination ℝ Finset.univ ![x, y]) ![2, -1] ∈ {x | ↑Hi_.f x = Hi_.α}
⊢ 2 • x - y ∈ {x | ↑Hi_.f x = Hi_.α} |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rw [Finset.affineCombination_eq_linear_combination _ _ _ h21, Fin.sum_univ_two, Matrix.cons_val_zero,
Matrix.cons_val_one, Matrix.head_cons, Matrix.cons_val_zero, Matrix.cons_val_one,
Matrix.head_cons, neg_one_smul, ← sub_eq_add_neg] at h2x_y | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
v : E
hv : v ∈ segment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hy : y ∈ {x | ↑Hi_.f x = Hi_.α}
hHi_2 : x ∈ {x | ↑Hi_.f x = Hi_.α}
h21 : Finset.sum Finset.univ ![2, -1] = 1
h2x_y : (Finset.affineCombination ℝ Finset.univ ![x, y]) ![2, -1] ∈ {x | ↑Hi_.f x = Hi_.α}
⊢ 2 • x - y ∈ {x | ↑Hi_.f x = Hi_.α} | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
v : E
hv : v ∈ segment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hy : y ∈ {x | ↑Hi_.f x = Hi_.α}
hHi_2 : x ∈ {x | ↑Hi_.f x = Hi_.α}
h21 : Finset.sum Finset.univ ![2, -1] = 1
h2x_y : 2 • x - y ∈ {x | ↑Hi_.f x = Hi_.α}
⊢ 2 • x - y ∈ {x | ↑Hi_.f x = Hi_.α} |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | exact h2x_y | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
v : E
hv : v ∈ segment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hy : y ∈ {x | ↑Hi_.f x = Hi_.α}
hHi_2 : x ∈ {x | ↑Hi_.f x = Hi_.α}
h21 : Finset.sum Finset.univ ![2, -1] = 1
h2x_y : 2 • x - y ∈ {x | ↑Hi_.f x = Hi_.α}
⊢ 2 • x - y ∈ {x | ↑Hi_.f x = Hi_.α} | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rw [Fin.sum_univ_two, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.head_cons] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
v : E
hv : v ∈ segment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hy : y ∈ {x | ↑Hi_.f x = Hi_.α}
hHi_2 : x ∈ {x | ↑Hi_.f x = Hi_.α}
⊢ Finset.sum Finset.univ ![2, -1] = 1 | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
v : E
hv : v ∈ segment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hy : y ∈ {x | ↑Hi_.f x = Hi_.α}
hHi_2 : x ∈ {x | ↑Hi_.f x = Hi_.α}
⊢ 2 + -1 = 1 |
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
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
v : E
hv : v ∈ segment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hy : y ∈ {x | ↑Hi_.f x = Hi_.α}
hHi_2 : x ∈ {x | ↑Hi_.f x = Hi_.α}
⊢ 2 + -1 = 1 | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rw [Matrix.range_cons, Matrix.range_cons, Matrix.range_empty, Set.union_empty] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
v : E
hv : v ∈ segment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hy : y ∈ {x | ↑Hi_.f x = Hi_.α}
hHi_2 : x ∈ {x | ↑Hi_.f x = Hi_.α}
h21 : Finset.sum Finset.univ ![2, -1] = 1
⊢ Set.range ![x, y] ⊆ {x | ↑Hi_.f x = Hi_.α} | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
v : E
hv : v ∈ segment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hy : y ∈ {x | ↑Hi_.f x = Hi_.α}
hHi_2 : x ∈ {x | ↑Hi_.f x = Hi_.α}
h21 : Finset.sum Finset.univ ![2, -1] = 1
⊢ {x} ∪ {y} ⊆ {x | ↑Hi_.f x = Hi_.α} |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | exact Set.union_subset (Set.singleton_subset_iff.mpr hHi_2) (Set.singleton_subset_iff.mpr hy) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
y : E
hyx : y ≠ x
hxyy : x ∈ openSegment ℝ (2 • x - y) y
hmemballmemIc : ∃ ε > 0, ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
v : E
hv : v ∈ segment ℝ (2 • x - y) y
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hy : y ∈ {x | ↑Hi_.f x = Hi_.α}
hHi_2 : x ∈ {x | ↑Hi_.f x = Hi_.α}
h21 : Finset.sum Finset.univ ![2, -1] = 1
⊢ {x} ∪ {y} ⊆ {x | ↑Hi_.f x = Hi_.α} | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | specialize hsub (left_mem_segment ℝ x1 x2) | case mp.intro.intro.intro.intro.intro.intro.intro.intro.refine_1.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
hsub : segment ℝ x1 x2 ⊆ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
⊢ x1 ∈ ↑Hi_ | case mp.intro.intro.intro.intro.intro.intro.intro.intro.refine_1.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
hsub : x1 ∈ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
⊢ x1 ∈ ↑Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rcases (em (Hi_ ∈ Hpolytope.I H_ x)) with (hinI | hninI) | case mp.intro.intro.intro.intro.intro.intro.intro.intro.refine_1.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
hsub : x1 ∈ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
⊢ x1 ∈ ↑Hi_ | case mp.intro.intro.intro.intro.intro.intro.intro.intro.refine_1.intro.intro.inl
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
hsub : x1 ∈ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hinI : Hi_ ∈ Hpolytope.I H_ x
⊢ x1 ∈ ↑Hi_
case mp.intro.intro.intro.intro.intro.intro.intro.intro.refine_1.intro.intro.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
hsub : x1 ∈ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hninI : Hi_ ∉ Hpolytope.I H_ x
⊢ x1 ∈ ↑Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | apply hmemsegmemI x1 ?_ Hi_ hinI | case mp.intro.intro.intro.intro.intro.intro.intro.intro.refine_1.intro.intro.inl
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
hsub : x1 ∈ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hinI : Hi_ ∈ Hpolytope.I H_ x
⊢ x1 ∈ ↑Hi_ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
hsub : x1 ∈ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hinI : Hi_ ∈ Hpolytope.I H_ x
⊢ x1 ∈ segment ℝ (2 • x - y) y |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | apply openSegment_subset_segment | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
hsub : x1 ∈ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hinI : Hi_ ∈ Hpolytope.I H_ x
⊢ x1 ∈ segment ℝ (2 • x - y) y | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
hsub : x1 ∈ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hinI : Hi_ ∈ Hpolytope.I H_ x
⊢ x1 ∈ openSegment ℝ (2 • x - y) y |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | exact Set.mem_of_mem_inter_left hsub | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
hsub : x1 ∈ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hinI : Hi_ ∈ Hpolytope.I H_ x
⊢ x1 ∈ openSegment ℝ (2 • x - y) y | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | have : Hi_ ∈ H_ \ Hpolytope.I H_ x := by
rw [Set.mem_diff]
exact ⟨ hHi_, hninI ⟩ | case mp.intro.intro.intro.intro.intro.intro.intro.intro.refine_1.intro.intro.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
hsub : x1 ∈ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hninI : Hi_ ∉ Hpolytope.I H_ x
⊢ x1 ∈ ↑Hi_ | case mp.intro.intro.intro.intro.intro.intro.intro.intro.refine_1.intro.intro.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
hsub : x1 ∈ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hninI : Hi_ ∉ Hpolytope.I H_ x
this : Hi_ ∈ H_ \ Hpolytope.I H_ x
⊢ x1 ∈ ↑Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | exact hmemballmemIc x1 (Set.mem_of_mem_inter_right hsub) Hi_ this | case mp.intro.intro.intro.intro.intro.intro.intro.intro.refine_1.intro.intro.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
hsub : x1 ∈ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hninI : Hi_ ∉ Hpolytope.I H_ x
this : Hi_ ∈ H_ \ Hpolytope.I H_ x
⊢ x1 ∈ ↑Hi_ | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rw [Set.mem_diff] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
hsub : x1 ∈ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hninI : Hi_ ∉ Hpolytope.I H_ x
⊢ Hi_ ∈ H_ \ Hpolytope.I H_ x | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
hsub : x1 ∈ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hninI : Hi_ ∉ Hpolytope.I H_ x
⊢ Hi_ ∈ H_ ∧ Hi_ ∉ Hpolytope.I H_ x |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | exact ⟨ hHi_, hninI ⟩ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
hsub : x1 ∈ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hninI : Hi_ ∉ Hpolytope.I H_ x
⊢ Hi_ ∈ H_ ∧ Hi_ ∉ Hpolytope.I H_ x | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | specialize hsub (right_mem_segment ℝ x1 x2) | case mp.intro.intro.intro.intro.intro.intro.intro.intro.refine_2.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
hsub : segment ℝ x1 x2 ⊆ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
⊢ x2 ∈ ↑Hi_ | case mp.intro.intro.intro.intro.intro.intro.intro.intro.refine_2.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
hsub : x2 ∈ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
⊢ x2 ∈ ↑Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rcases (em (Hi_ ∈ Hpolytope.I H_ x)) with (hinI | hninI) | case mp.intro.intro.intro.intro.intro.intro.intro.intro.refine_2.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
hsub : x2 ∈ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
⊢ x2 ∈ ↑Hi_ | case mp.intro.intro.intro.intro.intro.intro.intro.intro.refine_2.intro.intro.inl
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
hsub : x2 ∈ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hinI : Hi_ ∈ Hpolytope.I H_ x
⊢ x2 ∈ ↑Hi_
case mp.intro.intro.intro.intro.intro.intro.intro.intro.refine_2.intro.intro.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
hsub : x2 ∈ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hninI : Hi_ ∉ Hpolytope.I H_ x
⊢ x2 ∈ ↑Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | apply hmemsegmemI x2 ?_ Hi_ hinI | case mp.intro.intro.intro.intro.intro.intro.intro.intro.refine_2.intro.intro.inl
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
hsub : x2 ∈ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hinI : Hi_ ∈ Hpolytope.I H_ x
⊢ x2 ∈ ↑Hi_ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
hsub : x2 ∈ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hinI : Hi_ ∈ Hpolytope.I H_ x
⊢ x2 ∈ segment ℝ (2 • x - y) y |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | apply openSegment_subset_segment | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
hsub : x2 ∈ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hinI : Hi_ ∈ Hpolytope.I H_ x
⊢ x2 ∈ segment ℝ (2 • x - y) y | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
hsub : x2 ∈ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hinI : Hi_ ∈ Hpolytope.I H_ x
⊢ x2 ∈ openSegment ℝ (2 • x - y) y |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | exact Set.mem_of_mem_inter_left hsub | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
hsub : x2 ∈ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hinI : Hi_ ∈ Hpolytope.I H_ x
⊢ x2 ∈ openSegment ℝ (2 • x - y) y | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | have : Hi_ ∈ H_ \ Hpolytope.I H_ x := by
rw [Set.mem_diff]
exact ⟨ hHi_, hninI ⟩ | case mp.intro.intro.intro.intro.intro.intro.intro.intro.refine_2.intro.intro.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
hsub : x2 ∈ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hninI : Hi_ ∉ Hpolytope.I H_ x
⊢ x2 ∈ ↑Hi_ | case mp.intro.intro.intro.intro.intro.intro.intro.intro.refine_2.intro.intro.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
hsub : x2 ∈ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hninI : Hi_ ∉ Hpolytope.I H_ x
this : Hi_ ∈ H_ \ Hpolytope.I H_ x
⊢ x2 ∈ ↑Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | exact hmemballmemIc x2 (Set.mem_of_mem_inter_right hsub) Hi_ this | case mp.intro.intro.intro.intro.intro.intro.intro.intro.refine_2.intro.intro.inr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
hsub : x2 ∈ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hninI : Hi_ ∉ Hpolytope.I H_ x
this : Hi_ ∈ H_ \ Hpolytope.I H_ x
⊢ x2 ∈ ↑Hi_ | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rw [Set.mem_diff] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
hsub : x2 ∈ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hninI : Hi_ ∉ Hpolytope.I H_ x
⊢ Hi_ ∈ H_ \ Hpolytope.I H_ x | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
hsub : x2 ∈ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hninI : Hi_ ∉ Hpolytope.I H_ x
⊢ Hi_ ∈ H_ ∧ Hi_ ∉ Hpolytope.I H_ x |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | exact ⟨ hHi_, hninI ⟩ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x y : E
hmemsegmemI : ∀ v ∈ segment ℝ (2 • x - y) y, ∀ Hi_ ∈ Hpolytope.I H_ x, v ∈ ↑Hi_
ε : ℝ
hmemballmemIc : ∀ v ∈ Metric.ball x ε, ∀ Hi_ ∈ H_ \ Hpolytope.I H_ x, v ∈ ↑Hi_
x1 x2 : E
Hi_ : Halfspace E
hHi_ : Hi_ ∈ H_
hsub : x2 ∈ openSegment ℝ (2 • x - y) y ∩ Metric.ball x ε
hninI : Hi_ ∉ Hpolytope.I H_ x
⊢ Hi_ ∈ H_ ∧ Hi_ ∉ Hpolytope.I H_ x | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | intro hinterx | case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
⊢ ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x} → x ∈ Set.extremePoints ℝ (Hpolytope hH_) | case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hinterx : ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
⊢ x ∈ Set.extremePoints ℝ (Hpolytope hH_) |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rw [mem_extremePoints] | case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hinterx : ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
⊢ x ∈ Set.extremePoints ℝ (Hpolytope hH_) | case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hinterx : ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
⊢ x ∈ Hpolytope hH_ ∧ ∀ x₁ ∈ Hpolytope hH_, ∀ x₂ ∈ Hpolytope hH_, x ∈ openSegment ℝ x₁ x₂ → x₁ = x ∧ x₂ = x |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | refine ⟨ hxH, λ x1 hx1 x2 hx2 hx => ?_ ⟩ | case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hinterx : ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
⊢ x ∈ Hpolytope hH_ ∧ ∀ x₁ ∈ Hpolytope hH_, ∀ x₂ ∈ Hpolytope hH_, x ∈ openSegment ℝ x₁ x₂ → x₁ = x ∧ x₂ = x | case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hinterx : ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
x1 : E
hx1 : x1 ∈ Hpolytope hH_
x2 : E
hx2 : x2 ∈ Hpolytope hH_
hx : x ∈ openSegment ℝ x1 x2
⊢ x1 = x ∧ x2 = x |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | have : segment ℝ x1 x2 ⊆ {x} → x1 = x ∧ x2 = x := by
rw [Set.Nonempty.subset_singleton_iff (Set.nonempty_of_mem (left_mem_segment ℝ x1 x2)),
Set.eq_singleton_iff_unique_mem]
exact fun hseg => ⟨ hseg.2 x1 (left_mem_segment ℝ x1 x2),
hseg.2 x2 (right_mem_segment ℝ x1 x2) ⟩ | case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hinterx : ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
x1 : E
hx1 : x1 ∈ Hpolytope hH_
x2 : E
hx2 : x2 ∈ Hpolytope hH_
hx : x ∈ openSegment ℝ x1 x2
⊢ x1 = x ∧ x2 = x | case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hinterx : ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
x1 : E
hx1 : x1 ∈ Hpolytope hH_
x2 : E
hx2 : x2 ∈ Hpolytope hH_
hx : x ∈ openSegment ℝ x1 x2
this : segment ℝ x1 x2 ⊆ {x} → x1 = x ∧ x2 = x
⊢ x1 = x ∧ x2 = x |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | apply this | case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hinterx : ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
x1 : E
hx1 : x1 ∈ Hpolytope hH_
x2 : E
hx2 : x2 ∈ Hpolytope hH_
hx : x ∈ openSegment ℝ x1 x2
this : segment ℝ x1 x2 ⊆ {x} → x1 = x ∧ x2 = x
⊢ x1 = x ∧ x2 = x | case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hinterx : ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
x1 : E
hx1 : x1 ∈ Hpolytope hH_
x2 : E
hx2 : x2 ∈ Hpolytope hH_
hx : x ∈ openSegment ℝ x1 x2
this : segment ℝ x1 x2 ⊆ {x} → x1 = x ∧ x2 = x
⊢ segment ℝ x1 x2 ⊆ {x} |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | clear this | case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hinterx : ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
x1 : E
hx1 : x1 ∈ Hpolytope hH_
x2 : E
hx2 : x2 ∈ Hpolytope hH_
hx : x ∈ openSegment ℝ x1 x2
this : segment ℝ x1 x2 ⊆ {x} → x1 = x ∧ x2 = x
⊢ segment ℝ x1 x2 ⊆ {x} | case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hinterx : ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
x1 : E
hx1 : x1 ∈ Hpolytope hH_
x2 : E
hx2 : x2 ∈ Hpolytope hH_
hx : x ∈ openSegment ℝ x1 x2
⊢ segment ℝ x1 x2 ⊆ {x} |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rw [← hinterx, Set.subset_sInter_iff] | case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hinterx : ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
x1 : E
hx1 : x1 ∈ Hpolytope hH_
x2 : E
hx2 : x2 ∈ Hpolytope hH_
hx : x ∈ openSegment ℝ x1 x2
⊢ segment ℝ x1 x2 ⊆ {x} | case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hinterx : ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
x1 : E
hx1 : x1 ∈ Hpolytope hH_
x2 : E
hx2 : x2 ∈ Hpolytope hH_
hx : x ∈ openSegment ℝ x1 x2
⊢ ∀ t' ∈ (fun x => frontier ↑x) '' Hpolytope.I H_ x, segment ℝ x1 x2 ⊆ t' |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rintro HiS ⟨ Hi_, hHi_, rfl ⟩ | case mpr
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hinterx : ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
x1 : E
hx1 : x1 ∈ Hpolytope hH_
x2 : E
hx2 : x2 ∈ Hpolytope hH_
hx : x ∈ openSegment ℝ x1 x2
⊢ ∀ t' ∈ (fun x => frontier ↑x) '' Hpolytope.I H_ x, segment ℝ x1 x2 ⊆ t' | case mpr.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hinterx : ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
x1 : E
hx1 : x1 ∈ Hpolytope hH_
x2 : E
hx2 : x2 ∈ Hpolytope hH_
hx : x ∈ openSegment ℝ x1 x2
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
⊢ segment ℝ x1 x2 ⊆ (fun x => frontier ↑x) Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | simp only | case mpr.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hinterx : ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
x1 : E
hx1 : x1 ∈ Hpolytope hH_
x2 : E
hx2 : x2 ∈ Hpolytope hH_
hx : x ∈ openSegment ℝ x1 x2
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
⊢ segment ℝ x1 x2 ⊆ (fun x => frontier ↑x) Hi_ | case mpr.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hinterx : ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
x1 : E
hx1 : x1 ∈ Hpolytope hH_
x2 : E
hx2 : x2 ∈ Hpolytope hH_
hx : x ∈ openSegment ℝ x1 x2
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
⊢ segment ℝ x1 x2 ⊆ frontier ↑Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | have hfxα : Hi_.f.1 x = Hi_.α := by
have : x ∈ {x} := by
exact Set.mem_singleton x
rw [← hinterx, Set.mem_sInter] at this
specialize this (frontier <| SetLike.coe Hi_) ⟨ Hi_, hHi_, rfl ⟩
rw [frontierHalfspace_Hyperplane, Set.mem_setOf] at this
exact this | case mpr.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hinterx : ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
x1 : E
hx1 : x1 ∈ Hpolytope hH_
x2 : E
hx2 : x2 ∈ Hpolytope hH_
hx : x ∈ openSegment ℝ x1 x2
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
⊢ segment ℝ x1 x2 ⊆ frontier ↑Hi_ | case mpr.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hinterx : ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
x1 : E
hx1 : x1 ∈ Hpolytope hH_
x2 : E
hx2 : x2 ∈ Hpolytope hH_
hx : x ∈ openSegment ℝ x1 x2
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hfxα : ↑Hi_.f x = Hi_.α
⊢ segment ℝ x1 x2 ⊆ frontier ↑Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | clear hinterx hxH | case mpr.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hinterx : ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
x1 : E
hx1 : x1 ∈ Hpolytope hH_
x2 : E
hx2 : x2 ∈ Hpolytope hH_
hx : x ∈ openSegment ℝ x1 x2
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hfxα : ↑Hi_.f x = Hi_.α
⊢ segment ℝ x1 x2 ⊆ frontier ↑Hi_ | case mpr.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x x1 : E
hx1 : x1 ∈ Hpolytope hH_
x2 : E
hx2 : x2 ∈ Hpolytope hH_
hx : x ∈ openSegment ℝ x1 x2
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hfxα : ↑Hi_.f x = Hi_.α
⊢ segment ℝ x1 x2 ⊆ frontier ↑Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rw [mem_Hpolytope] at hx1 hx2 | case mpr.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x x1 : E
hx1 : x1 ∈ Hpolytope hH_
x2 : E
hx2 : x2 ∈ Hpolytope hH_
hx : x ∈ openSegment ℝ x1 x2
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hfxα : ↑Hi_.f x = Hi_.α
⊢ segment ℝ x1 x2 ⊆ frontier ↑Hi_ | case mpr.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x x1 : E
hx1 : ∀ Hi ∈ H_, ↑Hi.f x1 ≤ Hi.α
x2 : E
hx2 : ∀ Hi ∈ H_, ↑Hi.f x2 ≤ Hi.α
hx : x ∈ openSegment ℝ x1 x2
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hfxα : ↑Hi_.f x = Hi_.α
⊢ segment ℝ x1 x2 ⊆ frontier ↑Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | specialize hx1 Hi_ hHi_.1 | case mpr.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x x1 : E
hx1 : ∀ Hi ∈ H_, ↑Hi.f x1 ≤ Hi.α
x2 : E
hx2 : ∀ Hi ∈ H_, ↑Hi.f x2 ≤ Hi.α
hx : x ∈ openSegment ℝ x1 x2
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hfxα : ↑Hi_.f x = Hi_.α
⊢ segment ℝ x1 x2 ⊆ frontier ↑Hi_ | case mpr.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x x1 x2 : E
hx2 : ∀ Hi ∈ H_, ↑Hi.f x2 ≤ Hi.α
hx : x ∈ openSegment ℝ x1 x2
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hfxα : ↑Hi_.f x = Hi_.α
hx1 : ↑Hi_.f x1 ≤ Hi_.α
⊢ segment ℝ x1 x2 ⊆ frontier ↑Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | specialize hx2 Hi_ hHi_.1 | case mpr.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x x1 x2 : E
hx2 : ∀ Hi ∈ H_, ↑Hi.f x2 ≤ Hi.α
hx : x ∈ openSegment ℝ x1 x2
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hfxα : ↑Hi_.f x = Hi_.α
hx1 : ↑Hi_.f x1 ≤ Hi_.α
⊢ segment ℝ x1 x2 ⊆ frontier ↑Hi_ | case mpr.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x x1 x2 : E
hx : x ∈ openSegment ℝ x1 x2
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hfxα : ↑Hi_.f x = Hi_.α
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
⊢ segment ℝ x1 x2 ⊆ frontier ↑Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | clear hHi_ hH_ H_ | case mpr.intro.intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x x1 x2 : E
hx : x ∈ openSegment ℝ x1 x2
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
hfxα : ↑Hi_.f x = Hi_.α
hx1 : ↑Hi_.f x1 ≤ Hi_.α
hx2 : ↑Hi_.f x2 ≤ Hi_.α
⊢ segment ℝ x1 x2 ⊆ frontier ↑Hi_ | case mpr.intro.intro
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
⊢ segment ℝ x1 x2 ⊆ frontier ↑Hi_ |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rw [frontierHalfspace_Hyperplane] | case mpr.intro.intro
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
⊢ segment ℝ x1 x2 ⊆ frontier ↑Hi_ | case mpr.intro.intro
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
⊢ segment ℝ x1 x2 ⊆ {x | ↑Hi_.f x = Hi_.α} |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | have := Hyperplane_convex Hi_ | case mpr.intro.intro
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
⊢ segment ℝ x1 x2 ⊆ {x | ↑Hi_.f x = Hi_.α} | case mpr.intro.intro
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
this : Convex ℝ {x | ↑Hi_.f x = Hi_.α}
⊢ segment ℝ x1 x2 ⊆ {x | ↑Hi_.f x = Hi_.α} |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rw [convex_iff_segment_subset] at this | case mpr.intro.intro
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
this : Convex ℝ {x | ↑Hi_.f x = Hi_.α}
⊢ segment ℝ x1 x2 ⊆ {x | ↑Hi_.f x = Hi_.α} | case mpr.intro.intro
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
this :
∀ ⦃x : E⦄, x ∈ {x | ↑Hi_.f x = Hi_.α} → ∀ ⦃y : E⦄, y ∈ {x | ↑Hi_.f x = Hi_.α} → segment ℝ x y ⊆ {x | ↑Hi_.f x = Hi_.α}
⊢ segment ℝ x1 x2 ⊆ {x | ↑Hi_.f x = Hi_.α} |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | apply this <;>
clear this <;>
rw [Set.mem_setOf] <;>
by_contra h <;>
push_neg at h <;>
have hlt := lt_of_le_of_ne (by assumption) h <;>
clear h | case mpr.intro.intro
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
this :
∀ ⦃x : E⦄, x ∈ {x | ↑Hi_.f x = Hi_.α} → ∀ ⦃y : E⦄, y ∈ {x | ↑Hi_.f x = Hi_.α} → segment ℝ x y ⊆ {x | ↑Hi_.f x = Hi_.α}
⊢ segment ℝ x1 x2 ⊆ {x | ↑Hi_.f x = Hi_.α} | 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 ∈ 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 |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rw [Set.Nonempty.subset_singleton_iff (Set.nonempty_of_mem (left_mem_segment ℝ x1 x2)),
Set.eq_singleton_iff_unique_mem] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hinterx : ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
x1 : E
hx1 : x1 ∈ Hpolytope hH_
x2 : E
hx2 : x2 ∈ Hpolytope hH_
hx : x ∈ openSegment ℝ x1 x2
⊢ segment ℝ x1 x2 ⊆ {x} → x1 = x ∧ x2 = x | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hinterx : ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
x1 : E
hx1 : x1 ∈ Hpolytope hH_
x2 : E
hx2 : x2 ∈ Hpolytope hH_
hx : x ∈ openSegment ℝ x1 x2
⊢ (x ∈ segment ℝ x1 x2 ∧ ∀ x_1 ∈ segment ℝ x1 x2, x_1 = x) → x1 = x ∧ x2 = x |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | exact fun hseg => ⟨ hseg.2 x1 (left_mem_segment ℝ x1 x2),
hseg.2 x2 (right_mem_segment ℝ x1 x2) ⟩ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hinterx : ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
x1 : E
hx1 : x1 ∈ Hpolytope hH_
x2 : E
hx2 : x2 ∈ Hpolytope hH_
hx : x ∈ openSegment ℝ x1 x2
⊢ (x ∈ segment ℝ x1 x2 ∧ ∀ x_1 ∈ segment ℝ x1 x2, x_1 = x) → x1 = x ∧ x2 = x | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | have : x ∈ {x} := by
exact Set.mem_singleton x | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hinterx : ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
x1 : E
hx1 : x1 ∈ Hpolytope hH_
x2 : E
hx2 : x2 ∈ Hpolytope hH_
hx : x ∈ openSegment ℝ x1 x2
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
⊢ ↑Hi_.f x = Hi_.α | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hinterx : ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
x1 : E
hx1 : x1 ∈ Hpolytope hH_
x2 : E
hx2 : x2 ∈ Hpolytope hH_
hx : x ∈ openSegment ℝ x1 x2
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
this : x ∈ {x}
⊢ ↑Hi_.f x = Hi_.α |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rw [← hinterx, Set.mem_sInter] at this | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hinterx : ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
x1 : E
hx1 : x1 ∈ Hpolytope hH_
x2 : E
hx2 : x2 ∈ Hpolytope hH_
hx : x ∈ openSegment ℝ x1 x2
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
this : x ∈ {x}
⊢ ↑Hi_.f x = Hi_.α | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hinterx : ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
x1 : E
hx1 : x1 ∈ Hpolytope hH_
x2 : E
hx2 : x2 ∈ Hpolytope hH_
hx : x ∈ openSegment ℝ x1 x2
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
this : ∀ t ∈ (fun x => frontier ↑x) '' Hpolytope.I H_ x, x ∈ t
⊢ ↑Hi_.f x = Hi_.α |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | specialize this (frontier <| SetLike.coe Hi_) ⟨ Hi_, hHi_, rfl ⟩ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hinterx : ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
x1 : E
hx1 : x1 ∈ Hpolytope hH_
x2 : E
hx2 : x2 ∈ Hpolytope hH_
hx : x ∈ openSegment ℝ x1 x2
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
this : ∀ t ∈ (fun x => frontier ↑x) '' Hpolytope.I H_ x, x ∈ t
⊢ ↑Hi_.f x = Hi_.α | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hinterx : ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
x1 : E
hx1 : x1 ∈ Hpolytope hH_
x2 : E
hx2 : x2 ∈ Hpolytope hH_
hx : x ∈ openSegment ℝ x1 x2
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
this : x ∈ frontier ↑Hi_
⊢ ↑Hi_.f x = Hi_.α |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | rw [frontierHalfspace_Hyperplane, Set.mem_setOf] at this | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hinterx : ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
x1 : E
hx1 : x1 ∈ Hpolytope hH_
x2 : E
hx2 : x2 ∈ Hpolytope hH_
hx : x ∈ openSegment ℝ x1 x2
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
this : x ∈ frontier ↑Hi_
⊢ ↑Hi_.f x = Hi_.α | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hinterx : ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
x1 : E
hx1 : x1 ∈ Hpolytope hH_
x2 : E
hx2 : x2 ∈ Hpolytope hH_
hx : x ∈ openSegment ℝ x1 x2
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
this : ↑Hi_.f x = Hi_.α
⊢ ↑Hi_.f x = Hi_.α |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | exact this | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hinterx : ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
x1 : E
hx1 : x1 ∈ Hpolytope hH_
x2 : E
hx2 : x2 ∈ Hpolytope hH_
hx : x ∈ openSegment ℝ x1 x2
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
this : ↑Hi_.f x = Hi_.α
⊢ ↑Hi_.f x = Hi_.α | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | exact Set.mem_singleton x | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
H_ : Set (Halfspace E)
hH_ : Set.Finite H_
x : E
hxH : x ∈ Hpolytope hH_
hinterx : ⋂₀ ((fun x => frontier ↑x) '' Hpolytope.I H_ x) = {x}
x1 : E
hx1 : x1 ∈ Hpolytope hH_
x2 : E
hx2 : x2 ∈ Hpolytope hH_
hx : x ∈ openSegment ℝ x1 x2
Hi_ : Halfspace E
hHi_ : Hi_ ∈ Hpolytope.I H_ x
⊢ x ∈ {x} | no goals |
https://github.com/Jun2M/Main-theorem-of-polytopes.git | fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8 | src/MainTheorem.lean | ExtremePointsofHpolytope | [143, 1] | [338, 7] | assumption | 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
h : ↑Hi_.f x2 ≠ Hi_.α
⊢ ↑Hi_.f x2 ≤ Hi_.α | no goals |
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