lean2/examples/lean/ex5.lean

Push
   Theorem ReflIf (A : Type)
                  (R : A → A → Bool)
                  (Symmetry : Π x y, R x y → R y x)
                  (Transitivity : Π x y z, R x y → R y z → R x z)
                  (Linked : Π x, ∃ y, R x y)
                  :
                  Π x, R x x :=
       fun x, ExistsElim (Linked x)
             (fun (w : A) (H : R x w),
                 let L1 : R w x := Symmetry x w H
                 in Transitivity x w x H L1)
Pop

Push
  -- Same example but using ∀ instead of Π and ⇒ instead of →
  Theorem ReflIf (A : Type)
                 (R : A → A → Bool)
                 (Symmetry : ∀ x y, R x y ⇒ R y x)
                 (Transitivity : ∀ x y z, R x y ⇒ R y z ⇒ R x z)
                 (Linked : ∀ x, ∃ y, R x y)
                 :
                 ∀ x, R x x :=
    ForallIntro (fun x,
         ExistsElim (ForallElim Linked x)
            (fun (w : A) (H : R x w),
                let L1 : R w x := (MP (ForallElim (ForallElim Symmetry x) w) H)
                in (MP (MP (ForallElim (ForallElim (ForallElim Transitivity x) w) x) H) L1)))

    -- We can make the previous example less verbose by using custom notation

    Infixl 50 !  : ForallElim
    Infixl 30 << : MP

    Theorem ReflIf2 (A : Type)
                    (R : A → A → Bool)
                    (Symmetry : ∀ x y, R x y ⇒ R y x)
                    (Transitivity : ∀ x y z, R x y ⇒ R y z ⇒ R x z)
                    (Linked : ∀ x, ∃ y, R x y)
                    :
                    ∀ x, R x x :=
       ForallIntro (fun x,
          ExistsElim (Linked ! x)
             (fun (w : A) (H : R x w),
                  let L1 : R w x := Symmetry ! x ! w << H
                  in Transitivity ! x ! w ! x << H << L1))

    print Environment 1
Pop

Scope
  -- Same example again.
  Variable A : Type
  Variable R : A → A → Bool
  Axiom Symmetry  {x y : A} : R x y → R y x
  Axiom Transitivity {x y z : A} : R x y → R y z → R x z
  Axiom Linked (x : A) : ∃ y, R x y

  Theorem ReflIf (x : A) : R x x :=
      ExistsElim (Linked x) (fun (w : A) (H : R x w),
            let L1 : R w x := Symmetry H
            in Transitivity H L1)

  -- Even more compact proof of the same theorem
  Theorem ReflIf2 (x : A) : R x x :=
      ExistsElim (Linked x) (fun w H, Transitivity H (Symmetry H))

  -- The command EndScope exports both theorem to the main scope
  -- The variables and axioms in the scope become parameters to both theorems.
EndScope

-- Display the last two theorems
print Environment 2