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stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
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| start
stringlengths 6
10
| end
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https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | closure_eq_closure_interior | [139, 1] | [158, 16] | obtain ⟨z, hzA, hzx⟩ := hx | case intro
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x : E
hx : ∃ x_1, (∀ (n : ℕ), x_1 n ∈ s) ∧ Filter.Tendsto x_1 Filter.atTop (nhds x)
y : E
hy : y ∈ interior s
⊢ ∃ x_1, (∀ (n : ℕ), x_1 n ∈ interior s) ∧ Filter.Tendsto x_1 Filter.atTop (nhds x) | case intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ interior s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
⊢ ∃ x_1, (∀ (n : ℕ), x_1 n ∈ interior s) ∧ Filter.Tendsto x_1 Filter.atTop (nhds x) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | closure_eq_closure_interior | [139, 1] | [158, 16] | refine' ⟨fun n => (1 - 1 / (n + 2) : 𝕜) • z n + (1 / (n + 2) : 𝕜) • y, fun n => _, _⟩ | case intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ interior s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
⊢ ∃ x_1, (∀ (n : ℕ), x_1 n ∈ interior s) ∧ Filter.Tendsto x_1 Filter.atTop (nhds x) | case intro.intro.intro.refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ interior s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
n : ℕ
⊢ (fun n => (1 - 1 / (↑n + 2)) • z n + (1 / (↑n + 2)) • y) n ∈ interior s
case intro.intro.intro.refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ interior s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
⊢ Filter.Tendsto (fun n => (1 - 1 / (↑n + 2)) • z n + (1 / (↑n + 2)) • y) Filter.atTop (nhds x) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | closure_eq_closure_interior | [139, 1] | [158, 16] | have h : Filter.Tendsto (fun n : ℕ => 1 / ((n : 𝕜) + 2)) Filter.atTop (nhds (0 : 𝕜)) := by sorry | case intro.intro.intro.refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ interior s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
⊢ Filter.Tendsto (fun n => (1 - 1 / (↑n + 2)) • z n + (1 / (↑n + 2)) • y) Filter.atTop (nhds x) | case intro.intro.intro.refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ interior s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
h : Filter.Tendsto (fun n => 1 / (↑n + 2)) Filter.atTop (nhds 0)
⊢ Filter.Tendsto (fun n => (1 - 1 / (↑n + 2)) • z n + (1 / (↑n + 2)) • y) Filter.atTop (nhds x) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | closure_eq_closure_interior | [139, 1] | [158, 16] | rw [← add_zero x, ← one_smul 𝕜 x, ← zero_smul 𝕜 y] | case intro.intro.intro.refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ interior s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
h : Filter.Tendsto (fun n => 1 / (↑n + 2)) Filter.atTop (nhds 0)
⊢ Filter.Tendsto (fun n => (1 - 1 / (↑n + 2)) • z n + (1 / (↑n + 2)) • y) Filter.atTop (nhds x) | case intro.intro.intro.refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ interior s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
h : Filter.Tendsto (fun n => 1 / (↑n + 2)) Filter.atTop (nhds 0)
⊢ Filter.Tendsto (fun n => (1 - 1 / (↑n + 2)) • z n + (1 / (↑n + 2)) • y) Filter.atTop (nhds (1 • x + 0 • y)) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | closure_eq_closure_interior | [139, 1] | [158, 16] | convert ((h.const_sub _).smul hzx).add (h.smul_const _) | case intro.intro.intro.refine'_2
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ interior s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
h : Filter.Tendsto (fun n => 1 / (↑n + 2)) Filter.atTop (nhds 0)
⊢ Filter.Tendsto (fun n => (1 - 1 / (↑n + 2)) • z n + (1 / (↑n + 2)) • y) Filter.atTop (nhds (1 • x + 0 • y)) | case h.e'_5.h.e'_3.h.e'_5.h.e'_5
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ interior s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
h : Filter.Tendsto (fun n => 1 / (↑n + 2)) Filter.atTop (nhds 0)
⊢ 1 = 1 - 0 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | closure_eq_closure_interior | [139, 1] | [158, 16] | rw [sub_zero] | case h.e'_5.h.e'_3.h.e'_5.h.e'_5
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ interior s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
h : Filter.Tendsto (fun n => 1 / (↑n + 2)) Filter.atTop (nhds 0)
⊢ 1 = 1 - 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | closure_eq_closure_interior | [139, 1] | [158, 16] | rw [← closure_diff_frontier] at hy ⊢ | case intro.intro.intro.refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ interior s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
n : ℕ
⊢ (fun n => (1 - 1 / (↑n + 2)) • z n + (1 / (↑n + 2)) • y) n ∈ interior s | case intro.intro.intro.refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ closure s \ frontier s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
n : ℕ
⊢ (fun n => (1 - 1 / (↑n + 2)) • z n + (1 / (↑n + 2)) • y) n ∈ closure s \ frontier s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | closure_eq_closure_interior | [139, 1] | [158, 16] | have h₁ : (1 : 𝕜) < ↑n + 2 := by norm_cast; norm_num | case intro.intro.intro.refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ closure s \ frontier s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
n : ℕ
⊢ (fun n => (1 - 1 / (↑n + 2)) • z n + (1 / (↑n + 2)) • y) n ∈ closure s \ frontier s | case intro.intro.intro.refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ closure s \ frontier s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
n : ℕ
h₁ : 1 < ↑n + 2
⊢ (fun n => (1 - 1 / (↑n + 2)) • z n + (1 / (↑n + 2)) • y) n ∈ closure s \ frontier s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | closure_eq_closure_interior | [139, 1] | [158, 16] | have h₀ := zero_lt_one.trans h₁ | case intro.intro.intro.refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ closure s \ frontier s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
n : ℕ
h₁ : 1 < ↑n + 2
⊢ (fun n => (1 - 1 / (↑n + 2)) • z n + (1 / (↑n + 2)) • y) n ∈ closure s \ frontier s | case intro.intro.intro.refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ closure s \ frontier s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
n : ℕ
h₁ : 1 < ↑n + 2
h₀ : 0 < ↑n + 2
⊢ (fun n => (1 - 1 / (↑n + 2)) • z n + (1 / (↑n + 2)) • y) n ∈ closure s \ frontier s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | closure_eq_closure_interior | [139, 1] | [158, 16] | exact
(hAconv.closure.isExtreme_iff_openSegment_subset_diff.1
hAconv.frontier_extreme_to_closure).2
(subset_closure (hzA n)) hy
⟨1 - 1 / (n + 2), 1 / (n + 2), sub_pos.2 <| (div_lt_one h₀).2 h₁, div_pos zero_lt_one h₀,
sub_add_cancel _ _, rfl⟩ | case intro.intro.intro.refine'_1
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ closure s \ frontier s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
n : ℕ
h₁ : 1 < ↑n + 2
h₀ : 0 < ↑n + 2
⊢ (fun n => (1 - 1 / (↑n + 2)) • z n + (1 / (↑n + 2)) • y) n ∈ closure s \ frontier s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | closure_eq_closure_interior | [139, 1] | [158, 16] | norm_cast | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ closure s \ frontier s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
n : ℕ
⊢ 1 < ↑n + 2 | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ closure s \ frontier s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
n : ℕ
⊢ 1 < n + 2 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | closure_eq_closure_interior | [139, 1] | [158, 16] | norm_num | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ closure s \ frontier s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
n : ℕ
⊢ 1 < n + 2 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | closure_eq_closure_interior | [139, 1] | [158, 16] | sorry | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t : Set E
x✝ : E
s : Set E
hAconv : Convex 𝕜 s
x y : E
hy : y ∈ interior s
z : ℕ → E
hzA : ∀ (n : ℕ), z n ∈ s
hzx : Filter.Tendsto z Filter.atTop (nhds x)
⊢ Filter.Tendsto (fun n => 1 / (↑n + 2)) Filter.atTop (nhds 0) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | ConvexIndependent.subset_of_convexHull_eq_convexHull | [160, 1] | [166, 59] | rintro x hx | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t✝ : Set E
x : E
s t : Finset E
hs : ConvexIndependent 𝕜 Subtype.val
h : (convexHull 𝕜) ↑s = (convexHull 𝕜) ↑t
⊢ s ⊆ t | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t✝ : Set E
x✝ : E
s t : Finset E
hs : ConvexIndependent 𝕜 Subtype.val
h : (convexHull 𝕜) ↑s = (convexHull 𝕜) ↑t
x : E
hx : x ∈ s
⊢ x ∈ t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | ConvexIndependent.subset_of_convexHull_eq_convexHull | [160, 1] | [166, 59] | have hxextreme := (extremePoints_convexHull_eq_iff_convexIndependent.2 hs).symm.subset hx | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t✝ : Set E
x✝ : E
s t : Finset E
hs : ConvexIndependent 𝕜 Subtype.val
h : (convexHull 𝕜) ↑s = (convexHull 𝕜) ↑t
x : E
hx : x ∈ s
⊢ x ∈ t | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t✝ : Set E
x✝ : E
s t : Finset E
hs : ConvexIndependent 𝕜 Subtype.val
h : (convexHull 𝕜) ↑s = (convexHull 𝕜) ↑t
x : E
hx : x ∈ s
hxextreme : x ∈ extremePoints 𝕜 ((convexHull 𝕜) fun x => x ∈ s.val)
⊢ x ∈ t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | ConvexIndependent.subset_of_convexHull_eq_convexHull | [160, 1] | [166, 59] | erw [h] at hxextreme | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t✝ : Set E
x✝ : E
s t : Finset E
hs : ConvexIndependent 𝕜 Subtype.val
h : (convexHull 𝕜) ↑s = (convexHull 𝕜) ↑t
x : E
hx : x ∈ s
hxextreme : x ∈ extremePoints 𝕜 ((convexHull 𝕜) fun x => x ∈ s.val)
⊢ x ∈ t | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t✝ : Set E
x✝ : E
s t : Finset E
hs : ConvexIndependent 𝕜 Subtype.val
h : (convexHull 𝕜) ↑s = (convexHull 𝕜) ↑t
x : E
hx : x ∈ s
hxextreme : x ∈ extremePoints 𝕜 ((convexHull 𝕜) ↑t)
⊢ x ∈ t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Extreme.lean | ConvexIndependent.subset_of_convexHull_eq_convexHull | [160, 1] | [166, 59] | exact_mod_cast extremePoints_convexHull_subset hxextreme | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedLinearOrderedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s✝ t✝ : Set E
x✝ : E
s t : Finset E
hs : ConvexIndependent 𝕜 Subtype.val
h : (convexHull 𝕜) ↑s = (convexHull 𝕜) ↑t
x : E
hx : x ∈ s
hxextreme : x ∈ extremePoints 𝕜 ((convexHull 𝕜) ↑t)
⊢ x ∈ t | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | generateLine_minimal | [36, 1] | [44, 61] | intro x hx | V : Type u_1
inst✝ : MetricSpace V
u v w : V
S L : Set V
hL₀ : S ⊆ L
out_closed : ∀ {x y z : V}, x ∈ L → y ∈ L → sbtw x y z → z ∈ L
in_closed : ∀ {x y z : V}, x ∈ L → y ∈ L → sbtw x z y → z ∈ L
⊢ GenerateLine S ⊆ L | V : Type u_1
inst✝ : MetricSpace V
u v w : V
S L : Set V
hL₀ : S ⊆ L
out_closed : ∀ {x y z : V}, x ∈ L → y ∈ L → sbtw x y z → z ∈ L
in_closed : ∀ {x y z : V}, x ∈ L → y ∈ L → sbtw x z y → z ∈ L
x : V
hx : x ∈ GenerateLine S
⊢ x ∈ L |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | generateLine_minimal | [36, 1] | [44, 61] | induction hx with
| basic h => exact hL₀ h
| extend_out u v w _ _ hw hu hv => exact out_closed hu hv hw
| extend_in u v w _ _ hw hu hv => exact in_closed hu hv hw | V : Type u_1
inst✝ : MetricSpace V
u v w : V
S L : Set V
hL₀ : S ⊆ L
out_closed : ∀ {x y z : V}, x ∈ L → y ∈ L → sbtw x y z → z ∈ L
in_closed : ∀ {x y z : V}, x ∈ L → y ∈ L → sbtw x z y → z ∈ L
x : V
hx : x ∈ GenerateLine S
⊢ x ∈ L | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | generateLine_minimal | [36, 1] | [44, 61] | exact hL₀ h | case basic
V : Type u_1
inst✝ : MetricSpace V
u v w : V
S L : Set V
hL₀ : S ⊆ L
out_closed : ∀ {x y z : V}, x ∈ L → y ∈ L → sbtw x y z → z ∈ L
in_closed : ∀ {x y z : V}, x ∈ L → y ∈ L → sbtw x z y → z ∈ L
x x✝ : V
h : x✝ ∈ S
⊢ x✝ ∈ L | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | generateLine_minimal | [36, 1] | [44, 61] | exact out_closed hu hv hw | case extend_out
V : Type u_1
inst✝ : MetricSpace V
u✝ v✝ w✝ : V
S L : Set V
hL₀ : S ⊆ L
out_closed : ∀ {x y z : V}, x ∈ L → y ∈ L → sbtw x y z → z ∈ L
in_closed : ∀ {x y z : V}, x ∈ L → y ∈ L → sbtw x z y → z ∈ L
x u v w : V
a✝¹ : GenerateLine S u
a✝ : GenerateLine S v
hw : sbtw u v w
hu : u ∈ L
hv : v ∈ L
⊢ w ∈ L | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | generateLine_minimal | [36, 1] | [44, 61] | exact in_closed hu hv hw | case extend_in
V : Type u_1
inst✝ : MetricSpace V
u✝ v✝ w✝ : V
S L : Set V
hL₀ : S ⊆ L
out_closed : ∀ {x y z : V}, x ∈ L → y ∈ L → sbtw x y z → z ∈ L
in_closed : ∀ {x y z : V}, x ∈ L → y ∈ L → sbtw x z y → z ∈ L
x u v w : V
a✝¹ : GenerateLine S u
a✝ : GenerateLine S v
hw : sbtw u w v
hu : u ∈ L
hv : v ∈ L
⊢ w ∈ L | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | left_mem_Line | [48, 1] | [48, 88] | simp | V : Type u_1
inst✝ : MetricSpace V
u v w a b : V
⊢ a ∈ {a, b} | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | right_mem_Line | [49, 1] | [49, 89] | simp | V : Type u_1
inst✝ : MetricSpace V
u v w a b : V
⊢ b ∈ {a, b} | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | Line_comm | [57, 1] | [57, 75] | rw [Line, Line, Set.pair_comm] | V : Type u_1
inst✝ : MetricSpace V
u v w : V
⊢ Line u v = Line v u | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | Set.IsLine.close_right | [63, 1] | [66, 42] | obtain ⟨x, y, _, rfl⟩ := hL | V : Type u_1
inst✝ : MetricSpace V
u v w : V
L : Set V
hL : L.IsLine
a b c : V
ha : a ∈ L
hb : b ∈ L
hc : sbtw a b c
⊢ c ∈ L | case intro.intro.intro
V : Type u_1
inst✝ : MetricSpace V
u v w a b c : V
hc : sbtw a b c
x y : V
left✝ : x ≠ y
ha : a ∈ Line x y
hb : b ∈ Line x y
⊢ c ∈ Line x y |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | Set.IsLine.close_right | [63, 1] | [66, 42] | exact generateLine_close_right ha hb hc | case intro.intro.intro
V : Type u_1
inst✝ : MetricSpace V
u v w a b c : V
hc : sbtw a b c
x y : V
left✝ : x ≠ y
ha : a ∈ Line x y
hb : b ∈ Line x y
⊢ c ∈ Line x y | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | Set.IsLine.close_left | [68, 1] | [71, 41] | obtain ⟨x, y, _, rfl⟩ := hL | V : Type u_1
inst✝ : MetricSpace V
u v w : V
L : Set V
hL : L.IsLine
a b c : V
ha : a ∈ L
hb : b ∈ L
hc : sbtw c a b
⊢ c ∈ L | case intro.intro.intro
V : Type u_1
inst✝ : MetricSpace V
u v w a b c : V
hc : sbtw c a b
x y : V
left✝ : x ≠ y
ha : a ∈ Line x y
hb : b ∈ Line x y
⊢ c ∈ Line x y |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | Set.IsLine.close_left | [68, 1] | [71, 41] | exact generateLine_close_left ha hb hc | case intro.intro.intro
V : Type u_1
inst✝ : MetricSpace V
u v w a b c : V
hc : sbtw c a b
x y : V
left✝ : x ≠ y
ha : a ∈ Line x y
hb : b ∈ Line x y
⊢ c ∈ Line x y | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | Set.IsLine.close_middle | [73, 1] | [76, 43] | obtain ⟨x, y, _, rfl⟩ := hL | V : Type u_1
inst✝ : MetricSpace V
u v w : V
L : Set V
hL : L.IsLine
a b c : V
ha : a ∈ L
hb : b ∈ L
hc : sbtw a c b
⊢ c ∈ L | case intro.intro.intro
V : Type u_1
inst✝ : MetricSpace V
u v w a b c : V
hc : sbtw a c b
x y : V
left✝ : x ≠ y
ha : a ∈ Line x y
hb : b ∈ Line x y
⊢ c ∈ Line x y |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | Set.IsLine.close_middle | [73, 1] | [76, 43] | exact generateLine_close_middle ha hb hc | case intro.intro.intro
V : Type u_1
inst✝ : MetricSpace V
u v w a b c : V
hc : sbtw a c b
x y : V
left✝ : x ≠ y
ha : a ∈ Line x y
hb : b ∈ Line x y
⊢ c ∈ Line x y | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | exists_isLine | [86, 1] | [90, 48] | rcases ne_or_eq a b with h | rfl | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b : V
⊢ ∃ L, L.IsLine ∧ {a, b} ⊆ L | case inl
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b : V
h : a ≠ b
⊢ ∃ L, L.IsLine ∧ {a, b} ⊆ L
case inr
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a : V
⊢ ∃ L, L.IsLine ∧ {a, a} ⊆ L |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | exists_isLine | [86, 1] | [90, 48] | have ⟨b, h⟩ := exists_ne a | case inr
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a : V
⊢ ∃ L, L.IsLine ∧ {a, a} ⊆ L | case inr
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b : V
h : b ≠ a
⊢ ∃ L, L.IsLine ∧ {a, a} ⊆ L |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | exists_isLine | [86, 1] | [90, 48] | exact ⟨Line a b, Line_isLine h.symm, by simp⟩ | case inr
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b : V
h : b ≠ a
⊢ ∃ L, L.IsLine ∧ {a, a} ⊆ L | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | exists_isLine | [86, 1] | [90, 48] | exact ⟨Line a b, Line_isLine h, by simp⟩ | case inl
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b : V
h : a ≠ b
⊢ ∃ L, L.IsLine ∧ {a, b} ⊆ L | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | exists_isLine | [86, 1] | [90, 48] | simp | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b : V
h : a ≠ b
⊢ {a, b} ⊆ Line a b | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | exists_isLine | [86, 1] | [90, 48] | simp | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z a b : V
h : b ≠ a
⊢ {a, a} ⊆ Line a b | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | NotCollinear.mk | [98, 1] | [108, 10] | refine ⟨?_, ?_, ?_, hl⟩ | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, z} ⊆ l
⊢ NotCollinear x y z | case refine_1
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, z} ⊆ l
⊢ x ≠ y
case refine_2
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, z} ⊆ l
⊢ x ≠ z
case refine_3
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, z} ⊆ l
⊢ y ≠ z |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | NotCollinear.mk | [98, 1] | [108, 10] | rintro rfl | case refine_1
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, z} ⊆ l
⊢ x ≠ y | case refine_1
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x z : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, x, z} ⊆ l
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | NotCollinear.mk | [98, 1] | [108, 10] | obtain ⟨L, hL, hL'⟩ := exists_isLine x z | case refine_1
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x z : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, x, z} ⊆ l
⊢ False | case refine_1.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x z : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, x, z} ⊆ l
L : Set V
hL : L.IsLine
hL' : {x, z} ⊆ L
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | NotCollinear.mk | [98, 1] | [108, 10] | aesop | case refine_1.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x z : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, x, z} ⊆ l
L : Set V
hL : L.IsLine
hL' : {x, z} ⊆ L
⊢ False | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | NotCollinear.mk | [98, 1] | [108, 10] | rintro rfl | case refine_2
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, z} ⊆ l
⊢ x ≠ z | case refine_2
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, x} ⊆ l
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | NotCollinear.mk | [98, 1] | [108, 10] | obtain ⟨L, hL, hL'⟩ := exists_isLine x y | case refine_2
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, x} ⊆ l
⊢ False | case refine_2.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, x} ⊆ l
L : Set V
hL : L.IsLine
hL' : {x, y} ⊆ L
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | NotCollinear.mk | [98, 1] | [108, 10] | aesop | case refine_2.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, x} ⊆ l
L : Set V
hL : L.IsLine
hL' : {x, y} ⊆ L
⊢ False | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | NotCollinear.mk | [98, 1] | [108, 10] | rintro rfl | case refine_3
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, z} ⊆ l
⊢ y ≠ z | case refine_3
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, y} ⊆ l
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | NotCollinear.mk | [98, 1] | [108, 10] | obtain ⟨L, hL, hL'⟩ := exists_isLine x y | case refine_3
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, y} ⊆ l
⊢ False | case refine_3.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, y} ⊆ l
L : Set V
hL : L.IsLine
hL' : {x, y} ⊆ L
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | NotCollinear.mk | [98, 1] | [108, 10] | aesop | case refine_3.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y : V
hl : ∀ (l : Set V), l.IsLine → ¬{x, y, y} ⊆ l
L : Set V
hL : L.IsLine
hL' : {x, y} ⊆ L
⊢ False | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | NotCollinear.rotate | [110, 1] | [112, 54] | simp | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : NotCollinear u v w
l : Set V
h₁ : l.IsLine
h₂ : {v, w, u} ⊆ l
⊢ {u, v, w} ≤ {v, w, u} | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | NotCollinear.swap | [114, 1] | [116, 54] | simp | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : NotCollinear u v w
l : Set V
h₁ : l.IsLine
h₂ : {w, v, u} ⊆ l
⊢ {u, v, w} ≤ {w, v, u} | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | let S : Set (Set V) := setOf Set.IsLine | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
⊢ Set.univ.IsLine | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
⊢ Set.univ.IsLine |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | have : S.Nonempty := let ⟨x, y, hxy⟩ := exists_pair_ne V; ⟨_, Line_isLine hxy⟩ | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
⊢ Set.univ.IsLine | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
⊢ Set.univ.IsLine |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | obtain ⟨L, hL, hL'⟩ := S.toFinite.exists_maximal_wrt id S this | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
⊢ Set.univ.IsLine | case intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
L : Set V
hL : L ∈ S
hL' : ∀ a' ∈ S, id L ≤ id a' → id L = id a'
⊢ Set.univ.IsLine |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | dsimp at hL' | case intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
L : Set V
hL : L ∈ S
hL' : ∀ a' ∈ S, id L ≤ id a' → id L = id a'
⊢ Set.univ.IsLine | case intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
L : Set V
hL : L ∈ S
hL' : ∀ a' ∈ S, (∀ x ∈ L, x ∈ a') → L = a'
⊢ Set.univ.IsLine |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | suffices L = Set.univ by rwa [← this] | case intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
L : Set V
hL : L ∈ S
hL' : ∀ a' ∈ S, (∀ x ∈ L, x ∈ a') → L = a'
⊢ Set.univ.IsLine | case intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
L : Set V
hL : L ∈ S
hL' : ∀ a' ∈ S, (∀ x ∈ L, x ∈ a') → L = a'
⊢ L = Set.univ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | rw [Set.eq_univ_iff_forall] | case intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
L : Set V
hL : L ∈ S
hL' : ∀ a' ∈ S, (∀ x ∈ L, x ∈ a') → L = a'
⊢ L = Set.univ | case intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
L : Set V
hL : L ∈ S
hL' : ∀ a' ∈ S, (∀ x ∈ L, x ∈ a') → L = a'
⊢ ∀ (x : V), x ∈ L |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | by_contra! | case intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
L : Set V
hL : L ∈ S
hL' : ∀ a' ∈ S, (∀ x ∈ L, x ∈ a') → L = a'
⊢ ∀ (x : V), x ∈ L | case intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this✝ : S.Nonempty
L : Set V
hL : L ∈ S
hL' : ∀ a' ∈ S, (∀ x ∈ L, x ∈ a') → L = a'
this : ∃ x, x ∉ L
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | obtain ⟨a, b, hab, rfl⟩ := hL | case intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this✝ : S.Nonempty
L : Set V
hL : L ∈ S
hL' : ∀ a' ∈ S, (∀ x ∈ L, x ∈ a') → L = a'
this : ∃ x, x ∉ L
⊢ False | case intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this✝ : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
this : ∃ x, x ∉ Line a b
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | obtain ⟨c, hc'⟩ := this | case intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this✝ : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
this : ∃ x, x ∉ Line a b
⊢ False | case intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | have hac : a ≠ c := fun h => hc' (subset_generateLine _ (by simp [h])) | case intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
⊢ False | case intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
hac : a ≠ c
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | have hbc : b ≠ c := fun h => hc' (subset_generateLine _ (by simp [h])) | case intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
hac : a ≠ c
⊢ False | case intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
hac : a ≠ c
hbc : b ≠ c
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | simp only [NotCollinear, not_and, not_forall, not_not] at h | case intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
hac : a ≠ c
hbc : b ≠ c
⊢ False | case intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
hac : a ≠ c
hbc : b ≠ c
h : ∀ (x y z : V), x ≠ y → x ≠ z → y ≠ z → ∃ x_1, ∃ (_ : x_1.IsLine), {x, y, z} ⊆ x_1
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | obtain ⟨M, hM, habc⟩ := h a b c hab hac hbc | case intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
hac : a ≠ c
hbc : b ≠ c
h : ∀ (x y z : V), x ≠ y → x ≠ z → y ≠ z → ∃ x_1, ∃ (_ : x_1.IsLine), {x, y, z} ⊆ x_1
⊢ False | case intro.intro.intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
hac : a ≠ c
hbc : b ≠ c
h : ∀ (x y z : V), x ≠ y → x ≠ z → y ≠ z → ∃ x_1, ∃ (_ : x_1.IsLine), {x, y, z} ⊆ x_1
M : Set V
hM : M.IsLine
habc : {a, b, c} ⊆ M
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | have := hL' M hM (Set.IsLine.generateLine_subset (habc.trans' (by simp)) hM) | case intro.intro.intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
hac : a ≠ c
hbc : b ≠ c
h : ∀ (x y z : V), x ≠ y → x ≠ z → y ≠ z → ∃ x_1, ∃ (_ : x_1.IsLine), {x, y, z} ⊆ x_1
M : Set V
hM : M.IsLine
habc : {a, b, c} ⊆ M
⊢ False | case intro.intro.intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
S : Set (Set V) := setOf Set.IsLine
this✝ : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
hac : a ≠ c
hbc : b ≠ c
h : ∀ (x y z : V), x ≠ y → x ≠ z → y ≠ z → ∃ x_1, ∃ (_ : x_1.IsLine), {x, y, z} ⊆ x_1
M : Set V
hM : M.IsLine
habc : {a, b, c} ⊆ M
this : Line a b = M
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | rw [this] at hc' | case intro.intro.intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
S : Set (Set V) := setOf Set.IsLine
this✝ : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
hac : a ≠ c
hbc : b ≠ c
h : ∀ (x y z : V), x ≠ y → x ≠ z → y ≠ z → ∃ x_1, ∃ (_ : x_1.IsLine), {x, y, z} ⊆ x_1
M : Set V
hM : M.IsLine
habc : {a, b, c} ⊆ M
this : Line a b = M
⊢ False | case intro.intro.intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
S : Set (Set V) := setOf Set.IsLine
this✝ : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hac : a ≠ c
hbc : b ≠ c
h : ∀ (x y z : V), x ≠ y → x ≠ z → y ≠ z → ∃ x_1, ∃ (_ : x_1.IsLine), {x, y, z} ⊆ x_1
M : Set V
hc' : c ∉ M
hM : M.IsLine
habc : {a, b, c} ⊆ M
this : Line a b = M
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | exact hc' (habc (by simp)) | case intro.intro.intro.intro.intro.intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
S : Set (Set V) := setOf Set.IsLine
this✝ : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hac : a ≠ c
hbc : b ≠ c
h : ∀ (x y z : V), x ≠ y → x ≠ z → y ≠ z → ∃ x_1, ∃ (_ : x_1.IsLine), {x, y, z} ⊆ x_1
M : Set V
hc' : c ∉ M
hM : M.IsLine
habc : {a, b, c} ⊆ M
this : Line a b = M
⊢ False | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | rwa [← this] | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this✝ : S.Nonempty
L : Set V
hL : L ∈ S
hL' : ∀ a' ∈ S, (∀ x ∈ L, x ∈ a') → L = a'
this : L = Set.univ
⊢ Set.univ.IsLine | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | simp [h] | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h✝ : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
h : a = c
⊢ c ∈ {a, b} | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | simp [h] | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h✝ : ∀ (x y z : V), ¬NotCollinear x y z
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
hac : a ≠ c
h : b = c
⊢ c ∈ {a, b} | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | simp | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
S : Set (Set V) := setOf Set.IsLine
this : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hc' : c ∉ Line a b
hac : a ≠ c
hbc : b ≠ c
h : ∀ (x y z : V), x ≠ y → x ≠ z → y ≠ z → ∃ x_1, ∃ (_ : x_1.IsLine), {x, y, z} ⊆ x_1
M : Set V
hM : M.IsLine
habc : {a, b, c} ⊆ M
⊢ {a, b} ≤ {a, b, c} | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | thm_two | [118, 1] | [135, 29] | simp | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
S : Set (Set V) := setOf Set.IsLine
this✝ : S.Nonempty
a b : V
hab : a ≠ b
hL' : ∀ a' ∈ S, (∀ x ∈ Line a b, x ∈ a') → Line a b = a'
c : V
hac : a ≠ c
hbc : b ≠ c
h : ∀ (x y z : V), x ≠ y → x ≠ z → y ≠ z → ∃ x_1, ∃ (_ : x_1.IsLine), {x, y, z} ⊆ x_1
M : Set V
hc' : c ∉ M
hM : M.IsLine
habc : {a, b, c} ⊆ M
this : Line a b = M
⊢ c ∈ {a, b, c} | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | let S : Set (V × V × V) := setOf (fun ⟨a, b, c⟩ => NotCollinear a b c) | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
⊢ ∃ x y z, SimpleTriangle x y z | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
⊢ ∃ x y z, SimpleTriangle x y z |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | have : S.Nonempty := let ⟨x, y, z, hxyz⟩ := h; ⟨(x, y, z), hxyz⟩ | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
⊢ ∃ x y z, SimpleTriangle x y z | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
⊢ ∃ x y z, SimpleTriangle x y z |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | let f : V × V × V → ℝ := fun ⟨a, b, c⟩ => dist a b + dist b c + dist c a | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
⊢ ∃ x y z, SimpleTriangle x y z | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
⊢ ∃ x y z, SimpleTriangle x y z |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | obtain ⟨⟨a, b, c⟩, (h₁ : NotCollinear _ _ _), h₂⟩ := S.toFinite.exists_minimal_wrt f S this | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
⊢ ∃ x y z, SimpleTriangle x y z | case intro.mk.mk.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ a' ∈ S, f a' ≤ f (a, b, c) → f (a, b, c) = f a'
⊢ ∃ x y z, SimpleTriangle x y z |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | simp only [Prod.forall, Set.mem_setOf_eq] at h₂ | case intro.mk.mk.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ a' ∈ S, f a' ≤ f (a, b, c) → f (a, b, c) = f a'
⊢ ∃ x y z, SimpleTriangle x y z | case intro.mk.mk.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a_1 a_2 b_1 : V), (a_1, a_2, b_1) ∈ S → f (a_1, a_2, b_1) ≤ f (a, b, c) → f (a, b, c) = f (a_1, a_2, b_1)
⊢ ∃ x y z, SimpleTriangle x y z |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | replace h₂ : ∀ a' b' c' : V, NotCollinear a' b' c' →
dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a' := by
intro a' b' c' hL
by_contra! h
exact h.ne' (h₂ a' b' c' hL h.le) | case intro.mk.mk.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a_1 a_2 b_1 : V), (a_1, a_2, b_1) ∈ S → f (a_1, a_2, b_1) ≤ f (a, b, c) → f (a, b, c) = f (a_1, a_2, b_1)
⊢ ∃ x y z, SimpleTriangle x y z | case intro.mk.mk.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
⊢ ∃ x y z, SimpleTriangle x y z |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | simp only [SimpleTriangle] | case intro.mk.mk.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
⊢ ∃ x y z, SimpleTriangle x y z | case intro.mk.mk.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
⊢ ∃ x y z, SimpleEdges.Adj x y ∧ SimpleEdges.Adj y z ∧ SimpleEdges.Adj z x ∧ NotCollinear x y z |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | by_contra! cont | case intro.mk.mk.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
⊢ ∃ x y z, SimpleEdges.Adj x y ∧ SimpleEdges.Adj y z ∧ SimpleEdges.Adj z x ∧ NotCollinear x y z | case intro.mk.mk.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | wlog hab : ¬ SimpleEdges.Adj a b generalizing a b c | case intro.mk.mk.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
⊢ False | case intro.mk.mk.intro.inr
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
this :
∀ (a b c : V) (h₁ : NotCollinear a b c)
(h₂ :
∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'),
¬SimpleEdges.Adj a b → False
hab : ¬¬SimpleEdges.Adj a b
⊢ False
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
hab : ¬SimpleEdges.Adj a b
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | simp only [SimpleEdges_adj, h₁.1, ne_eq, not_false_eq_true, true_and, not_forall, not_not] at hab | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
hab : ¬SimpleEdges.Adj a b
⊢ False | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
hab : ∃ x, sbtw a x b
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | obtain ⟨d, adb⟩ := hab | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
hab : ∃ x, sbtw a x b
⊢ False | case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | have habc : c ∉ Line a b := fun hc ↦ h₁.2.2.2 (Line a b) (Line_isLine h₁.1)
(by simp [left_mem_Line, right_mem_Line, hc]) | case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
⊢ False | case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | have hdab : d ∈ Line a b := middle_extend_mem_Line adb | case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
⊢ False | case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | have : dist d c < dist d b + dist b c := by
by_contra!
refine habc (generateLine_close_right hdab right_mem_Line ?_)
exact ⟨adb.ne23, hcd.symm, h₁.2.2.1, this.antisymm (dist_triangle _ _ _)⟩ | case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
⊢ False | case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
this : dist d c < dist d b + dist b c
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | replace : dist a d + dist d c + dist c a < dist a b + dist b c + dist c a := by
linarith only [this, adb.2.2.2] | case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
this : dist d c < dist d b + dist b c
⊢ False | case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
this : dist a d + dist d c + dist c a < dist a b + dist b c + dist c a
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | replace : ¬ NotCollinear a d c := fun h => (h₂ a d c h).not_lt this | case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
this : dist a d + dist d c + dist c a < dist a b + dist b c + dist c a
⊢ False | case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
this : ¬NotCollinear a d c
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | simp only [notCollinear_iff, adb.ne12, hcd.symm, h₁.2.1, true_and, not_and, forall_true_left,
ne_eq, not_forall, not_not, exists_prop, not_false_eq_true] at this | case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
this : ¬NotCollinear a d c
⊢ False | case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
this : ∃ x, x.IsLine ∧ {a, d, c} ⊆ x
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | obtain ⟨L, hL, hL'⟩ := this | case intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
this : ∃ x, x.IsLine ∧ {a, d, c} ⊆ x
⊢ False | case intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
L : Set V
hL : L.IsLine
hL' : {a, d, c} ⊆ L
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | simp only [Set.subset_def, Set.mem_singleton_iff, Set.mem_insert_iff, forall_eq_or_imp,
forall_eq] at hL' | case intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
L : Set V
hL : L.IsLine
hL' : {a, d, c} ⊆ L
⊢ False | case intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
L : Set V
hL : L.IsLine
hL' : a ∈ L ∧ d ∈ L ∧ c ∈ L
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | have : b ∈ L := hL.close_right hL'.1 hL'.2.1 adb | case intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
L : Set V
hL : L.IsLine
hL' : a ∈ L ∧ d ∈ L ∧ c ∈ L
⊢ False | case intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
L : Set V
hL : L.IsLine
hL' : a ∈ L ∧ d ∈ L ∧ c ∈ L
this : b ∈ L
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | refine h₁.2.2.2 L hL ?_ | case intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
L : Set V
hL : L.IsLine
hL' : a ∈ L ∧ d ∈ L ∧ c ∈ L
this : b ∈ L
⊢ False | case intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
L : Set V
hL : L.IsLine
hL' : a ∈ L ∧ d ∈ L ∧ c ∈ L
this : b ∈ L
⊢ {a, b, c} ⊆ L |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | simp [this, hL'.1, hL'.2.2] | case intro.intro.intro
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
d : V
adb : sbtw a d b
habc : c ∉ Line a b
hdab : d ∈ Line a b
hcd : c ≠ d
L : Set V
hL : L.IsLine
hL' : a ∈ L ∧ d ∈ L ∧ c ∈ L
this : b ∈ L
⊢ {a, b, c} ⊆ L | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | intro a' b' c' hL | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a_1 a_2 b_1 : V), (a_1, a_2, b_1) ∈ S → f (a_1, a_2, b_1) ≤ f (a, b, c) → f (a, b, c) = f (a_1, a_2, b_1)
⊢ ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a' | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a_1 a_2 b_1 : V), (a_1, a_2, b_1) ∈ S → f (a_1, a_2, b_1) ≤ f (a, b, c) → f (a, b, c) = f (a_1, a_2, b_1)
a' b' c' : V
hL : NotCollinear a' b' c'
⊢ dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a' |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | by_contra! h | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a_1 a_2 b_1 : V), (a_1, a_2, b_1) ∈ S → f (a_1, a_2, b_1) ≤ f (a, b, c) → f (a, b, c) = f (a_1, a_2, b_1)
a' b' c' : V
hL : NotCollinear a' b' c'
⊢ dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a' | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h✝ : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a_1 a_2 b_1 : V), (a_1, a_2, b_1) ∈ S → f (a_1, a_2, b_1) ≤ f (a, b, c) → f (a, b, c) = f (a_1, a_2, b_1)
a' b' c' : V
hL : NotCollinear a' b' c'
h : dist a' b' + dist b' c' + dist c' a' < dist a b + dist b c + dist c a
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | exact h.ne' (h₂ a' b' c' hL h.le) | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h✝ : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a_1 a_2 b_1 : V), (a_1, a_2, b_1) ∈ S → f (a_1, a_2, b_1) ≤ f (a, b, c) → f (a, b, c) = f (a_1, a_2, b_1)
a' b' c' : V
hL : NotCollinear a' b' c'
h : dist a' b' + dist b' c' + dist c' a' < dist a b + dist b c + dist c a
⊢ False | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | rw [not_not] at hab | case intro.mk.mk.intro.inr
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
this :
∀ (a b c : V) (h₁ : NotCollinear a b c)
(h₂ :
∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'),
¬SimpleEdges.Adj a b → False
hab : ¬¬SimpleEdges.Adj a b
⊢ False | case intro.mk.mk.intro.inr
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
this :
∀ (a b c : V) (h₁ : NotCollinear a b c)
(h₂ :
∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'),
¬SimpleEdges.Adj a b → False
hab : SimpleEdges.Adj a b
⊢ False
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
⊢ (a b c : V) →
NotCollinear a b c →
(∀ (a' b' c' : V),
NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a') →
MetricSpace V |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | refine cont a b c hab ?_ ?_ h₁ | case intro.mk.mk.intro.inr
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
this :
∀ (a b c : V) (h₁ : NotCollinear a b c)
(h₂ :
∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'),
¬SimpleEdges.Adj a b → False
hab : SimpleEdges.Adj a b
⊢ False
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
⊢ (a b c : V) →
NotCollinear a b c →
(∀ (a' b' c' : V),
NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a') →
MetricSpace V | case intro.mk.mk.intro.inr.refine_1
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
this :
∀ (a b c : V),
NotCollinear a b c →
(∀ (a' b' c' : V),
NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a') →
¬SimpleEdges.Adj a b → False
hab : SimpleEdges.Adj a b
⊢ SimpleEdges.Adj b c
case intro.mk.mk.intro.inr.refine_2
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
this :
∀ (a b c : V),
NotCollinear a b c →
(∀ (a' b' c' : V),
NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a') →
¬SimpleEdges.Adj a b → False
hab : SimpleEdges.Adj a b
⊢ SimpleEdges.Adj c a |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | exact not_not.1 <| this b c a h₁.rotate <|
fun _ _ _ h => (h₂ _ _ _ h).trans_eq' <| by ring | case intro.mk.mk.intro.inr.refine_1
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
this :
∀ (a b c : V),
NotCollinear a b c →
(∀ (a' b' c' : V),
NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a') →
¬SimpleEdges.Adj a b → False
hab : SimpleEdges.Adj a b
⊢ SimpleEdges.Adj b c | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | ring | V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h✝ : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
this :
∀ (a b c : V),
NotCollinear a b c →
(∀ (a' b' c' : V),
NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a') →
¬SimpleEdges.Adj a b → False
hab : SimpleEdges.Adj a b
x✝² x✝¹ x✝ : V
h : NotCollinear x✝² x✝¹ x✝
⊢ dist b c + dist c a + dist a b = dist a b + dist b c + dist c a | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SylvesterChvatal.lean | one_implies_two | [153, 1] | [194, 30] | exact not_not.1 <| this c a b h₁.rotate.rotate <|
fun _ _ _ h => (h₂ _ _ _ h).trans_eq' <| by ring | case intro.mk.mk.intro.inr.refine_2
V : Type u_1
inst✝² : MetricSpace V
u v w : V
inst✝¹ : Finite V
inst✝ : Nontrivial V
x y z : V
h : ∃ x y z, NotCollinear x y z
S : Set (V × V × V) := {(a, b, c) | NotCollinear a b c}
this✝ : S.Nonempty
f : V × V × V → ℝ :=
fun x =>
match x with
| (a, b, c) => dist a b + dist b c + dist c a
a b c : V
h₁ : NotCollinear a b c
h₂ : ∀ (a' b' c' : V), NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a'
cont : ∀ (x y z : V), SimpleEdges.Adj x y → SimpleEdges.Adj y z → SimpleEdges.Adj z x → ¬NotCollinear x y z
this :
∀ (a b c : V),
NotCollinear a b c →
(∀ (a' b' c' : V),
NotCollinear a' b' c' → dist a b + dist b c + dist c a ≤ dist a' b' + dist b' c' + dist c' a') →
¬SimpleEdges.Adj a b → False
hab : SimpleEdges.Adj a b
⊢ SimpleEdges.Adj c a | no goals |
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