lean2/examples/lean/ex5.lean

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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)
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(*
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))
Show Environment 1
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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 *)
Show Environment 2