234 lines
6.1 KiB
Plaintext
234 lines
6.1 KiB
Plaintext
(* Author: Tobias Nipkow
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Copyright 1998 TUM
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To generate a regular expression, the alphabet must be finite.
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regexp needs to be supplied with an 'a list for a unique order
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add_Atom d i j r a = (if d a i = j then Union r (Atom a) else r)
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atoms d i j as = foldl (add_Atom d i j) Empty as
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regexp as d i j 0 = (if i=j then Union (Star Empty) (atoms d i j as)
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else atoms d i j as
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*)
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section "From deterministic automata to regular sets"
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theory RegSet_of_nat_DA
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imports "Regular-Sets.Regular_Set" DA
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begin
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type_synonym 'a nat_next = "'a => nat => nat"
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abbreviation
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deltas :: "'a nat_next => 'a list => nat => nat" where
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"deltas == foldl2"
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primrec trace :: "'a nat_next => nat => 'a list => nat list" where
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"trace d i [] = []" |
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"trace d i (x#xs) = d x i # trace d (d x i) xs"
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(* conversion a la Warshall *)
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primrec regset :: "'a nat_next => nat => nat => nat => 'a list set" where
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"regset d i j 0 = (if i=j then insert [] {[a] | a. d a i = j}
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else {[a] | a. d a i = j})" |
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"regset d i j (Suc k) =
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regset d i j k Un
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(regset d i k k) @@ (star(regset d k k k)) @@ (regset d k j k)"
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definition
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regset_of_DA :: "('a,nat)da => nat => 'a list set" where
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"regset_of_DA A k = (UN j:{j. j<k & fin A j}. regset (next A) (start A) j k)"
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definition
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bounded :: "'a nat_next => nat => bool" where
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"bounded d k = (!n. n < k --> (!x. d x n < k))"
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declare
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in_set_butlast_appendI[simp,intro] less_SucI[simp] image_eqI[simp]
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(* Lists *)
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lemma butlast_empty[iff]:
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"(butlast xs = []) = (case xs of [] => True | y#ys => ys=[])"
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by (cases xs) simp_all
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lemma in_set_butlast_concatI:
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"x:set(butlast xs) ==> xs:set xss ==> x:set(butlast(concat xss))"
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apply (induct "xss")
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apply simp
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apply (simp add: butlast_append del: ball_simps)
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apply (rule conjI)
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apply (clarify)
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apply (erule disjE)
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apply (blast)
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apply (subgoal_tac "xs=[]")
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apply simp
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apply (blast)
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apply (blast dest: in_set_butlastD)
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done
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(* Regular sets *)
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(* The main lemma:
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how to decompose a trace into a prefix, a list of loops and a suffix.
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*)
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lemma decompose[rule_format]:
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"!i. k : set(trace d i xs) --> (EX pref mids suf.
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xs = pref @ concat mids @ suf &
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deltas d pref i = k & (!n:set(butlast(trace d i pref)). n ~= k) &
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(!mid:set mids. (deltas d mid k = k) &
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(!n:set(butlast(trace d k mid)). n ~= k)) &
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(!n:set(butlast(trace d k suf)). n ~= k))"
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apply (induct "xs")
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apply (simp)
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apply (rename_tac a as)
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apply (intro strip)
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apply (case_tac "d a i = k")
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apply (rule_tac x = "[a]" in exI)
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apply simp
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apply (case_tac "k : set(trace d (d a i) as)")
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apply (erule allE)
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apply (erule impE)
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apply (assumption)
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apply (erule exE)+
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apply (rule_tac x = "pref#mids" in exI)
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apply (rule_tac x = "suf" in exI)
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apply simp
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apply (rule_tac x = "[]" in exI)
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apply (rule_tac x = "as" in exI)
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apply simp
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apply (blast dest: in_set_butlastD)
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apply simp
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apply (erule allE)
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apply (erule impE)
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apply (assumption)
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apply (erule exE)+
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apply (rule_tac x = "a#pref" in exI)
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apply (rule_tac x = "mids" in exI)
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apply (rule_tac x = "suf" in exI)
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apply simp
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done
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lemma length_trace[simp]: "!!i. length(trace d i xs) = length xs"
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by (induct "xs") simp_all
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lemma deltas_append[simp]:
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"!!i. deltas d (xs@ys) i = deltas d ys (deltas d xs i)"
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by (induct "xs") simp_all
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lemma trace_append[simp]:
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"!!i. trace d i (xs@ys) = trace d i xs @ trace d (deltas d xs i) ys"
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by (induct "xs") simp_all
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lemma trace_concat[simp]:
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"(!xs: set xss. deltas d xs i = i) ==>
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trace d i (concat xss) = concat (map (trace d i) xss)"
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by (induct "xss") simp_all
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lemma trace_is_Nil[simp]: "!!i. (trace d i xs = []) = (xs = [])"
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by (case_tac "xs") simp_all
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lemma trace_is_Cons_conv[simp]:
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"(trace d i xs = n#ns) =
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(case xs of [] => False | y#ys => n = d y i & ns = trace d n ys)"
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apply (case_tac "xs")
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apply simp_all
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apply (blast)
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done
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lemma set_trace_conv:
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"!!i. set(trace d i xs) =
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(if xs=[] then {} else insert(deltas d xs i)(set(butlast(trace d i xs))))"
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apply (induct "xs")
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apply (simp)
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apply (simp add: insert_commute)
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done
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lemma deltas_concat[simp]:
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"(!mid:set mids. deltas d mid k = k) ==> deltas d (concat mids) k = k"
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by (induct mids) simp_all
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lemma lem: "[| n < Suc k; n ~= k |] ==> n < k"
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by arith
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lemma regset_spec:
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"!!i j xs. xs : regset d i j k =
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((!n:set(butlast(trace d i xs)). n < k) & deltas d xs i = j)"
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apply (induct k)
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apply(simp split: list.split)
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apply(fastforce)
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apply (simp add: conc_def)
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apply (rule iffI)
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apply (erule disjE)
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apply simp
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apply (erule exE conjE)+
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apply simp
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apply (subgoal_tac
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"(!m:set(butlast(trace d k xsb)). m < Suc k) & deltas d xsb k = k")
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apply (simp add: set_trace_conv butlast_append ball_Un)
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apply (erule star_induct)
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apply (simp)
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apply (simp add: set_trace_conv butlast_append ball_Un)
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apply (case_tac "k : set(butlast(trace d i xs))")
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prefer 2 apply (rule disjI1)
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apply (blast intro:lem)
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apply (rule disjI2)
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apply (drule in_set_butlastD[THEN decompose])
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apply (clarify)
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apply (rule_tac x = "pref" in exI)
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apply simp
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apply (rule conjI)
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apply (rule ballI)
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apply (rule lem)
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prefer 2 apply simp
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apply (drule bspec) prefer 2 apply assumption
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apply simp
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apply (rule_tac x = "concat mids" in exI)
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apply (simp)
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apply (rule conjI)
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apply (rule concat_in_star)
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apply (clarsimp simp: subset_iff)
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apply (rule lem)
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prefer 2 apply simp
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apply (drule bspec) prefer 2 apply assumption
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apply (simp add: image_eqI in_set_butlast_concatI)
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apply (rule ballI)
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apply (rule lem)
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apply auto
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done
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lemma trace_below:
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"bounded d k ==> !i. i < k --> (!n:set(trace d i xs). n < k)"
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apply (unfold bounded_def)
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apply (induct "xs")
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apply simp
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apply (simp (no_asm))
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apply (blast)
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done
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lemma regset_below:
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"[| bounded d k; i < k; j < k |] ==>
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regset d i j k = {xs. deltas d xs i = j}"
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apply (rule set_eqI)
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apply (simp add: regset_spec)
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apply (blast dest: trace_below in_set_butlastD)
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done
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lemma deltas_below:
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"!!i. bounded d k ==> i < k ==> deltas d w i < k"
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apply (unfold bounded_def)
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apply (induct "w")
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apply simp_all
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done
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lemma regset_DA_equiv:
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"[| bounded (next A) k; start A < k; j < k |] ==>
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w : regset_of_DA A k = accepts A w"
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apply(unfold regset_of_DA_def)
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apply (simp cong: conj_cong
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add: regset_below deltas_below accepts_def delta_def)
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done
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end
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