l3lab/ntp-mathlib huggingface dataset:
@[simp]
theorem openSegment_same (x : E) : openSegment 𝕜 x x = {x} :=
  Set.ext fun z =>
    ⟨fun ⟨a, b, _, _, hab, hz⟩ => by
      simpa only [← add_smul, mem_singleton_iff, hab, one_smul, eq_comm] using hz,
    fun h : z = x => by
      obtain ⟨a, ha₀, ha₁⟩ := DenselyOrdered.dense (0 : 𝕜) 1 zero_lt_one
      refine' ⟨a, 1 - a, ha₀, sub_pos_of_lt ha₁, add_sub_cancel'_right _ _, _⟩
      

Proof state:
case intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁷ : OrderedRing 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : AddCommGroup F
inst✝⁴ : AddCommGroup G
inst✝³ : Module 𝕜 E
inst✝² : Module 𝕜 F
inst✝¹ : Nontrivial 𝕜
inst✝ : DenselyOrdered 𝕜
x z : E
h : z = x
a : 𝕜
ha₀ : 0 < a
ha₁ : a < 1
⊢ a • x + (1 - a) • x = z

Next tactic:
rw [← add_smul, add_sub_cancel'_right, one_smul, h]
theorem segment_eq_image (x y : E) :
    [x -[𝕜] y] = (fun θ : 𝕜 => (1 - θ) • x + θ • y) '' Icc (0 : 𝕜) 1 :=
  Set.ext fun z =>
    ⟨fun ⟨a, b, ha, hb, hab, hz⟩ =>
      ⟨b, ⟨hb, hab ▸ le_add_of_nonneg_left ha⟩, hab ▸ hz ▸ by 

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
x y z : E
x✝ : z ∈ [x-[𝕜]y]
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
hz : a • x + b • y = z
⊢ (fun θ => (a + b - θ) • x + θ • y) b = a • x + b • y

Next tactic:
simp only [add_sub_cancel]
theorem openSegment_eq_image (x y : E) :
    openSegment 𝕜 x y = (fun θ : 𝕜 => (1 - θ) • x + θ • y) '' Ioo (0 : 𝕜) 1 :=
  Set.ext fun z =>
    ⟨fun ⟨a, b, ha, hb, hab, hz⟩ =>
      ⟨b, ⟨hb, hab ▸ lt_add_of_pos_left _ ha⟩, hab ▸ hz ▸ by 

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
x y z : E
x✝ : z ∈ openSegment 𝕜 x y
a b : 𝕜
ha : 0 < a
hb : 0 < b
hab : a + b = 1
hz : a • x + b • y = z
⊢ (fun θ => (a + b - θ) • x + θ • y) b = a • x + b • y

Next tactic:
simp only [add_sub_cancel]
theorem segment_eq_image' (x y : E) :
    [x -[𝕜] y] = (fun θ : 𝕜 => x + θ • (y - x)) '' Icc (0 : 𝕜) 1 := by
  

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
x y : E
⊢ [x-[𝕜]y] = (fun θ => x + θ • (y - x)) '' Icc 0 1

Next tactic:
convert segment_eq_image 𝕜 x y using 2
theorem segment_eq_image' (x y : E) :
    [x -[𝕜] y] = (fun θ : 𝕜 => x + θ • (y - x)) '' Icc (0 : 𝕜) 1 := by
  convert segment_eq_image 𝕜 x y using 2
  

Proof state:
case h.e'_3.h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
x y : E
a✝¹ : 𝕜
a✝ : a✝¹ ∈ Icc 0 1
⊢ x + a✝¹ • (y - x) = (1 - a✝¹) • x + a✝¹ • y

Next tactic:
simp only [smul_sub, sub_smul, one_smul]
theorem segment_eq_image' (x y : E) :
    [x -[𝕜] y] = (fun θ : 𝕜 => x + θ • (y - x)) '' Icc (0 : 𝕜) 1 := by
  convert segment_eq_image 𝕜 x y using 2
  simp only [smul_sub, sub_smul, one_smul]
  

Proof state:
case h.e'_3.h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
x y : E
a✝¹ : 𝕜
a✝ : a✝¹ ∈ Icc 0 1
⊢ x + (a✝¹ • y - a✝¹ • x) = x - a✝¹ • x + a✝¹ • y

Next tactic:
abel
theorem openSegment_eq_image' (x y : E) :
    openSegment 𝕜 x y = (fun θ : 𝕜 => x + θ • (y - x)) '' Ioo (0 : 𝕜) 1 := by
  

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
x y : E
⊢ openSegment 𝕜 x y = (fun θ => x + θ • (y - x)) '' Ioo 0 1

Next tactic:
convert openSegment_eq_image 𝕜 x y using 2
theorem openSegment_eq_image' (x y : E) :
    openSegment 𝕜 x y = (fun θ : 𝕜 => x + θ • (y - x)) '' Ioo (0 : 𝕜) 1 := by
  convert openSegment_eq_image 𝕜 x y using 2
  

Proof state:
case h.e'_3.h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
x y : E
a✝¹ : 𝕜
a✝ : a✝¹ ∈ Ioo 0 1
⊢ x + a✝¹ • (y - x) = (1 - a✝¹) • x + a✝¹ • y

Next tactic:
simp only [smul_sub, sub_smul, one_smul]
theorem openSegment_eq_image' (x y : E) :
    openSegment 𝕜 x y = (fun θ : 𝕜 => x + θ • (y - x)) '' Ioo (0 : 𝕜) 1 := by
  convert openSegment_eq_image 𝕜 x y using 2
  simp only [smul_sub, sub_smul, one_smul]
  

Proof state:
case h.e'_3.h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
x y : E
a✝¹ : 𝕜
a✝ : a✝¹ ∈ Ioo 0 1
⊢ x + (a✝¹ • y - a✝¹ • x) = x - a✝¹ • x + a✝¹ • y

Next tactic:
abel
theorem segment_eq_image_lineMap (x y : E) : [x -[𝕜] y] =
    AffineMap.lineMap x y '' Icc (0 : 𝕜) 1 := by
  

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
x y : E
⊢ [x-[𝕜]y] = ⇑(AffineMap.lineMap x y) '' Icc 0 1

Next tactic:
convert segment_eq_image 𝕜 x y using 2
theorem segment_eq_image_lineMap (x y : E) : [x -[𝕜] y] =
    AffineMap.lineMap x y '' Icc (0 : 𝕜) 1 := by
  convert segment_eq_image 𝕜 x y using 2
  

Proof state:
case h.e'_3.h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
x y : E
a✝¹ : 𝕜
a✝ : a✝¹ ∈ Icc 0 1
⊢ (AffineMap.lineMap x y) a✝¹ = (1 - a✝¹) • x + a✝¹ • y

Next tactic:
exact AffineMap.lineMap_apply_module _ _ _
theorem openSegment_eq_image_lineMap (x y : E) :
    openSegment 𝕜 x y = AffineMap.lineMap x y '' Ioo (0 : 𝕜) 1 := by
  

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
x y : E
⊢ openSegment 𝕜 x y = ⇑(AffineMap.lineMap x y) '' Ioo 0 1

Next tactic:
convert openSegment_eq_image 𝕜 x y using 2
theorem openSegment_eq_image_lineMap (x y : E) :
    openSegment 𝕜 x y = AffineMap.lineMap x y '' Ioo (0 : 𝕜) 1 := by
  convert openSegment_eq_image 𝕜 x y using 2
  

Proof state:
case h.e'_3.h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
x y : E
a✝¹ : 𝕜
a✝ : a✝¹ ∈ Ioo 0 1
⊢ (AffineMap.lineMap x y) a✝¹ = (1 - a✝¹) • x + a✝¹ • y

Next tactic:
exact AffineMap.lineMap_apply_module _ _ _
@[simp]
theorem image_segment (f : E →ᵃ[𝕜] F) (a b : E) : f '' [a -[𝕜] b] = [f a -[𝕜] f b] :=
  Set.ext fun x => by
    

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
f : E →ᵃ[𝕜] F
a b : E
x : F
⊢ x ∈ ⇑f '' [a-[𝕜]b] ↔ x ∈ [f a-[𝕜]f b]

Next tactic:
simp_rw [segment_eq_image_lineMap, mem_image, exists_exists_and_eq_and, AffineMap.apply_lineMap]
@[simp]
theorem image_openSegment (f : E →ᵃ[𝕜] F) (a b : E) :
    f '' openSegment 𝕜 a b = openSegment 𝕜 (f a) (f b) :=
  Set.ext fun x => by
    

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
f : E →ᵃ[𝕜] F
a b : E
x : F
⊢ x ∈ ⇑f '' openSegment 𝕜 a b ↔ x ∈ openSegment 𝕜 (f a) (f b)

Next tactic:
simp_rw [openSegment_eq_image_lineMap, mem_image, exists_exists_and_eq_and,
      AffineMap.apply_lineMap]
@[simp]
theorem mem_segment_translate (a : E) {x b c} : a + x ∈ [a + b -[𝕜] a + c] ↔ x ∈ [b -[𝕜] c] := by
  

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
a x b c : E
⊢ a + x ∈ [a + b-[𝕜]a + c] ↔ x ∈ [b-[𝕜]c]

Next tactic:
simp_rw [← vadd_eq_add, ← vadd_segment, vadd_mem_vadd_set_iff]
@[simp]
theorem mem_openSegment_translate (a : E) {x b c : E} :
    a + x ∈ openSegment 𝕜 (a + b) (a + c) ↔ x ∈ openSegment 𝕜 b c := by
  

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
a x b c : E
⊢ a + x ∈ openSegment 𝕜 (a + b) (a + c) ↔ x ∈ openSegment 𝕜 b c

Next tactic:
simp_rw [← vadd_eq_add, ← vadd_openSegment, vadd_mem_vadd_set_iff]
lemma segment_inter_eq_endpoint_of_linearIndependent_sub
    {c x y : E} (h : LinearIndependent 𝕜 ![x - c, y - c]) :
    [c -[𝕜] x] ∩ [c -[𝕜] y] = {c} := by
  

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
c x y : E
h : LinearIndependent 𝕜 ![x - c, y - c]
⊢ [c-[𝕜]x] ∩ [c-[𝕜]y] = {c}

Next tactic:
apply Subset.antisymm
lemma segment_inter_eq_endpoint_of_linearIndependent_sub
    {c x y : E} (h : LinearIndependent 𝕜 ![x - c, y - c]) :
    [c -[𝕜] x] ∩ [c -[𝕜] y] = {c} := by
  apply Subset.antisymm; 

Proof state:
case h₁
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
c x y : E
h : LinearIndependent 𝕜 ![x - c, y - c]
⊢ [c-[𝕜]x] ∩ [c-[𝕜]y] ⊆ {c}
case h₂
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
c x y : E
h : LinearIndependent 𝕜 ![x - c, y - c]
⊢ {c} ⊆ [c-[𝕜]x] ∩ [c-[𝕜]y]

Next tactic:
swap
lemma segment_inter_eq_endpoint_of_linearIndependent_sub
    {c x y : E} (h : LinearIndependent 𝕜 ![x - c, y - c]) :
    [c -[𝕜] x] ∩ [c -[𝕜] y] = {c} := by
  apply Subset.antisymm; swap
  · 

Proof state:
case h₂
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
c x y : E
h : LinearIndependent 𝕜 ![x - c, y - c]
⊢ {c} ⊆ [c-[𝕜]x] ∩ [c-[𝕜]y]

Next tactic:
simp [singleton_subset_iff, left_mem_segment]
lemma segment_inter_eq_endpoint_of_linearIndependent_sub
    {c x y : E} (h : LinearIndependent 𝕜 ![x - c, y - c]) :
    [c -[𝕜] x] ∩ [c -[𝕜] y] = {c} := by
  apply Subset.antisymm; swap
  · simp [singleton_subset_iff, left_mem_segment]
  

Proof state:
case h₁
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
c x y : E
h : LinearIndependent 𝕜 ![x - c, y - c]
⊢ [c-[𝕜]x] ∩ [c-[𝕜]y] ⊆ {c}

Next tactic:
intro z ⟨hzt, hzs⟩
lemma segment_inter_eq_endpoint_of_linearIndependent_sub
    {c x y : E} (h : LinearIndependent 𝕜 ![x - c, y - c]) :
    [c -[𝕜] x] ∩ [c -[𝕜] y] = {c} := by
  apply Subset.antisymm; swap
  · simp [singleton_subset_iff, left_mem_segment]
  intro z ⟨hzt, hzs⟩
  

Proof state:
case h₁
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
c x y : E
h : LinearIndependent 𝕜 ![x - c, y - c]
z : E
hzt : z ∈ [c-[𝕜]x]
hzs : z ∈ [c-[𝕜]y]
⊢ z ∈ {c}

Next tactic:
rw [segment_eq_image, mem_image] at hzt hzs
lemma segment_inter_eq_endpoint_of_linearIndependent_sub
    {c x y : E} (h : LinearIndependent 𝕜 ![x - c, y - c]) :
    [c -[𝕜] x] ∩ [c -[𝕜] y] = {c} := by
  apply Subset.antisymm; swap
  · simp [singleton_subset_iff, left_mem_segment]
  intro z ⟨hzt, hzs⟩
  rw [segment_eq_image, mem_image] at hzt hzs
  

Proof state:
case h₁
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
c x y : E
h : LinearIndependent 𝕜 ![x - c, y - c]
z : E
hzt : ∃ x_1 ∈ Icc 0 1, (1 - x_1) • c + x_1 • x = z
hzs : ∃ x ∈ Icc 0 1, (1 - x) • c + x • y = z
⊢ z ∈ {c}

Next tactic:
rcases hzt with ⟨p, ⟨p0, p1⟩, rfl⟩
lemma segment_inter_eq_endpoint_of_linearIndependent_sub
    {c x y : E} (h : LinearIndependent 𝕜 ![x - c, y - c]) :
    [c -[𝕜] x] ∩ [c -[𝕜] y] = {c} := by
  apply Subset.antisymm; swap
  · simp [singleton_subset_iff, left_mem_segment]
  intro z ⟨hzt, hzs⟩
  rw [segment_eq_image, mem_image] at hzt hzs
  rcases hzt with ⟨p, ⟨p0, p1⟩, rfl⟩
  

Proof state:
case h₁.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
c x y : E
h : LinearIndependent 𝕜 ![x - c, y - c]
p : 𝕜
p0 : 0 ≤ p
p1 : p ≤ 1
hzs : ∃ x_1 ∈ Icc 0 1, (1 - x_1) • c + x_1 • y = (1 - p) • c + p • x
⊢ (1 - p) • c + p • x ∈ {c}

Next tactic:
rcases hzs with ⟨q, ⟨q0, q1⟩, H⟩
lemma segment_inter_eq_endpoint_of_linearIndependent_sub
    {c x y : E} (h : LinearIndependent 𝕜 ![x - c, y - c]) :
    [c -[𝕜] x] ∩ [c -[𝕜] y] = {c} := by
  apply Subset.antisymm; swap
  · simp [singleton_subset_iff, left_mem_segment]
  intro z ⟨hzt, hzs⟩
  rw [segment_eq_image, mem_image] at hzt hzs
  rcases hzt with ⟨p, ⟨p0, p1⟩, rfl⟩
  rcases hzs with ⟨q, ⟨q0, q1⟩, H⟩
  

Proof state:
case h₁.intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
c x y : E
h : LinearIndependent 𝕜 ![x - c, y - c]
p : 𝕜
p0 : 0 ≤ p
p1 : p ≤ 1
q : 𝕜
H : (1 - q) • c + q • y = (1 - p) • c + p • x
q0 : 0 ≤ q
q1 : q ≤ 1
⊢ (1 - p) • c + p • x ∈ {c}

Next tactic:
have Hx : x = (x - c) + c := by abel
lemma segment_inter_eq_endpoint_of_linearIndependent_sub
    {c x y : E} (h : LinearIndependent 𝕜 ![x - c, y - c]) :
    [c -[𝕜] x] ∩ [c -[𝕜] y] = {c} := by
  apply Subset.antisymm; swap
  · simp [singleton_subset_iff, left_mem_segment]
  intro z ⟨hzt, hzs⟩
  rw [segment_eq_image, mem_image] at hzt hzs
  rcases hzt with ⟨p, ⟨p0, p1⟩, rfl⟩
  rcases hzs with ⟨q, ⟨q0, q1⟩, H⟩
  have Hx : x = (x - c) + c := by 

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
c x y : E
h : LinearIndependent 𝕜 ![x - c, y - c]
p : 𝕜
p0 : 0 ≤ p
p1 : p ≤ 1
q : 𝕜
H : (1 - q) • c + q • y = (1 - p) • c + p • x
q0 : 0 ≤ q
q1 : q ≤ 1
⊢ x = x - c + c

Next tactic:
abel
lemma segment_inter_eq_endpoint_of_linearIndependent_sub
    {c x y : E} (h : LinearIndependent 𝕜 ![x - c, y - c]) :
    [c -[𝕜] x] ∩ [c -[𝕜] y] = {c} := by
  apply Subset.antisymm; swap
  · simp [singleton_subset_iff, left_mem_segment]
  intro z ⟨hzt, hzs⟩
  rw [segment_eq_image, mem_image] at hzt hzs
  rcases hzt with ⟨p, ⟨p0, p1⟩, rfl⟩
  rcases hzs with ⟨q, ⟨q0, q1⟩, H⟩
  have Hx : x = (x - c) + c := by abel
  

Proof state:
case h₁.intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
c x y : E
h : LinearIndependent 𝕜 ![x - c, y - c]
p : 𝕜
p0 : 0 ≤ p
p1 : p ≤ 1
q : 𝕜
H : (1 - q) • c + q • y = (1 - p) • c + p • x
q0 : 0 ≤ q
q1 : q ≤ 1
Hx : x = x - c + c
⊢ (1 - p) • c + p • x ∈ {c}

Next tactic:
have Hy : y = (y - c) + c := by abel
lemma segment_inter_eq_endpoint_of_linearIndependent_sub
    {c x y : E} (h : LinearIndependent 𝕜 ![x - c, y - c]) :
    [c -[𝕜] x] ∩ [c -[𝕜] y] = {c} := by
  apply Subset.antisymm; swap
  · simp [singleton_subset_iff, left_mem_segment]
  intro z ⟨hzt, hzs⟩
  rw [segment_eq_image, mem_image] at hzt hzs
  rcases hzt with ⟨p, ⟨p0, p1⟩, rfl⟩
  rcases hzs with ⟨q, ⟨q0, q1⟩, H⟩
  have Hx : x = (x - c) + c := by abel
  have Hy : y = (y - c) + c := by 

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
c x y : E
h : LinearIndependent 𝕜 ![x - c, y - c]
p : 𝕜
p0 : 0 ≤ p
p1 : p ≤ 1
q : 𝕜
H : (1 - q) • c + q • y = (1 - p) • c + p • x
q0 : 0 ≤ q
q1 : q ≤ 1
Hx : x = x - c + c
⊢ y = y - c + c

Next tactic:
abel
lemma segment_inter_eq_endpoint_of_linearIndependent_sub
    {c x y : E} (h : LinearIndependent 𝕜 ![x - c, y - c]) :
    [c -[𝕜] x] ∩ [c -[𝕜] y] = {c} := by
  apply Subset.antisymm; swap
  · simp [singleton_subset_iff, left_mem_segment]
  intro z ⟨hzt, hzs⟩
  rw [segment_eq_image, mem_image] at hzt hzs
  rcases hzt with ⟨p, ⟨p0, p1⟩, rfl⟩
  rcases hzs with ⟨q, ⟨q0, q1⟩, H⟩
  have Hx : x = (x - c) + c := by abel
  have Hy : y = (y - c) + c := by abel
  

Proof state:
case h₁.intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
c x y : E
h : LinearIndependent 𝕜 ![x - c, y - c]
p : 𝕜
p0 : 0 ≤ p
p1 : p ≤ 1
q : 𝕜
H : (1 - q) • c + q • y = (1 - p) • c + p • x
q0 : 0 ≤ q
q1 : q ≤ 1
Hx : x = x - c + c
Hy : y = y - c + c
⊢ (1 - p) • c + p • x ∈ {c}

Next tactic:
rw [Hx, Hy, smul_add, smul_add] at H
lemma segment_inter_eq_endpoint_of_linearIndependent_sub
    {c x y : E} (h : LinearIndependent 𝕜 ![x - c, y - c]) :
    [c -[𝕜] x] ∩ [c -[𝕜] y] = {c} := by
  apply Subset.antisymm; swap
  · simp [singleton_subset_iff, left_mem_segment]
  intro z ⟨hzt, hzs⟩
  rw [segment_eq_image, mem_image] at hzt hzs
  rcases hzt with ⟨p, ⟨p0, p1⟩, rfl⟩
  rcases hzs with ⟨q, ⟨q0, q1⟩, H⟩
  have Hx : x = (x - c) + c := by abel
  have Hy : y = (y - c) + c := by abel
  rw [Hx, Hy, smul_add, smul_add] at H
  

Proof state:
case h₁.intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
c x y : E
h : LinearIndependent 𝕜 ![x - c, y - c]
p : 𝕜
p0 : 0 ≤ p
p1 : p ≤ 1
q : 𝕜
H : (1 - q) • c + (q • (y - c) + q • c) = (1 - p) • c + (p • (x - c) + p • c)
q0 : 0 ≤ q
q1 : q ≤ 1
Hx : x = x - c + c
Hy : y = y - c + c
⊢ (1 - p) • c + p • x ∈ {c}

Next tactic:
have : c + q • (y - c) = c + p • (x - c) := by
    convert H using 1 <;> simp [sub_smul]
lemma segment_inter_eq_endpoint_of_linearIndependent_sub
    {c x y : E} (h : LinearIndependent 𝕜 ![x - c, y - c]) :
    [c -[𝕜] x] ∩ [c -[𝕜] y] = {c} := by
  apply Subset.antisymm; swap
  · simp [singleton_subset_iff, left_mem_segment]
  intro z ⟨hzt, hzs⟩
  rw [segment_eq_image, mem_image] at hzt hzs
  rcases hzt with ⟨p, ⟨p0, p1⟩, rfl⟩
  rcases hzs with ⟨q, ⟨q0, q1⟩, H⟩
  have Hx : x = (x - c) + c := by abel
  have Hy : y = (y - c) + c := by abel
  rw [Hx, Hy, smul_add, smul_add] at H
  have : c + q • (y - c) = c + p • (x - c) := by
    

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
c x y : E
h : LinearIndependent 𝕜 ![x - c, y - c]
p : 𝕜
p0 : 0 ≤ p
p1 : p ≤ 1
q : 𝕜
H : (1 - q) • c + (q • (y - c) + q • c) = (1 - p) • c + (p • (x - c) + p • c)
q0 : 0 ≤ q
q1 : q ≤ 1
Hx : x = x - c + c
Hy : y = y - c + c
⊢ c + q • (y - c) = c + p • (x - c)

Next tactic:
convert H using 1
lemma segment_inter_eq_endpoint_of_linearIndependent_sub
    {c x y : E} (h : LinearIndependent 𝕜 ![x - c, y - c]) :
    [c -[𝕜] x] ∩ [c -[𝕜] y] = {c} := by
  apply Subset.antisymm; swap
  · simp [singleton_subset_iff, left_mem_segment]
  intro z ⟨hzt, hzs⟩
  rw [segment_eq_image, mem_image] at hzt hzs
  rcases hzt with ⟨p, ⟨p0, p1⟩, rfl⟩
  rcases hzs with ⟨q, ⟨q0, q1⟩, H⟩
  have Hx : x = (x - c) + c := by abel
  have Hy : y = (y - c) + c := by abel
  rw [Hx, Hy, smul_add, smul_add] at H
  have : c + q • (y - c) = c + p • (x - c) := by
    convert H using 1 <;> 

Proof state:
case h.e'_2
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
c x y : E
h : LinearIndependent 𝕜 ![x - c, y - c]
p : 𝕜
p0 : 0 ≤ p
p1 : p ≤ 1
q : 𝕜
H : (1 - q) • c + (q • (y - c) + q • c) = (1 - p) • c + (p • (x - c) + p • c)
q0 : 0 ≤ q
q1 : q ≤ 1
Hx : x = x - c + c
Hy : y = y - c + c
⊢ c + q • (y - c) = (1 - q) • c + (q • (y - c) + q • c)

Next tactic:
simp [sub_smul]
lemma segment_inter_eq_endpoint_of_linearIndependent_sub
    {c x y : E} (h : LinearIndependent 𝕜 ![x - c, y - c]) :
    [c -[𝕜] x] ∩ [c -[𝕜] y] = {c} := by
  apply Subset.antisymm; swap
  · simp [singleton_subset_iff, left_mem_segment]
  intro z ⟨hzt, hzs⟩
  rw [segment_eq_image, mem_image] at hzt hzs
  rcases hzt with ⟨p, ⟨p0, p1⟩, rfl⟩
  rcases hzs with ⟨q, ⟨q0, q1⟩, H⟩
  have Hx : x = (x - c) + c := by abel
  have Hy : y = (y - c) + c := by abel
  rw [Hx, Hy, smul_add, smul_add] at H
  have : c + q • (y - c) = c + p • (x - c) := by
    convert H using 1 <;> 

Proof state:
case h.e'_3
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
c x y : E
h : LinearIndependent 𝕜 ![x - c, y - c]
p : 𝕜
p0 : 0 ≤ p
p1 : p ≤ 1
q : 𝕜
H : (1 - q) • c + (q • (y - c) + q • c) = (1 - p) • c + (p • (x - c) + p • c)
q0 : 0 ≤ q
q1 : q ≤ 1
Hx : x = x - c + c
Hy : y = y - c + c
⊢ c + p • (x - c) = (1 - p) • c + (p • (x - c) + p • c)

Next tactic:
simp [sub_smul]
lemma segment_inter_eq_endpoint_of_linearIndependent_sub
    {c x y : E} (h : LinearIndependent 𝕜 ![x - c, y - c]) :
    [c -[𝕜] x] ∩ [c -[𝕜] y] = {c} := by
  apply Subset.antisymm; swap
  · simp [singleton_subset_iff, left_mem_segment]
  intro z ⟨hzt, hzs⟩
  rw [segment_eq_image, mem_image] at hzt hzs
  rcases hzt with ⟨p, ⟨p0, p1⟩, rfl⟩
  rcases hzs with ⟨q, ⟨q0, q1⟩, H⟩
  have Hx : x = (x - c) + c := by abel
  have Hy : y = (y - c) + c := by abel
  rw [Hx, Hy, smul_add, smul_add] at H
  have : c + q • (y - c) = c + p • (x - c) := by
    convert H using 1 <;> simp [sub_smul]
  

Proof state:
case h₁.intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
c x y : E
h : LinearIndependent 𝕜 ![x - c, y - c]
p : 𝕜
p0 : 0 ≤ p
p1 : p ≤ 1
q : 𝕜
H : (1 - q) • c + (q • (y - c) + q • c) = (1 - p) • c + (p • (x - c) + p • c)
q0 : 0 ≤ q
q1 : q ≤ 1
Hx : x = x - c + c
Hy : y = y - c + c
this : c + q • (y - c) = c + p • (x - c)
⊢ (1 - p) • c + p • x ∈ {c}

Next tactic:
obtain ⟨rfl, rfl⟩ : p = 0 ∧ q = 0 := h.eq_zero_of_pair' ((add_right_inj c).1 this).symm