src/HOL/Induct/Tree.thy
 changeset 11046 b5f5942781a0 parent 7018 ae18bb3075c3 child 11649 dfb59b9954a6
```--- a/src/HOL/Induct/Tree.thy	Sat Feb 03 15:22:57 2001 +0100
+++ b/src/HOL/Induct/Tree.thy	Sat Feb 03 17:40:16 2001 +0100
@@ -2,26 +2,39 @@
ID:         \$Id\$
Author:     Stefan Berghofer,  TU Muenchen
-
-Infinitely branching trees
*)

-Tree = Main +
+header {* Infinitely branching trees *}
+
+theory Tree = Main:

-datatype 'a tree = Atom 'a | Branch "nat => 'a tree"
+datatype 'a tree =
+    Atom 'a
+  | Branch "nat => 'a tree"

consts
map_tree :: "('a => 'b) => 'a tree => 'b tree"
-
primrec
"map_tree f (Atom a) = Atom (f a)"
-  "map_tree f (Branch ts) = Branch (%x. map_tree f (ts x))"
+  "map_tree f (Branch ts) = Branch (\<lambda>x. map_tree f (ts x))"
+
+lemma tree_map_compose: "map_tree g (map_tree f t) = map_tree (g \<circ> f) t"
+  apply (induct t)
+   apply simp_all
+  done

consts
exists_tree :: "('a => bool) => 'a tree => bool"
-
primrec
"exists_tree P (Atom a) = P a"
-  "exists_tree P (Branch ts) = (? x. exists_tree P (ts x))"
+  "exists_tree P (Branch ts) = (\<exists>x. exists_tree P (ts x))"
+
+lemma exists_map:
+  "(!!x. P x ==> Q (f x)) ==>
+    exists_tree P ts ==> exists_tree Q (map_tree f ts)"
+  apply (induct ts)
+   apply simp_all
+  apply blast
+  done

end```