src/HOL/Lex/RegSet_of_nat_DA.thy

 
 regexp as d i j 0 = (if i=j then Union (Star Empty) (atoms d i j as)
                         else atoms d i j as
 *)
 
-RegSet_of_nat_DA = RegSet + DA +
+theory RegSet_of_nat_DA = RegSet + DA:
 
 types 'a nat_next = "'a => nat => nat"
 
-syntax deltas :: 'a nat_next => 'a list => nat => nat
+syntax deltas :: "'a nat_next => 'a list => nat => nat"
 translations "deltas" == "foldl2"
 
-consts trace :: 'a nat_next => nat => 'a list => nat list
+consts trace :: "'a nat_next => nat => 'a list => nat list"
 primrec
 "trace d i [] = []"
 "trace d i (x#xs) = d x i # trace d (d x i) xs"
 
 (* conversion a la Warshall *)
 
-consts regset :: 'a nat_next => nat => nat => nat => 'a list set
+consts regset :: "'a nat_next => nat => nat => nat => 'a list set"
 primrec
 "regset d i j 0 = (if i=j then insert [] {[a] | a. d a i = j}
                           else {[a] | a. d a i = j})"
 "regset d i j (Suc k) = regset d i j k Un
                         conc (regset d i k k)
                              (conc (star(regset d k k k))
                                    (regset d k j k))"
 
 constdefs
- regset_of_DA :: ('a,nat)da => nat => 'a list set
+ regset_of_DA :: "('a,nat)da => nat => 'a list set"
 "regset_of_DA A k == UN j:{j. j<k & fin A j}. regset (next A) (start A) j k"
 
  bounded :: "'a => nat => bool"
 "bounded d k == !n. n < k --> (!x. d x n < k)"
 
+declare
+  in_set_butlast_appendI[simp,intro] less_SucI[simp] image_eqI[simp]
+
+(* Lists *)
+
+lemma butlast_empty[iff]:
+  "(butlast xs = []) = (case xs of [] => True | y#ys => ys=[])"
+by (case_tac "xs") simp_all
+
+lemma in_set_butlast_concatI:
+ "x:set(butlast xs) ==> xs:set xss ==> x:set(butlast(concat xss))"
+apply (induct "xss")
+ apply simp
+apply (simp add: butlast_append del: ball_simps)
+apply (rule conjI)
+ apply (clarify)
+ apply (erule disjE)
+  apply (blast)
+ apply (subgoal_tac "xs=[]")
+  apply simp
+ apply (blast)
+apply (blast dest: in_set_butlastD)
+done
+
+(* Regular sets *)
+
+(* The main lemma:
+   how to decompose a trace into a prefix, a list of loops and a suffix.
+*)
+lemma decompose[rule_format]:
+ "!i. k : set(trace d i xs) --> (EX pref mids suf.
+  xs = pref @ concat mids @ suf &
+  deltas d pref i = k & (!n:set(butlast(trace d i pref)). n ~= k) &
+  (!mid:set mids. (deltas d mid k = k) &
+                  (!n:set(butlast(trace d k mid)). n ~= k)) &
+  (!n:set(butlast(trace d k suf)). n ~= k))"
+apply (induct "xs")
+ apply (simp)
+apply (rename_tac a as)
+apply (intro strip)
+apply (case_tac "d a i = k")
+ apply (rule_tac x = "[a]" in exI)
+ apply simp
+ apply (case_tac "k : set(trace d (d a i) as)")
+  apply (erule allE)
+  apply (erule impE)
+   apply (assumption)
+  apply (erule exE)+
+  apply (rule_tac x = "pref#mids" in exI)
+  apply (rule_tac x = "suf" in exI)
+  apply simp
+ apply (rule_tac x = "[]" in exI)
+ apply (rule_tac x = "as" in exI)
+ apply simp
+ apply (blast dest: in_set_butlastD)
+apply simp
+apply (erule allE)
+apply (erule impE)
+ apply (assumption)
+apply (erule exE)+
+apply (rule_tac x = "a#pref" in exI)
+apply (rule_tac x = "mids" in exI)
+apply (rule_tac x = "suf" in exI)
+apply simp
+done
+
+lemma length_trace[simp]: "!!i. length(trace d i xs) = length xs"
+by (induct "xs") simp_all
+
+lemma deltas_append[simp]:
+  "!!i. deltas d (xs@ys) i = deltas d ys (deltas d xs i)"
+by (induct "xs") simp_all
+
+lemma trace_append[simp]:
+  "!!i. trace d i (xs@ys) = trace d i xs @ trace d (deltas d xs i) ys"
+by (induct "xs") simp_all
+
+lemma trace_concat[simp]:
+ "(!xs: set xss. deltas d xs i = i) ==>
+  trace d i (concat xss) = concat (map (trace d i) xss)"
+by (induct "xss") simp_all
+
+lemma trace_is_Nil[simp]: "!!i. (trace d i xs = []) = (xs = [])"
+by (case_tac "xs") simp_all
+
+lemma trace_is_Cons_conv[simp]:
+ "(trace d i xs = n#ns) =
+  (case xs of [] => False | y#ys => n = d y i & ns = trace d n ys)"
+apply (case_tac "xs")
+apply simp_all
+apply (blast)
+done
+
+lemma set_trace_conv:
+ "!!i. set(trace d i xs) =
+  (if xs=[] then {} else insert(deltas d xs i)(set(butlast(trace d i xs))))"
+apply (induct "xs")
+ apply (simp)
+apply (simp add: insert_commute)
+done
+
+lemma deltas_concat[simp]:
+ "(!mid:set mids. deltas d mid k = k) ==> deltas d (concat mids) k = k"
+by (induct mids) simp_all
+
+lemma lem: "[| n < Suc k; n ~= k |] ==> n < k"
+by arith
+
+lemma regset_spec:
+ "!!i j xs. xs : regset d i j k =
+        ((!n:set(butlast(trace d i xs)). n < k) & deltas d xs i = j)"
+apply (induct k)
+ apply(simp split: list.split)
+ apply(fastsimp)
+apply (simp add: conc_def)
+apply (rule iffI)
+ apply (erule disjE)
+  apply simp
+ apply (erule exE conjE)+
+ apply simp
+ apply (subgoal_tac
+      "(!m:set(butlast(trace d n xsb)). m < Suc n) & deltas d xsb n = n")
+  apply (simp add: set_trace_conv butlast_append ball_Un)
+ apply (erule star.induct)
+  apply (simp)
+ apply (simp add: set_trace_conv butlast_append ball_Un)
+apply (case_tac "n : set(butlast(trace d i xs))")
+ prefer 2 apply (rule disjI1)
+ apply (blast intro:lem)
+apply (rule disjI2)
+apply (drule in_set_butlastD[THEN decompose])
+apply (clarify)
+apply (rule_tac x = "pref" in exI)
+apply simp
+apply (rule conjI)
+ apply (rule ballI)
+ apply (rule lem)
+  prefer 2 apply simp
+ apply (drule bspec) prefer 2 apply assumption
+ apply simp
+apply (rule_tac x = "concat mids" in exI)
+apply (simp)
+apply (rule conjI)
+ apply (rule concat_in_star)
+ apply simp
+ apply (rule ballI)
+ apply (rule ballI)
+ apply (rule lem)
+  prefer 2 apply simp
+ apply (drule bspec) prefer 2 apply assumption
+ apply (simp add: image_eqI in_set_butlast_concatI)
+apply (rule ballI)
+apply (rule lem)
+ apply auto
+done
+
+lemma trace_below:
+ "bounded d k ==> !i. i < k --> (!n:set(trace d i xs). n < k)"
+apply (unfold bounded_def)
+apply (induct "xs")
+ apply simp
+apply (simp (no_asm))
+apply (blast)
+done
+
+lemma regset_below:
+ "[| bounded d k; i < k; j < k |] ==>
+  regset d i j k = {xs. deltas d xs i = j}"
+apply (rule set_ext)
+apply (simp add: regset_spec)
+apply (blast dest: trace_below in_set_butlastD)
+done
+
+lemma deltas_below:
+ "!!i. bounded d k ==> i < k ==> deltas d w i < k"
+apply (unfold bounded_def)
+apply (induct "w")
+ apply simp_all
+done
+
+lemma regset_DA_equiv:
+ "[| bounded (next A) k; start A < k; j < k |] ==> \
+\ w : regset_of_DA A k = accepts A w"
+apply(unfold regset_of_DA_def)
+apply (simp cong: conj_cong
+            add: regset_below deltas_below accepts_def delta_def)
+done
+
 end