--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal/activities/cas09/Example.thy	Wed Mar 30 17:27:34 2016 +0100
@@ -0,0 +1,47 @@
+theory Example
+imports Main
+begin
+
+text {* 
+  A small example theory proving a few facts
+  about list append and list reverse.
+*}
+
+datatype 'a list2 =
+    nil2                   ("[]")
+  | cons2 "'a"  "'a list2" ("_ ## _")
+
+abbreviation
+  singleton :: "'a \<Rightarrow> 'a list2" ("[_]")
+where
+  "[x] \<equiv> x##[]"
+
+fun
+  append2 :: "'a list2 \<Rightarrow> 'a list2 \<Rightarrow> 'a list2" ("_ @@ _")
+where
+  append2_nil2:  "[] @@ ys = ys"
+| append2_cons2: "(x##xs) @@ ys = x##(xs @@ ys)"
+
+fun
+  rev2 :: "'a list2 \<Rightarrow> 'a list2"
+where
+  "rev2 [] = []"
+| "rev2 (x##xs) = (rev2 xs) @@ (x##[])"
+
+lemma append2_nil2R[simp]:
+  shows "xs @@ [] = xs"
+by (induct xs) (auto)
+
+lemma append2_assoc[simp]:
+  shows "(xs @@ ys) @@ zs = xs @@ (ys @@ zs)"
+by (induct xs) (auto)
+
+lemma rev2_append2[simp]: 
+  shows "rev2 (xs @@ ys) = (rev2 ys) @@ (rev2 xs)"
+by (induct xs) (auto)
+
+lemma rev2_rev2:
+  shows "rev2 (rev2 xs) = xs"
+by (induct xs) (auto)
+
+end
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