File: .lake/packages/proofwidgets/ProofWidgets/Demos/Euclidean.lean

full_name: IncidenceGeometry.inCircle
code: class IncidenceGeometry where
  Point : Type u₁
  Line : Type u₂
  Circle : Type u₃

  between : Point → Point → Point → Prop -- implies colinearity
  onLine : Point → Line → Prop
  onCircle : Point → Circle → Prop
  inCircle : Point → Circle → Prop
  centerCircle : Point → Circle → Prop
  circlesInter : Circle → Circle → Prop

  ne_23_of_between : ∀ {a b c : Point}, between a b c → b ≠ c
  line_unique_of_pts : ∀ {a b : Point}, ∀ {L M : Line},
    a ≠ b → onLine a L → onLine b L → onLine a M → onLine b M → L = M
  onLine_2_of_between : ∀ {a b c : Point}, ∀ {L : Line},
    between a b c → onLine a L → onLine c L → onLine b L
  line_of_pts : ∀ a b, ∃ L, onLine a L ∧ onLine b L
  circle_of_ne : ∀ a b, a ≠ b → ∃ C, centerCircle a C ∧ onCircle b C
  circlesInter_of_onCircle_inCircle :
    ∀ {a b α β}, onCircle b α → onCircle a β → inCircle a α →
    inCircle b β → circlesInter α β
  pts_of_circlesInter : ∀ {α β}, circlesInter α β →
    ∃ a b, a ≠ b ∧ onCircle a α ∧ onCircle a β ∧ onCircle b α ∧ onCircle b β
  inCircle_of_centerCircle : ∀ {a α}, centerCircle a α → inCircle a α
goal: IncidenceGeometry.Point → IncidenceGeometry.Circle → Prop

full_name: IncidenceGeometry.Point
code: class IncidenceGeometry where
  Point : Type u₁
  Line : Type u₂
  Circle : Type u₃

  between : Point → Point → Point → Prop -- implies colinearity
  onLine : Point → Line → Prop
  onCircle : Point → Circle → Prop
  inCircle : Point → Circle → Prop
  centerCircle : Point → Circle → Prop
  circlesInter : Circle → Circle → Prop

  ne_23_of_between : ∀ {a b c : Point}, between a b c → b ≠ c
  line_unique_of_pts : ∀ {a b : Point}, ∀ {L M : Line},
    a ≠ b → onLine a L → onLine b L → onLine a M → onLine b M → L = M
  onLine_2_of_between : ∀ {a b c : Point}, ∀ {L : Line},
    between a b c → onLine a L → onLine c L → onLine b L
  line_of_pts : ∀ a b, ∃ L, onLine a L ∧ onLine b L
  circle_of_ne : ∀ a b, a ≠ b → ∃ C, centerCircle a C ∧ onCircle b C
  circlesInter_of_onCircle_inCircle :
    ∀ {a b α β}, onCircle b α → onCircle a β → inCircle a α →
    inCircle b β → circlesInter α β
  pts_of_circlesInter : ∀ {α β}, circlesInter α β →
    ∃ a b, a ≠ b ∧ onCircle a α ∧ onCircle a β ∧ onCircle b α ∧ onCircle b β
  inCircle_of_centerCircle : ∀ {a α}, centerCircle a α → inCircle a α
goal: Type u₁

full_name: IncidenceGeometry.circlesInter
code: class IncidenceGeometry where
  Point : Type u₁
  Line : Type u₂
  Circle : Type u₃

  between : Point → Point → Point → Prop -- implies colinearity
  onLine : Point → Line → Prop
  onCircle : Point → Circle → Prop
  inCircle : Point → Circle → Prop
  centerCircle : Point → Circle → Prop
  circlesInter : Circle → Circle → Prop

  ne_23_of_between : ∀ {a b c : Point}, between a b c → b ≠ c
  line_unique_of_pts : ∀ {a b : Point}, ∀ {L M : Line},
    a ≠ b → onLine a L → onLine b L → onLine a M → onLine b M → L = M
  onLine_2_of_between : ∀ {a b c : Point}, ∀ {L : Line},
    between a b c → onLine a L → onLine c L → onLine b L
  line_of_pts : ∀ a b, ∃ L, onLine a L ∧ onLine b L
  circle_of_ne : ∀ a b, a ≠ b → ∃ C, centerCircle a C ∧ onCircle b C
  circlesInter_of_onCircle_inCircle :
    ∀ {a b α β}, onCircle b α → onCircle a β → inCircle a α →
    inCircle b β → circlesInter α β
  pts_of_circlesInter : ∀ {α β}, circlesInter α β →
    ∃ a b, a ≠ b ∧ onCircle a α ∧ onCircle a β ∧ onCircle b α ∧ onCircle b β
  inCircle_of_centerCircle : ∀ {a α}, centerCircle a α → inCircle a α
goal: IncidenceGeometry.Circle → IncidenceGeometry.Circle → Prop

full_name: IncidenceGeometry.pts_of_circlesInter
code: class IncidenceGeometry where
  Point : Type u₁
  Line : Type u₂
  Circle : Type u₃

  between : Point → Point → Point → Prop -- implies colinearity
  onLine : Point → Line → Prop
  onCircle : Point → Circle → Prop
  inCircle : Point → Circle → Prop
  centerCircle : Point → Circle → Prop
  circlesInter : Circle → Circle → Prop

  ne_23_of_between : ∀ {a b c : Point}, between a b c → b ≠ c
  line_unique_of_pts : ∀ {a b : Point}, ∀ {L M : Line},
    a ≠ b → onLine a L → onLine b L → onLine a M → onLine b M → L = M
  onLine_2_of_between : ∀ {a b c : Point}, ∀ {L : Line},
    between a b c → onLine a L → onLine c L → onLine b L
  line_of_pts : ∀ a b, ∃ L, onLine a L ∧ onLine b L
  circle_of_ne : ∀ a b, a ≠ b → ∃ C, centerCircle a C ∧ onCircle b C
  circlesInter_of_onCircle_inCircle :
    ∀ {a b α β}, onCircle b α → onCircle a β → inCircle a α →
    inCircle b β → circlesInter α β
  pts_of_circlesInter : ∀ {α β}, circlesInter α β →
    ∃ a b, a ≠ b ∧ onCircle a α ∧ onCircle a β ∧ onCircle b α ∧ onCircle b β
  inCircle_of_centerCircle : ∀ {a α}, centerCircle a α → inCircle a α
goal: IncidenceGeometry.circlesInter α β →
  ∃ a b,
    a ≠ b ∧
      IncidenceGeometry.onCircle a α ∧
        IncidenceGeometry.onCircle a β ∧ IncidenceGeometry.onCircle b α ∧ IncidenceGeometry.onCircle b β

full_name: IncidenceGeometry.between
code: class IncidenceGeometry where
  Point : Type u₁
  Line : Type u₂
  Circle : Type u₃

  between : Point → Point → Point → Prop -- implies colinearity
  onLine : Point → Line → Prop
  onCircle : Point → Circle → Prop
  inCircle : Point → Circle → Prop
  centerCircle : Point → Circle → Prop
  circlesInter : Circle → Circle → Prop

  ne_23_of_between : ∀ {a b c : Point}, between a b c → b ≠ c
  line_unique_of_pts : ∀ {a b : Point}, ∀ {L M : Line},
    a ≠ b → onLine a L → onLine b L → onLine a M → onLine b M → L = M
  onLine_2_of_between : ∀ {a b c : Point}, ∀ {L : Line},
    between a b c → onLine a L → onLine c L → onLine b L
  line_of_pts : ∀ a b, ∃ L, onLine a L ∧ onLine b L
  circle_of_ne : ∀ a b, a ≠ b → ∃ C, centerCircle a C ∧ onCircle b C
  circlesInter_of_onCircle_inCircle :
    ∀ {a b α β}, onCircle b α → onCircle a β → inCircle a α →
    inCircle b β → circlesInter α β
  pts_of_circlesInter : ∀ {α β}, circlesInter α β →
    ∃ a b, a ≠ b ∧ onCircle a α ∧ onCircle a β ∧ onCircle b α ∧ onCircle b β
  inCircle_of_centerCircle : ∀ {a α}, centerCircle a α → inCircle a α
goal: IncidenceGeometry.Point → IncidenceGeometry.Point → IncidenceGeometry.Point → Prop

full_name: IncidenceGeometry.onLine_2_of_between
code: class IncidenceGeometry where
  Point : Type u₁
  Line : Type u₂
  Circle : Type u₃

  between : Point → Point → Point → Prop -- implies colinearity
  onLine : Point → Line → Prop
  onCircle : Point → Circle → Prop
  inCircle : Point → Circle → Prop
  centerCircle : Point → Circle → Prop
  circlesInter : Circle → Circle → Prop

  ne_23_of_between : ∀ {a b c : Point}, between a b c → b ≠ c
  line_unique_of_pts : ∀ {a b : Point}, ∀ {L M : Line},
    a ≠ b → onLine a L → onLine b L → onLine a M → onLine b M → L = M
  onLine_2_of_between : ∀ {a b c : Point}, ∀ {L : Line},
    between a b c → onLine a L → onLine c L → onLine b L
  line_of_pts : ∀ a b, ∃ L, onLine a L ∧ onLine b L
  circle_of_ne : ∀ a b, a ≠ b → ∃ C, centerCircle a C ∧ onCircle b C
  circlesInter_of_onCircle_inCircle :
    ∀ {a b α β}, onCircle b α → onCircle a β → inCircle a α →
    inCircle b β → circlesInter α β
  pts_of_circlesInter : ∀ {α β}, circlesInter α β →
    ∃ a b, a ≠ b ∧ onCircle a α ∧ onCircle a β ∧ onCircle b α ∧ onCircle b β
  inCircle_of_centerCircle : ∀ {a α}, centerCircle a α → inCircle a α
goal: IncidenceGeometry.between a b c →
  IncidenceGeometry.onLine a L → IncidenceGeometry.onLine c L → IncidenceGeometry.onLine b L

full_name: IncidenceGeometry.Circle
code: class IncidenceGeometry where
  Point : Type u₁
  Line : Type u₂
  Circle : Type u₃

  between : Point → Point → Point → Prop -- implies colinearity
  onLine : Point → Line → Prop
  onCircle : Point → Circle → Prop
  inCircle : Point → Circle → Prop
  centerCircle : Point → Circle → Prop
  circlesInter : Circle → Circle → Prop

  ne_23_of_between : ∀ {a b c : Point}, between a b c → b ≠ c
  line_unique_of_pts : ∀ {a b : Point}, ∀ {L M : Line},
    a ≠ b → onLine a L → onLine b L → onLine a M → onLine b M → L = M
  onLine_2_of_between : ∀ {a b c : Point}, ∀ {L : Line},
    between a b c → onLine a L → onLine c L → onLine b L
  line_of_pts : ∀ a b, ∃ L, onLine a L ∧ onLine b L
  circle_of_ne : ∀ a b, a ≠ b → ∃ C, centerCircle a C ∧ onCircle b C
  circlesInter_of_onCircle_inCircle :
    ∀ {a b α β}, onCircle b α → onCircle a β → inCircle a α →
    inCircle b β → circlesInter α β
  pts_of_circlesInter : ∀ {α β}, circlesInter α β →
    ∃ a b, a ≠ b ∧ onCircle a α ∧ onCircle a β ∧ onCircle b α ∧ onCircle b β
  inCircle_of_centerCircle : ∀ {a α}, centerCircle a α → inCircle a α
goal: Type u₃

full_name: IncidenceGeometry.onLine
code: class IncidenceGeometry where
  Point : Type u₁
  Line : Type u₂
  Circle : Type u₃

  between : Point → Point → Point → Prop -- implies colinearity
  onLine : Point → Line → Prop
  onCircle : Point → Circle → Prop
  inCircle : Point → Circle → Prop
  centerCircle : Point → Circle → Prop
  circlesInter : Circle → Circle → Prop

  ne_23_of_between : ∀ {a b c : Point}, between a b c → b ≠ c
  line_unique_of_pts : ∀ {a b : Point}, ∀ {L M : Line},
    a ≠ b → onLine a L → onLine b L → onLine a M → onLine b M → L = M
  onLine_2_of_between : ∀ {a b c : Point}, ∀ {L : Line},
    between a b c → onLine a L → onLine c L → onLine b L
  line_of_pts : ∀ a b, ∃ L, onLine a L ∧ onLine b L
  circle_of_ne : ∀ a b, a ≠ b → ∃ C, centerCircle a C ∧ onCircle b C
  circlesInter_of_onCircle_inCircle :
    ∀ {a b α β}, onCircle b α → onCircle a β → inCircle a α →
    inCircle b β → circlesInter α β
  pts_of_circlesInter : ∀ {α β}, circlesInter α β →
    ∃ a b, a ≠ b ∧ onCircle a α ∧ onCircle a β ∧ onCircle b α ∧ onCircle b β
  inCircle_of_centerCircle : ∀ {a α}, centerCircle a α → inCircle a α
goal: IncidenceGeometry.Point → IncidenceGeometry.Line → Prop

full_name: IncidenceGeometry.onCircle
code: class IncidenceGeometry where
  Point : Type u₁
  Line : Type u₂
  Circle : Type u₃

  between : Point → Point → Point → Prop -- implies colinearity
  onLine : Point → Line → Prop
  onCircle : Point → Circle → Prop
  inCircle : Point → Circle → Prop
  centerCircle : Point → Circle → Prop
  circlesInter : Circle → Circle → Prop

  ne_23_of_between : ∀ {a b c : Point}, between a b c → b ≠ c
  line_unique_of_pts : ∀ {a b : Point}, ∀ {L M : Line},
    a ≠ b → onLine a L → onLine b L → onLine a M → onLine b M → L = M
  onLine_2_of_between : ∀ {a b c : Point}, ∀ {L : Line},
    between a b c → onLine a L → onLine c L → onLine b L
  line_of_pts : ∀ a b, ∃ L, onLine a L ∧ onLine b L
  circle_of_ne : ∀ a b, a ≠ b → ∃ C, centerCircle a C ∧ onCircle b C
  circlesInter_of_onCircle_inCircle :
    ∀ {a b α β}, onCircle b α → onCircle a β → inCircle a α →
    inCircle b β → circlesInter α β
  pts_of_circlesInter : ∀ {α β}, circlesInter α β →
    ∃ a b, a ≠ b ∧ onCircle a α ∧ onCircle a β ∧ onCircle b α ∧ onCircle b β
  inCircle_of_centerCircle : ∀ {a α}, centerCircle a α → inCircle a α
goal: IncidenceGeometry.Point → IncidenceGeometry.Circle → Prop

full_name: IncidenceGeometry.Line
code: class IncidenceGeometry where
  Point : Type u₁
  Line : Type u₂
  Circle : Type u₃

  between : Point → Point → Point → Prop -- implies colinearity
  onLine : Point → Line → Prop
  onCircle : Point → Circle → Prop
  inCircle : Point → Circle → Prop
  centerCircle : Point → Circle → Prop
  circlesInter : Circle → Circle → Prop

  ne_23_of_between : ∀ {a b c : Point}, between a b c → b ≠ c
  line_unique_of_pts : ∀ {a b : Point}, ∀ {L M : Line},
    a ≠ b → onLine a L → onLine b L → onLine a M → onLine b M → L = M
  onLine_2_of_between : ∀ {a b c : Point}, ∀ {L : Line},
    between a b c → onLine a L → onLine c L → onLine b L
  line_of_pts : ∀ a b, ∃ L, onLine a L ∧ onLine b L
  circle_of_ne : ∀ a b, a ≠ b → ∃ C, centerCircle a C ∧ onCircle b C
  circlesInter_of_onCircle_inCircle :
    ∀ {a b α β}, onCircle b α → onCircle a β → inCircle a α →
    inCircle b β → circlesInter α β
  pts_of_circlesInter : ∀ {α β}, circlesInter α β →
    ∃ a b, a ≠ b ∧ onCircle a α ∧ onCircle a β ∧ onCircle b α ∧ onCircle b β
  inCircle_of_centerCircle : ∀ {a α}, centerCircle a α → inCircle a α
goal: Type u₂

full_name: IncidenceGeometry
code: class IncidenceGeometry where
  Point : Type u₁
  Line : Type u₂
  Circle : Type u₃

  between : Point → Point → Point → Prop -- implies colinearity
  onLine : Point → Line → Prop
  onCircle : Point → Circle → Prop
  inCircle : Point → Circle → Prop
  centerCircle : Point → Circle → Prop
  circlesInter : Circle → Circle → Prop

  ne_23_of_between : ∀ {a b c : Point}, between a b c → b ≠ c
  line_unique_of_pts : ∀ {a b : Point}, ∀ {L M : Line},
    a ≠ b → onLine a L → onLine b L → onLine a M → onLine b M → L = M
  onLine_2_of_between : ∀ {a b c : Point}, ∀ {L : Line},
    between a b c → onLine a L → onLine c L → onLine b L
  line_of_pts : ∀ a b, ∃ L, onLine a L ∧ onLine b L
  circle_of_ne : ∀ a b, a ≠ b → ∃ C, centerCircle a C ∧ onCircle b C
  circlesInter_of_onCircle_inCircle :
    ∀ {a b α β}, onCircle b α → onCircle a β → inCircle a α →
    inCircle b β → circlesInter α β
  pts_of_circlesInter : ∀ {α β}, circlesInter α β →
    ∃ a b, a ≠ b ∧ onCircle a α ∧ onCircle a β ∧ onCircle b α ∧ onCircle b β
  inCircle_of_centerCircle : ∀ {a α}, centerCircle a α → inCircle a α
goal: Type (max (max (u₁ + 1) (u₂ + 1)) (u₃ + 1))

full_name: IncidenceGeometry.line_unique_of_pts
code: class IncidenceGeometry where
  Point : Type u₁
  Line : Type u₂
  Circle : Type u₃

  between : Point → Point → Point → Prop -- implies colinearity
  onLine : Point → Line → Prop
  onCircle : Point → Circle → Prop
  inCircle : Point → Circle → Prop
  centerCircle : Point → Circle → Prop
  circlesInter : Circle → Circle → Prop

  ne_23_of_between : ∀ {a b c : Point}, between a b c → b ≠ c
  line_unique_of_pts : ∀ {a b : Point}, ∀ {L M : Line},
    a ≠ b → onLine a L → onLine b L → onLine a M → onLine b M → L = M
  onLine_2_of_between : ∀ {a b c : Point}, ∀ {L : Line},
    between a b c → onLine a L → onLine c L → onLine b L
  line_of_pts : ∀ a b, ∃ L, onLine a L ∧ onLine b L
  circle_of_ne : ∀ a b, a ≠ b → ∃ C, centerCircle a C ∧ onCircle b C
  circlesInter_of_onCircle_inCircle :
    ∀ {a b α β}, onCircle b α → onCircle a β → inCircle a α →
    inCircle b β → circlesInter α β
  pts_of_circlesInter : ∀ {α β}, circlesInter α β →
    ∃ a b, a ≠ b ∧ onCircle a α ∧ onCircle a β ∧ onCircle b α ∧ onCircle b β
  inCircle_of_centerCircle : ∀ {a α}, centerCircle a α → inCircle a α
goal: a ≠ b →
  IncidenceGeometry.onLine a L →
    IncidenceGeometry.onLine b L → IncidenceGeometry.onLine a M → IncidenceGeometry.onLine b M → L = M

full_name: IncidenceGeometry.circlesInter_of_onCircle_inCircle
code: class IncidenceGeometry where
  Point : Type u₁
  Line : Type u₂
  Circle : Type u₃

  between : Point → Point → Point → Prop -- implies colinearity
  onLine : Point → Line → Prop
  onCircle : Point → Circle → Prop
  inCircle : Point → Circle → Prop
  centerCircle : Point → Circle → Prop
  circlesInter : Circle → Circle → Prop

  ne_23_of_between : ∀ {a b c : Point}, between a b c → b ≠ c
  line_unique_of_pts : ∀ {a b : Point}, ∀ {L M : Line},
    a ≠ b → onLine a L → onLine b L → onLine a M → onLine b M → L = M
  onLine_2_of_between : ∀ {a b c : Point}, ∀ {L : Line},
    between a b c → onLine a L → onLine c L → onLine b L
  line_of_pts : ∀ a b, ∃ L, onLine a L ∧ onLine b L
  circle_of_ne : ∀ a b, a ≠ b → ∃ C, centerCircle a C ∧ onCircle b C
  circlesInter_of_onCircle_inCircle :
    ∀ {a b α β}, onCircle b α → onCircle a β → inCircle a α →
    inCircle b β → circlesInter α β
  pts_of_circlesInter : ∀ {α β}, circlesInter α β →
    ∃ a b, a ≠ b ∧ onCircle a α ∧ onCircle a β ∧ onCircle b α ∧ onCircle b β
  inCircle_of_centerCircle : ∀ {a α}, centerCircle a α → inCircle a α
goal: IncidenceGeometry.onCircle b α →
  IncidenceGeometry.onCircle a β →
    IncidenceGeometry.inCircle a α → IncidenceGeometry.inCircle b β → IncidenceGeometry.circlesInter α β

full_name: IncidenceGeometry.ne_23_of_between
code: class IncidenceGeometry where
  Point : Type u₁
  Line : Type u₂
  Circle : Type u₃

  between : Point → Point → Point → Prop -- implies colinearity
  onLine : Point → Line → Prop
  onCircle : Point → Circle → Prop
  inCircle : Point → Circle → Prop
  centerCircle : Point → Circle → Prop
  circlesInter : Circle → Circle → Prop

  ne_23_of_between : ∀ {a b c : Point}, between a b c → b ≠ c
  line_unique_of_pts : ∀ {a b : Point}, ∀ {L M : Line},
    a ≠ b → onLine a L → onLine b L → onLine a M → onLine b M → L = M
  onLine_2_of_between : ∀ {a b c : Point}, ∀ {L : Line},
    between a b c → onLine a L → onLine c L → onLine b L
  line_of_pts : ∀ a b, ∃ L, onLine a L ∧ onLine b L
  circle_of_ne : ∀ a b, a ≠ b → ∃ C, centerCircle a C ∧ onCircle b C
  circlesInter_of_onCircle_inCircle :
    ∀ {a b α β}, onCircle b α → onCircle a β → inCircle a α →
    inCircle b β → circlesInter α β
  pts_of_circlesInter : ∀ {α β}, circlesInter α β →
    ∃ a b, a ≠ b ∧ onCircle a α ∧ onCircle a β ∧ onCircle b α ∧ onCircle b β
  inCircle_of_centerCircle : ∀ {a α}, centerCircle a α → inCircle a α
goal: IncidenceGeometry.between a b c → b ≠ c

full_name: IncidenceGeometry.centerCircle
code: class IncidenceGeometry where
  Point : Type u₁
  Line : Type u₂
  Circle : Type u₃

  between : Point → Point → Point → Prop -- implies colinearity
  onLine : Point → Line → Prop
  onCircle : Point → Circle → Prop
  inCircle : Point → Circle → Prop
  centerCircle : Point → Circle → Prop
  circlesInter : Circle → Circle → Prop

  ne_23_of_between : ∀ {a b c : Point}, between a b c → b ≠ c
  line_unique_of_pts : ∀ {a b : Point}, ∀ {L M : Line},
    a ≠ b → onLine a L → onLine b L → onLine a M → onLine b M → L = M
  onLine_2_of_between : ∀ {a b c : Point}, ∀ {L : Line},
    between a b c → onLine a L → onLine c L → onLine b L
  line_of_pts : ∀ a b, ∃ L, onLine a L ∧ onLine b L
  circle_of_ne : ∀ a b, a ≠ b → ∃ C, centerCircle a C ∧ onCircle b C
  circlesInter_of_onCircle_inCircle :
    ∀ {a b α β}, onCircle b α → onCircle a β → inCircle a α →
    inCircle b β → circlesInter α β
  pts_of_circlesInter : ∀ {α β}, circlesInter α β →
    ∃ a b, a ≠ b ∧ onCircle a α ∧ onCircle a β ∧ onCircle b α ∧ onCircle b β
  inCircle_of_centerCircle : ∀ {a α}, centerCircle a α → inCircle a α
goal: IncidenceGeometry.Point → IncidenceGeometry.Circle → Prop

full_name: IncidenceGeometry.inCircle_of_centerCircle
code: class IncidenceGeometry where
  Point : Type u₁
  Line : Type u₂
  Circle : Type u₃

  between : Point → Point → Point → Prop -- implies colinearity
  onLine : Point → Line → Prop
  onCircle : Point → Circle → Prop
  inCircle : Point → Circle → Prop
  centerCircle : Point → Circle → Prop
  circlesInter : Circle → Circle → Prop

  ne_23_of_between : ∀ {a b c : Point}, between a b c → b ≠ c
  line_unique_of_pts : ∀ {a b : Point}, ∀ {L M : Line},
    a ≠ b → onLine a L → onLine b L → onLine a M → onLine b M → L = M
  onLine_2_of_between : ∀ {a b c : Point}, ∀ {L : Line},
    between a b c → onLine a L → onLine c L → onLine b L
  line_of_pts : ∀ a b, ∃ L, onLine a L ∧ onLine b L
  circle_of_ne : ∀ a b, a ≠ b → ∃ C, centerCircle a C ∧ onCircle b C
  circlesInter_of_onCircle_inCircle :
    ∀ {a b α β}, onCircle b α → onCircle a β → inCircle a α →
    inCircle b β → circlesInter α β
  pts_of_circlesInter : ∀ {α β}, circlesInter α β →
    ∃ a b, a ≠ b ∧ onCircle a α ∧ onCircle a β ∧ onCircle b α ∧ onCircle b β
  inCircle_of_centerCircle : ∀ {a α}, centerCircle a α → inCircle a α
goal: IncidenceGeometry.centerCircle a α → IncidenceGeometry.inCircle a α

full_name: IncidenceGeometry.line_of_pts
code: class IncidenceGeometry where
  Point : Type u₁
  Line : Type u₂
  Circle : Type u₃

  between : Point → Point → Point → Prop -- implies colinearity
  onLine : Point → Line → Prop
  onCircle : Point → Circle → Prop
  inCircle : Point → Circle → Prop
  centerCircle : Point → Circle → Prop
  circlesInter : Circle → Circle → Prop

  ne_23_of_between : ∀ {a b c : Point}, between a b c → b ≠ c
  line_unique_of_pts : ∀ {a b : Point}, ∀ {L M : Line},
    a ≠ b → onLine a L → onLine b L → onLine a M → onLine b M → L = M
  onLine_2_of_between : ∀ {a b c : Point}, ∀ {L : Line},
    between a b c → onLine a L → onLine c L → onLine b L
  line_of_pts : ∀ a b, ∃ L, onLine a L ∧ onLine b L
  circle_of_ne : ∀ a b, a ≠ b → ∃ C, centerCircle a C ∧ onCircle b C
  circlesInter_of_onCircle_inCircle :
    ∀ {a b α β}, onCircle b α → onCircle a β → inCircle a α →
    inCircle b β → circlesInter α β
  pts_of_circlesInter : ∀ {α β}, circlesInter α β →
    ∃ a b, a ≠ b ∧ onCircle a α ∧ onCircle a β ∧ onCircle b α ∧ onCircle b β
  inCircle_of_centerCircle : ∀ {a α}, centerCircle a α → inCircle a α
goal: ∃ L, IncidenceGeometry.onLine a L ∧ IncidenceGeometry.onLine b L

full_name: IncidenceGeometry.circle_of_ne
code: class IncidenceGeometry where
  Point : Type u₁
  Line : Type u₂
  Circle : Type u₃

  between : Point → Point → Point → Prop -- implies colinearity
  onLine : Point → Line → Prop
  onCircle : Point → Circle → Prop
  inCircle : Point → Circle → Prop
  centerCircle : Point → Circle → Prop
  circlesInter : Circle → Circle → Prop

  ne_23_of_between : ∀ {a b c : Point}, between a b c → b ≠ c
  line_unique_of_pts : ∀ {a b : Point}, ∀ {L M : Line},
    a ≠ b → onLine a L → onLine b L → onLine a M → onLine b M → L = M
  onLine_2_of_between : ∀ {a b c : Point}, ∀ {L : Line},
    between a b c → onLine a L → onLine c L → onLine b L
  line_of_pts : ∀ a b, ∃ L, onLine a L ∧ onLine b L
  circle_of_ne : ∀ a b, a ≠ b → ∃ C, centerCircle a C ∧ onCircle b C
  circlesInter_of_onCircle_inCircle :
    ∀ {a b α β}, onCircle b α → onCircle a β → inCircle a α →
    inCircle b β → circlesInter α β
  pts_of_circlesInter : ∀ {α β}, circlesInter α β →
    ∃ a b, a ≠ b ∧ onCircle a α ∧ onCircle a β ∧ onCircle b α ∧ onCircle b β
  inCircle_of_centerCircle : ∀ {a α}, centerCircle a α → inCircle a α
goal: a ≠ b → ∃ C, IncidenceGeometry.centerCircle a C ∧ IncidenceGeometry.onCircle b C

full_name: isBetweenPred?
code: /-- If `e == between a b c` return `some (a, b, c)`, otherwise `none`. -/
def isBetweenPred? (e : Expr) : Option (Expr × Expr × Expr) 
goal: Option (Lean.Expr × Lean.Expr × Lean.Expr)

full_name: isOnLinePred?
code: /-- If `e == onLine a L` return `some (a, L)`, otherwise `none`. -/
def isOnLinePred? (e : Expr) : Option (Expr × Expr) 
goal: Option (Lean.Expr × Lean.Expr)

full_name: isOnCirclePred?
code: /-- If `e == onCircle a C` return `some (a, C)`, otherwise `none`. -/
def isOnCirclePred? (e : Expr) : Option (Expr × Expr) 
goal: Option (Lean.Expr × Lean.Expr)

full_name: isInCirclePred?
code: /-- If `e == inCircle a C` return `some (a, C)`, otherwise `none`. -/
def isInCirclePred? (e : Expr) : Option (Expr × Expr) 
goal: Option (Lean.Expr × Lean.Expr)

full_name: isCenterCirclePred?
code: /-- If `e == centerCircle a C` return `some (a, C)`, otherwise `none`. -/
def isCenterCirclePred? (e : Expr) : Option (Expr × Expr) 
goal: Option (Lean.Expr × Lean.Expr)

full_name: isCirclesInterPred?
code: /-- If `e == circlesInter a C` return `some (a, C)`, otherwise `none`. -/
def isCirclesInterPred? (e : Expr) : Option (Expr × Expr) 
goal: Option (Lean.Expr × Lean.Expr)

full_name: isPoint?
code: def isPoint? (e : Expr) : Bool 
goal: Bool

full_name: isLine?
code: def isLine? (e : Expr) : Bool 
goal: Bool

full_name: addHypotheses
code: def addHypotheses (hyps : Array LocalDecl) : DiagramBuilderM Unit 
goal: ProofWidgets.Penrose.DiagramBuilderM Unit

full_name: EuclideanDisplay.dsl
code: def EuclideanDisplay.dsl 
goal: String

full_name: EuclideanDisplay.sty
code: def EuclideanDisplay.sty 
goal: String

full_name: EuclideanDisplay.rpc
code: @[server_rpc_method]
def EuclideanDisplay.rpc (props : PanelWidgetProps) : RequestM (RequestTask Html) 
goal: Lean.Server.RequestM (Lean.Server.RequestTask ProofWidgets.Html)

full_name: EuclideanDisplay
code: @[widget_module]
def EuclideanDisplay : Component PanelWidgetProps 
goal: ProofWidgets.Component ProofWidgets.PanelWidgetProps

full_name: constructLines
code: /-- Add every possible line between any two points in `hyps`
to the diagram.
Lines are labelled with links to insert them into the proof script. -/
def constructLines (hyps : Array LocalDecl) (meta : Server.DocumentMeta) (cursorPos : Lsp.Position)
    : DiagramBuilderM Unit 
goal: ProofWidgets.Penrose.DiagramBuilderM Unit

full_name: EuclideanConstructions.rpc
code: @[server_rpc_method]
def EuclideanConstructions.rpc (props : PanelWidgetProps) : RequestM (RequestTask Html) 
goal: Lean.Server.RequestM (Lean.Server.RequestTask ProofWidgets.Html)

full_name: EuclideanConstructions
code: @[widget_module]
def EuclideanConstructions : Component PanelWidgetProps 
goal: ProofWidgets.Component ProofWidgets.PanelWidgetProps

full_name: test_sorry
code: axiom test_sorry {α} : α
goal: α