doc(examples/lean): new example

Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>

examples/lean/set.lean

@@ -0,0 +1,30 @@
+Definition set (A : Type) : Type := A -> Bool
+
+Definition element {A : Type} (x : A) (s : set A) := s x
+Infix 60 ∈ : element
+
+Definition subset {A : Type} (s1 : set A) (s2 : set A) := ∀ x, x ∈ s1 ⇒ x ∈ s2
+Infix 50 ⊆ : subset
+
+Theorem SubsetProp {A : Type} {s1 s2 : set A} {x : A} (H1 : s1 ⊆ s2) (H2 : x ∈ s1) : x ∈ s2 :=
+   MP (ForallElim H1 x) H2
+
+Theorem SubsetTrans {A : Type} {s1 s2 s3 : set A} (H1 : s1 ⊆ s2) (H2 : s2 ⊆ s3) : s1 ⊆ s3 :=
+   ForallIntro (λ x,
+       Discharge (λ Hin : x ∈ s1,
+          let L1 : x ∈ s2 := SubsetProp H1 Hin,
+              L2 : x ∈ s3 := SubsetProp H2 L1
+          in  L2)).
+
+Definition transitive {A : Type} (R : A → A → Bool) := ∀ x y z, R x y ⇒ R y z ⇒ R x z
+
+Theorem SubsetTrans2 {A : Type} : transitive (subset::explicit A) :=
+   ForallIntro (λ s1, ForallIntro (λ s2, ForallIntro (λ s3,
+      Discharge (λ H1, (Discharge (λ H2,
+         SubsetTrans H1 H2)))))).
+
+Theorem SubsetRefl {A : Type} (s : set A) : s ⊆ s :=
+   ForallIntro (λ x, Discharge (λ H : x ∈ s, H))
+
+Definition union {A : Type} (s1 : set A) (s2 : set A) := λ x, x ∈ s1 ∨ x ∈ s2
+Infix 55 ∪ : union

src/frontends/lean/parser.cpp

@@ -1654,8 +1654,14 @@ class parser::imp {
         } else {
             mk_scope scope(*this);
             parse_object_bindings(bindings);
-            check_colon_next("invalid definition, ':' expected");
-            expr type_body = parse_expr();
+            expr type_body;
+            if (curr_is_colon()) {
+                next();
+                type_body = parse_expr();
+            } else {
+                auto p = pos();
+                type_body = save(mk_placeholder(), p);
+            }
             pre_type  = mk_abstraction(false, bindings, type_body);
             if (!is_definition && curr_is_period()) {
                 pre_val = mk_abstraction(true, bindings, mk_placeholder());