642 lines
17 KiB
Plaintext
642 lines
17 KiB
Plaintext
(* Author: Tobias Nipkow
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Copyright 1998 TUM
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*)
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section "From regular expressions to nondeterministic automata with epsilon"
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theory RegExp2NAe
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imports "Regular-Sets.Regular_Exp" NAe
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begin
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type_synonym 'a bitsNAe = "('a,bool list)nae"
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definition
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epsilon :: "'a bitsNAe" where
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"epsilon = ([],%a s. {}, %s. s=[])"
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definition
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"atom" :: "'a => 'a bitsNAe" where
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"atom a = ([True],
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%b s. if s=[True] & b=Some a then {[False]} else {},
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%s. s=[False])"
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definition
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or :: "'a bitsNAe => 'a bitsNAe => 'a bitsNAe" where
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"or = (%(ql,dl,fl)(qr,dr,fr).
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([],
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%a s. case s of
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[] => if a=None then {True#ql,False#qr} else {}
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| left#s => if left then True ## dl a s
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else False ## dr a s,
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%s. case s of [] => False | left#s => if left then fl s else fr s))"
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definition
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conc :: "'a bitsNAe => 'a bitsNAe => 'a bitsNAe" where
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"conc = (%(ql,dl,fl)(qr,dr,fr).
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(True#ql,
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%a s. case s of
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[] => {}
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| left#s => if left then (True ## dl a s) Un
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(if fl s & a=None then {False#qr} else {})
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else False ## dr a s,
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%s. case s of [] => False | left#s => ~left & fr s))"
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definition
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star :: "'a bitsNAe => 'a bitsNAe" where
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"star = (%(q,d,f).
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([],
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%a s. case s of
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[] => if a=None then {True#q} else {}
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| left#s => if left then (True ## d a s) Un
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(if f s & a=None then {True#q} else {})
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else {},
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%s. case s of [] => True | left#s => left & f s))"
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primrec rexp2nae :: "'a rexp => 'a bitsNAe" where
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"rexp2nae Zero = ([], %a s. {}, %s. False)" |
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"rexp2nae One = epsilon" |
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"rexp2nae(Atom a) = atom a" |
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"rexp2nae(Plus r s) = or (rexp2nae r) (rexp2nae s)" |
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"rexp2nae(Times r s) = conc (rexp2nae r) (rexp2nae s)" |
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"rexp2nae(Star r) = star (rexp2nae r)"
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declare split_paired_all[simp]
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(******************************************************)
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(* epsilon *)
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(******************************************************)
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lemma step_epsilon[simp]: "step epsilon a = {}"
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by(simp add:epsilon_def step_def)
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lemma steps_epsilon: "((p,q) : steps epsilon w) = (w=[] & p=q)"
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by (induct "w") auto
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lemma accepts_epsilon[simp]: "accepts epsilon w = (w = [])"
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apply (simp add: steps_epsilon accepts_def)
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apply (simp add: epsilon_def)
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done
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(******************************************************)
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(* atom *)
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(******************************************************)
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lemma fin_atom: "(fin (atom a) q) = (q = [False])"
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by(simp add:atom_def)
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lemma start_atom: "start (atom a) = [True]"
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by(simp add:atom_def)
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(* Use {x. False} = {}? *)
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lemma eps_atom[simp]:
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"eps(atom a) = {}"
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by (simp add:atom_def step_def)
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lemma in_step_atom_Some[simp]:
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"(p,q) : step (atom a) (Some b) = (p=[True] & q=[False] & b=a)"
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by (simp add:atom_def step_def)
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lemma False_False_in_steps_atom:
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"([False],[False]) : steps (atom a) w = (w = [])"
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apply (induct "w")
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apply (simp)
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apply (simp add: relcomp_unfold)
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done
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lemma start_fin_in_steps_atom:
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"(start (atom a), [False]) : steps (atom a) w = (w = [a])"
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apply (induct "w")
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apply (simp add: start_atom rtrancl_empty)
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apply (simp add: False_False_in_steps_atom relcomp_unfold start_atom)
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done
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lemma accepts_atom: "accepts (atom a) w = (w = [a])"
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by (simp add: accepts_def start_fin_in_steps_atom fin_atom)
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(******************************************************)
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(* or *)
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(******************************************************)
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(***** lift True/False over fin *****)
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lemma fin_or_True[iff]:
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"!!L R. fin (or L R) (True#p) = fin L p"
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by(simp add:or_def)
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lemma fin_or_False[iff]:
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"!!L R. fin (or L R) (False#p) = fin R p"
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by(simp add:or_def)
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(***** lift True/False over step *****)
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lemma True_in_step_or[iff]:
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"!!L R. (True#p,q) : step (or L R) a = (? r. q = True#r & (p,r) : step L a)"
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apply (simp add:or_def step_def)
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apply blast
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done
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lemma False_in_step_or[iff]:
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"!!L R. (False#p,q) : step (or L R) a = (? r. q = False#r & (p,r) : step R a)"
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apply (simp add:or_def step_def)
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apply blast
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done
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(***** lift True/False over epsclosure *****)
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lemma lemma1a:
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"(tp,tq) : (eps(or L R))^* ==>
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(!!p. tp = True#p ==> ? q. (p,q) : (eps L)^* & tq = True#q)"
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apply (induct rule:rtrancl_induct)
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apply (blast)
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apply (clarify)
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apply (simp)
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apply (blast intro: rtrancl_into_rtrancl)
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done
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lemma lemma1b:
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"(tp,tq) : (eps(or L R))^* ==>
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(!!p. tp = False#p ==> ? q. (p,q) : (eps R)^* & tq = False#q)"
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apply (induct rule:rtrancl_induct)
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apply (blast)
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apply (clarify)
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apply (simp)
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apply (blast intro: rtrancl_into_rtrancl)
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done
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lemma lemma2a:
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"(p,q) : (eps L)^* ==> (True#p, True#q) : (eps(or L R))^*"
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apply (induct rule: rtrancl_induct)
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apply (blast)
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apply (blast intro: rtrancl_into_rtrancl)
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done
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lemma lemma2b:
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"(p,q) : (eps R)^* ==> (False#p, False#q) : (eps(or L R))^*"
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apply (induct rule: rtrancl_induct)
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apply (blast)
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apply (blast intro: rtrancl_into_rtrancl)
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done
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lemma True_epsclosure_or[iff]:
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"(True#p,q) : (eps(or L R))^* = (? r. q = True#r & (p,r) : (eps L)^*)"
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by (blast dest: lemma1a lemma2a)
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lemma False_epsclosure_or[iff]:
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"(False#p,q) : (eps(or L R))^* = (? r. q = False#r & (p,r) : (eps R)^*)"
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by (blast dest: lemma1b lemma2b)
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(***** lift True/False over steps *****)
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lemma lift_True_over_steps_or[iff]:
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"!!p. (True#p,q):steps (or L R) w = (? r. q = True # r & (p,r):steps L w)"
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apply (induct "w")
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apply auto
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apply force
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done
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lemma lift_False_over_steps_or[iff]:
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"!!p. (False#p,q):steps (or L R) w = (? r. q = False#r & (p,r):steps R w)"
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apply (induct "w")
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apply auto
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apply (force)
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done
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(***** Epsilon closure of start state *****)
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lemma unfold_rtrancl2:
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"R^* = Id Un (R O R^*)"
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apply (rule set_eqI)
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apply (simp)
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apply (rule iffI)
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apply (erule rtrancl_induct)
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apply (blast)
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apply (blast intro: rtrancl_into_rtrancl)
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apply (blast intro: converse_rtrancl_into_rtrancl)
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done
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lemma in_unfold_rtrancl2:
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"(p,q) : R^* = (q = p | (? r. (p,r) : R & (r,q) : R^*))"
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apply (rule unfold_rtrancl2[THEN equalityE])
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apply (blast)
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done
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lemmas [iff] = in_unfold_rtrancl2[where ?p = "start(or L R)"] for L R
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lemma start_eps_or[iff]:
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"!!L R. (start(or L R),q) : eps(or L R) =
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(q = True#start L | q = False#start R)"
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by (simp add:or_def step_def)
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lemma not_start_step_or_Some[iff]:
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"!!L R. (start(or L R),q) ~: step (or L R) (Some a)"
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by (simp add:or_def step_def)
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lemma steps_or:
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"(start(or L R), q) : steps (or L R) w =
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( (w = [] & q = start(or L R)) |
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(? p. q = True # p & (start L,p) : steps L w |
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q = False # p & (start R,p) : steps R w) )"
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apply (case_tac "w")
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apply (simp)
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apply (blast)
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apply (simp)
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apply (blast)
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done
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lemma start_or_not_final[iff]:
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"!!L R. ~ fin (or L R) (start(or L R))"
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by (simp add:or_def)
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lemma accepts_or:
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"accepts (or L R) w = (accepts L w | accepts R w)"
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apply (simp add:accepts_def steps_or)
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apply auto
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done
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(******************************************************)
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(* conc *)
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(******************************************************)
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(** True/False in fin **)
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lemma in_conc_True[iff]:
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"!!L R. fin (conc L R) (True#p) = False"
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by (simp add:conc_def)
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lemma fin_conc_False[iff]:
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"!!L R. fin (conc L R) (False#p) = fin R p"
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by (simp add:conc_def)
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(** True/False in step **)
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lemma True_step_conc[iff]:
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"!!L R. (True#p,q) : step (conc L R) a =
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((? r. q=True#r & (p,r): step L a) |
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(fin L p & a=None & q=False#start R))"
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by (simp add:conc_def step_def) (blast)
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lemma False_step_conc[iff]:
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"!!L R. (False#p,q) : step (conc L R) a =
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(? r. q = False#r & (p,r) : step R a)"
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by (simp add:conc_def step_def) (blast)
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(** False in epsclosure **)
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lemma lemma1b':
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"(tp,tq) : (eps(conc L R))^* ==>
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(!!p. tp = False#p ==> ? q. (p,q) : (eps R)^* & tq = False#q)"
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apply (induct rule: rtrancl_induct)
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apply (blast)
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apply (blast intro: rtrancl_into_rtrancl)
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done
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lemma lemma2b':
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"(p,q) : (eps R)^* ==> (False#p, False#q) : (eps(conc L R))^*"
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apply (induct rule: rtrancl_induct)
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apply (blast)
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apply (blast intro: rtrancl_into_rtrancl)
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done
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lemma False_epsclosure_conc[iff]:
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"((False # p, q) : (eps (conc L R))^*) =
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(? r. q = False # r & (p, r) : (eps R)^*)"
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apply (rule iffI)
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apply (blast dest: lemma1b')
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apply (blast dest: lemma2b')
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done
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(** False in steps **)
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lemma False_steps_conc[iff]:
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"!!p. (False#p,q): steps (conc L R) w = (? r. q=False#r & (p,r): steps R w)"
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apply (induct "w")
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apply (simp)
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apply (simp)
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apply (fast) (*MUCH faster than blast*)
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done
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(** True in epsclosure **)
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lemma True_True_eps_concI:
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"(p,q): (eps L)^* ==> (True#p,True#q) : (eps(conc L R))^*"
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apply (induct rule: rtrancl_induct)
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apply (blast)
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apply (blast intro: rtrancl_into_rtrancl)
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done
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lemma True_True_steps_concI:
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"!!p. (p,q) : steps L w ==> (True#p,True#q) : steps (conc L R) w"
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apply (induct "w")
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apply (simp add: True_True_eps_concI)
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apply (simp)
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apply (blast intro: True_True_eps_concI)
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done
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lemma lemma1a':
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"(tp,tq) : (eps(conc L R))^* ==>
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(!!p. tp = True#p ==>
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(? q. tq = True#q & (p,q) : (eps L)^*) |
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(? q r. tq = False#q & (p,r):(eps L)^* & fin L r & (start R,q) : (eps R)^*))"
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apply (induct rule: rtrancl_induct)
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apply (blast)
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apply (blast intro: rtrancl_into_rtrancl)
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done
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lemma lemma2a':
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"(p, q) : (eps L)^* ==> (True#p, True#q) : (eps(conc L R))^*"
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apply (induct rule: rtrancl_induct)
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apply (blast)
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apply (blast intro: rtrancl_into_rtrancl)
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done
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lemma lem:
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"!!L R. (p,q) : step R None ==> (False#p, False#q) : step (conc L R) None"
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by(simp add: conc_def step_def)
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lemma lemma2b'':
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"(p,q) : (eps R)^* ==> (False#p, False#q) : (eps(conc L R))^*"
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apply (induct rule: rtrancl_induct)
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apply (blast)
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apply (drule lem)
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apply (blast intro: rtrancl_into_rtrancl)
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done
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lemma True_False_eps_concI:
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"!!L R. fin L p ==> (True#p, False#start R) : eps(conc L R)"
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by(simp add: conc_def step_def)
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lemma True_epsclosure_conc[iff]:
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"((True#p,q) : (eps(conc L R))^*) =
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((? r. (p,r) : (eps L)^* & q = True#r) |
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(? r. (p,r) : (eps L)^* & fin L r &
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(? s. (start R, s) : (eps R)^* & q = False#s)))"
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apply (rule iffI)
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apply (blast dest: lemma1a')
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apply (erule disjE)
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apply (blast intro: lemma2a')
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apply (clarify)
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apply (rule rtrancl_trans)
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apply (erule lemma2a')
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apply (rule converse_rtrancl_into_rtrancl)
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apply (erule True_False_eps_concI)
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apply (erule lemma2b'')
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done
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(** True in steps **)
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lemma True_steps_concD[rule_format]:
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"!p. (True#p,q) : steps (conc L R) w -->
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((? r. (p,r) : steps L w & q = True#r) |
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(? u v. w = u@v & (? r. (p,r) : steps L u & fin L r &
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(? s. (start R,s) : steps R v & q = False#s))))"
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apply (induct "w")
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apply (simp)
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apply (simp)
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apply (clarify del: disjCI)
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apply (erule disjE)
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apply (clarify del: disjCI)
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apply (erule disjE)
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apply (clarify del: disjCI)
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apply (erule allE, erule impE, assumption)
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apply (erule disjE)
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apply (blast)
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apply (rule disjI2)
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apply (clarify)
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apply (simp)
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apply (rule_tac x = "a#u" in exI)
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apply (simp)
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apply (blast)
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apply (blast)
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apply (rule disjI2)
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apply (clarify)
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apply (simp)
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apply (rule_tac x = "[]" in exI)
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apply (simp)
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apply (blast)
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done
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lemma True_steps_conc:
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"(True#p,q) : steps (conc L R) w =
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((? r. (p,r) : steps L w & q = True#r) |
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(? u v. w = u@v & (? r. (p,r) : steps L u & fin L r &
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(? s. (start R,s) : steps R v & q = False#s))))"
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by (blast dest: True_steps_concD
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intro: True_True_steps_concI in_steps_epsclosure)
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(** starting from the start **)
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lemma start_conc:
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"!!L R. start(conc L R) = True#start L"
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by (simp add: conc_def)
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lemma final_conc:
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"!!L R. fin(conc L R) p = (? s. p = False#s & fin R s)"
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by (simp add:conc_def split: list.split)
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lemma accepts_conc:
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"accepts (conc L R) w = (? u v. w = u@v & accepts L u & accepts R v)"
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apply (simp add: accepts_def True_steps_conc final_conc start_conc)
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apply (blast)
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done
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(******************************************************)
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(* star *)
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(******************************************************)
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lemma True_in_eps_star[iff]:
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"!!A. (True#p,q) : eps(star A) =
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( (? r. q = True#r & (p,r) : eps A) | (fin A p & q = True#start A) )"
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by (simp add:star_def step_def) (blast)
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lemma True_True_step_starI:
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"!!A. (p,q) : step A a ==> (True#p, True#q) : step (star A) a"
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by (simp add:star_def step_def)
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lemma True_True_eps_starI:
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"(p,r) : (eps A)^* ==> (True#p, True#r) : (eps(star A))^*"
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apply (induct rule: rtrancl_induct)
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apply (blast)
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apply (blast intro: True_True_step_starI rtrancl_into_rtrancl)
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done
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lemma True_start_eps_starI:
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"!!A. fin A p ==> (True#p,True#start A) : eps(star A)"
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by (simp add:star_def step_def)
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lemma lem':
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"(tp,s) : (eps(star A))^* ==> (! p. tp = True#p -->
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(? r. ((p,r) : (eps A)^* |
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(? q. (p,q) : (eps A)^* & fin A q & (start A,r) : (eps A)^*)) &
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s = True#r))"
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apply (induct rule: rtrancl_induct)
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apply (simp)
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apply (clarify)
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apply (simp)
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apply (blast intro: rtrancl_into_rtrancl)
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done
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lemma True_eps_star[iff]:
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"((True#p,s) : (eps(star A))^*) =
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(? r. ((p,r) : (eps A)^* |
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(? q. (p,q) : (eps A)^* & fin A q & (start A,r) : (eps A)^*)) &
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s = True#r)"
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apply (rule iffI)
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apply (drule lem')
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apply (blast)
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(* Why can't blast do the rest? *)
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apply (clarify)
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apply (erule disjE)
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apply (erule True_True_eps_starI)
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apply (clarify)
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apply (rule rtrancl_trans)
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apply (erule True_True_eps_starI)
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apply (rule rtrancl_trans)
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apply (rule r_into_rtrancl)
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apply (erule True_start_eps_starI)
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apply (erule True_True_eps_starI)
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done
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(** True in step Some **)
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lemma True_step_star[iff]:
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"!!A. (True#p,r): step (star A) (Some a) =
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(? q. (p,q): step A (Some a) & r=True#q)"
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by (simp add:star_def step_def) (blast)
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(** True in steps **)
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(* reverse list induction! Complicates matters for conc? *)
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lemma True_start_steps_starD[rule_format]:
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"!rr. (True#start A,rr) : steps (star A) w -->
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(? us v. w = concat us @ v &
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(!u:set us. accepts A u) &
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(? r. (start A,r) : steps A v & rr = True#r))"
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apply (induct w rule: rev_induct)
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apply (simp)
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apply (clarify)
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apply (rule_tac x = "[]" in exI)
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apply (erule disjE)
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apply (simp)
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apply (clarify)
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apply (simp)
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apply (simp add: O_assoc[symmetric] epsclosure_steps)
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apply (clarify)
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apply (erule allE, erule impE, assumption)
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apply (clarify)
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apply (erule disjE)
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apply (rule_tac x = "us" in exI)
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apply (rule_tac x = "v@[x]" in exI)
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apply (simp add: O_assoc[symmetric] epsclosure_steps)
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apply (blast)
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apply (clarify)
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apply (rule_tac x = "us@[v@[x]]" in exI)
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apply (rule_tac x = "[]" in exI)
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apply (simp add: accepts_def)
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apply (blast)
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done
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lemma True_True_steps_starI:
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"!!p. (p,q) : steps A w ==> (True#p,True#q) : steps (star A) w"
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apply (induct "w")
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apply (simp)
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apply (simp)
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apply (blast intro: True_True_eps_starI True_True_step_starI)
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done
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lemma steps_star_cycle:
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"(!u : set us. accepts A u) ==>
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(True#start A,True#start A) : steps (star A) (concat us)"
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apply (induct "us")
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apply (simp add:accepts_def)
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apply (simp add:accepts_def)
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by(blast intro: True_True_steps_starI True_start_eps_starI in_epsclosure_steps)
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(* Better stated directly with start(star A)? Loop in star A back to start(star A)?*)
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lemma True_start_steps_star:
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"(True#start A,rr) : steps (star A) w =
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(? us v. w = concat us @ v &
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(!u:set us. accepts A u) &
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(? r. (start A,r) : steps A v & rr = True#r))"
|
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apply (rule iffI)
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apply (erule True_start_steps_starD)
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apply (clarify)
|
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apply (blast intro: steps_star_cycle True_True_steps_starI)
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|
done
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(** the start state **)
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lemma start_step_star[iff]:
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"!!A. (start(star A),r) : step (star A) a = (a=None & r = True#start A)"
|
|
by (simp add:star_def step_def)
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|
|
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lemmas epsclosure_start_step_star =
|
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in_unfold_rtrancl2[where ?p = "start (star A)"] for A
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|
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lemma start_steps_star:
|
|
"(start(star A),r) : steps (star A) w =
|
|
((w=[] & r= start(star A)) | (True#start A,r) : steps (star A) w)"
|
|
apply (rule iffI)
|
|
apply (case_tac "w")
|
|
apply (simp add: epsclosure_start_step_star)
|
|
apply (simp)
|
|
apply (clarify)
|
|
apply (simp add: epsclosure_start_step_star)
|
|
apply (blast)
|
|
apply (erule disjE)
|
|
apply (simp)
|
|
apply (blast intro: in_steps_epsclosure)
|
|
done
|
|
|
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lemma fin_star_True[iff]: "!!A. fin (star A) (True#p) = fin A p"
|
|
by (simp add:star_def)
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|
|
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lemma fin_star_start[iff]: "!!A. fin (star A) (start(star A))"
|
|
by (simp add:star_def)
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|
|
|
(* too complex! Simpler if loop back to start(star A)? *)
|
|
lemma accepts_star:
|
|
"accepts (star A) w =
|
|
(? us. (!u : set(us). accepts A u) & (w = concat us) )"
|
|
apply(unfold accepts_def)
|
|
apply (simp add: start_steps_star True_start_steps_star)
|
|
apply (rule iffI)
|
|
apply (clarify)
|
|
apply (erule disjE)
|
|
apply (clarify)
|
|
apply (simp)
|
|
apply (rule_tac x = "[]" in exI)
|
|
apply (simp)
|
|
apply (clarify)
|
|
apply (rule_tac x = "us@[v]" in exI)
|
|
apply (simp add: accepts_def)
|
|
apply (blast)
|
|
apply (clarify)
|
|
apply (rule_tac xs = "us" in rev_exhaust)
|
|
apply (simp)
|
|
apply (blast)
|
|
apply (clarify)
|
|
apply (simp add: accepts_def)
|
|
apply (blast)
|
|
done
|
|
|
|
|
|
(***** Correctness of r2n *****)
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|
|
|
lemma accepts_rexp2nae:
|
|
"!!w. accepts (rexp2nae r) w = (w : lang r)"
|
|
apply (induct "r")
|
|
apply (simp add: accepts_def)
|
|
apply simp
|
|
apply (simp add: accepts_atom)
|
|
apply (simp add: accepts_or)
|
|
apply (simp add: accepts_conc Regular_Set.conc_def)
|
|
apply (simp add: accepts_star in_star_iff_concat subset_iff Ball_def)
|
|
done
|
|
|
|
end
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