--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/thys/Turing2.thy	Tue May 21 13:49:31 2013 +0100
@@ -0,0 +1,69 @@
+(* Title: thys/Turing.thy
+   Author: Jian Xu, Xingyuan Zhang, and Christian Urban
+*)
+
+header {* Turing Machines *}
+
+theory Turing2
+imports Main
+begin
+
+section {* Basic definitions of Turing machine *}
+
+datatype action = W0 | W1 | L | R | Nop
+
+datatype cell = Bk | Oc
+
+type_synonym tape = "cell list \<times> cell list"
+
+type_synonym state = nat
+
+type_synonym instr = "action \<times> state"
+
+type_synonym tprog = "(instr \<times> instr) list"
+
+type_synonym config = "state \<times> tape"
+
+fun nth_of where
+  "nth_of xs i = (if i \<ge> length xs then None else Some (xs ! i))"
+
+fun 
+  fetch :: "tprog \<Rightarrow> state \<Rightarrow> cell \<Rightarrow> instr"
+where
+  "fetch p 0 b = (Nop, 0)"
+| "fetch p (Suc s) b = 
+     (case nth_of p s of
+        Some i \<Rightarrow> (case b of Bk \<Rightarrow> fst i | Oc \<Rightarrow> snd i)
+      | None \<Rightarrow> (Nop, 0))"
+
+fun 
+  update :: "action \<Rightarrow> tape \<Rightarrow> tape"
+where 
+  "update W0 (l, r) = (l, Bk # (tl r))" 
+| "update W1 (l, r) = (l, Oc # (tl r))"
+| "update L  (l, r) = (if l = [] then ([], Bk # r) else (tl l, (hd l) # r))" 
+| "update R  (l, r) = (if r = [] then (Bk # l, []) else ((hd r) # l, tl r))" 
+| "update Nop (l, r) = (l, r)"
+
+abbreviation 
+  "read r == if (r = []) then Bk else hd r"
+
+fun step :: "config \<Rightarrow> tprog \<Rightarrow> config"
+  where 
+  "step (s, l, r) p = 
+     (let (a, s') = fetch p s (read r) in (s', update a (l, r)))"
+
+fun steps :: "config \<Rightarrow> tprog \<Rightarrow> nat \<Rightarrow> config"
+  where
+  "steps c p 0 = c" |
+  "steps c p (Suc n) = steps (step c p) p n"
+
+(* well-formedness of Turing machine programs *)
+
+fun 
+  tm_wf :: "tprog \<Rightarrow> bool"
+where
+  "tm_wf p = (length p \<ge> 1 \<and> (\<forall>((_, s1), (_, s2)) \<in> set p. s1 \<le> length p \<and> s2 \<le> length p))"
+
+end
+