# Theory RegExp2NA

```(*  Author:     Tobias Nipkow
*)

section "From regular expressions directly to nondeterministic automata"

theory RegExp2NA
imports "Regular-Sets.Regular_Exp" NA
begin

type_synonym 'a bitsNA = "('a,bool list)na"

definition
"atom"  :: "'a ⇒ 'a bitsNA" where
"atom a = ([True],
λb s. if s=[True] ∧ b=a then {[False]} else {},
λs. s=[False])"

definition
or :: "'a bitsNA ⇒ 'a bitsNA ⇒ 'a bitsNA" where
"or = (λ(ql,dl,fl)(qr,dr,fr).
([],
λa s. case s of
[] ⇒ (True ## dl a ql) ∪ (False ## dr a qr)
| left#s ⇒ if left then True ## dl a s
else False ## dr a s,
λs. case s of [] ⇒ (fl ql | fr qr)
| left#s ⇒ if left then fl s else fr s))"

definition
conc :: "'a bitsNA ⇒ 'a bitsNA ⇒ 'a bitsNA" where
"conc = (λ(ql,dl,fl)(qr,dr,fr).
(True#ql,
λa s. case s of
[] ⇒ {}
| left#s ⇒ if left then (True ## dl a s) ∪
(if fl s then False ## dr a qr else {})
else False ## dr a s,
λs. case s of [] ⇒ False | left#s ⇒ left ∧ fl s ∧ fr qr | ¬left ∧ fr s))"

definition
epsilon :: "'a bitsNA" where
"epsilon = ([],λa s. {}, λs. s=[])"

definition
plus :: "'a bitsNA ⇒ 'a bitsNA" where
"plus = (λ(q,d,f). (q, λa s. d a s ∪ (if f s then d a q else {}), f))"

definition
star :: "'a bitsNA ⇒ 'a bitsNA" where
"star A = or epsilon (plus A)"

primrec rexp2na :: "'a rexp ⇒ 'a bitsNA" where
"rexp2na Zero       = ([], λa s. {}, λs. False)" |
"rexp2na One        = epsilon" |
"rexp2na(Atom a)    = atom a" |
"rexp2na(Plus r s)  = or (rexp2na r) (rexp2na s)" |
"rexp2na(Times r s) = conc (rexp2na r) (rexp2na s)" |
"rexp2na(Star r)    = star (rexp2na r)"

declare split_paired_all[simp]

(******************************************************)
(*                       atom                         *)
(******************************************************)

lemma fin_atom: "(fin (atom a) q) = (q = [False])"

lemma start_atom: "start (atom a) = [True]"

lemma in_step_atom_Some[simp]:
"(p,q) : step (atom a) b = (p=[True] ∧ q=[False] ∧ b=a)"

lemma False_False_in_steps_atom:
"([False],[False]) : steps (atom a) w = (w = [])"
apply (induct "w")
apply simp
done

lemma start_fin_in_steps_atom:
"(start (atom a), [False]) : steps (atom a) w = (w = [a])"
apply (induct "w")
apply (simp add: False_False_in_steps_atom relcomp_unfold start_atom)
done

lemma accepts_atom:
"accepts (atom a) w = (w = [a])"
by (simp add: accepts_conv_steps start_fin_in_steps_atom fin_atom)

(******************************************************)
(*                      or                            *)
(******************************************************)

(***** lift True/False over fin *****)

lemma fin_or_True[iff]:
"⋀L R. fin (or L R) (True#p) = fin L p"

lemma fin_or_False[iff]:
"⋀L R. fin (or L R) (False#p) = fin R p"

(***** lift True/False over step *****)

lemma True_in_step_or[iff]:
"⋀L R. (True#p,q) : step (or L R) a = (∃r. q = True#r ∧ (p,r) ∈ step L a)"
apply blast
done

lemma False_in_step_or[iff]:
"⋀L R. (False#p,q) : step (or L R) a = (∃r. q = False#r ∧ (p,r) ∈ step R a)"
apply blast
done

(***** lift True/False over steps *****)

lemma lift_True_over_steps_or[iff]:
"⋀p. (True#p,q)∈steps (or L R) w = (∃r. q = True # r ∧ (p,r) ∈ steps L w)"
apply (induct "w")
apply force
apply force
done

lemma lift_False_over_steps_or[iff]:
"⋀p. (False#p,q)∈steps (or L R) w = (∃r. q = False#r ∧ (p,r)∈steps R w)"
apply (induct "w")
apply force
apply force
done

(** From the start  **)

lemma start_step_or[iff]:
"⋀L R. (start(or L R),q) : step(or L R) a =
(∃p. (q = True#p ∧ (start L,p) : step L a) |
(q = False#p ∧ (start R,p) : step R a))"
apply blast
done

lemma steps_or:
"(start(or L R), q) : steps (or L R) w =
( (w = [] ∧ q = start(or L R)) |
(w ≠ [] ∧ (∃p.  q = True  # p ∧ (start L,p) : steps L w |
q = False # p ∧ (start R,p) : steps R w)))"
apply (case_tac "w")
apply (simp)
apply blast
apply (simp)
apply blast
done

lemma fin_start_or[iff]:
"⋀L R. fin (or L R) (start(or L R)) = (fin L (start L) | fin R (start R))"

lemma accepts_or[iff]:
"accepts (or L R) w = (accepts L w | accepts R w)"
(* get rid of case_tac: *)
apply (case_tac "w = []")
apply auto
done

(******************************************************)
(*                      conc                        *)
(******************************************************)

(** True/False in fin **)

lemma fin_conc_True[iff]:
"⋀L R. fin (conc L R) (True#p) = (fin L p ∧ fin R (start R))"

lemma fin_conc_False[iff]:
"⋀L R. fin (conc L R) (False#p) = fin R p"

(** True/False in step **)

lemma True_step_conc[iff]:
"⋀L R. (True#p,q) : step (conc L R) a =
((∃r. q=True#r ∧ (p,r): step L a) |
(fin L p ∧ (∃r. q=False#r ∧ (start R,r) : step R a)))"
apply blast
done

lemma False_step_conc[iff]:
"⋀L R. (False#p,q) : step (conc L R) a =
(∃r. q = False#r ∧ (p,r) : step R a)"
apply blast
done

(** False in steps **)

lemma False_steps_conc[iff]:
"⋀p. (False#p,q): steps (conc L R) w = (∃r. q=False#r ∧ (p,r): steps R w)"
apply (induct "w")
apply fastforce
apply force
done

(** True in steps **)

lemma True_True_steps_concI:
"⋀L R p. (p,q) : steps L w ⟹ (True#p,True#q) : steps (conc L R) w"
apply (induct "w")
apply simp
apply simp
apply fast
done

lemma True_False_step_conc[iff]:
"⋀L R. (True#p,False#q) : step (conc L R) a =
(fin L p ∧ (start R,q) : step R a)"
by simp

lemma True_steps_concD[rule_format]:
"∀p. (True#p,q) : steps (conc L R) w ⟶
((∃r. (p,r) : steps L w ∧ q = True#r)  ∨
(∃u a v. w = u@a#v ∧
(∃r. (p,r) : steps L u ∧ fin L r ∧
(∃s. (start R,s) : step R a ∧
(∃t. (s,t) : steps R v ∧ q = False#t)))))"
apply (induct "w")
apply simp
apply simp
apply (clarify del:disjCI)
apply (erule disjE)
apply (clarify del:disjCI)
apply (erule allE, erule impE, assumption)
apply (erule disjE)
apply blast
apply (rule disjI2)
apply (clarify)
apply simp
apply (rule_tac x = "a#u" in exI)
apply simp
apply blast
apply (rule disjI2)
apply (clarify)
apply simp
apply (rule_tac x = "[]" in exI)
apply simp
apply blast
done

lemma True_steps_conc:
"(True#p,q) : steps (conc L R) w =
((∃r. (p,r) : steps L w ∧ q = True#r)  ∨
(∃u a v. w = u@a#v ∧
(∃r. (p,r) : steps L u ∧ fin L r ∧
(∃s. (start R,s) : step R a ∧
(∃t. (s,t) : steps R v ∧ q = False#t)))))"
by(force dest!: True_steps_concD intro!: True_True_steps_concI)

(** starting from the start **)

lemma start_conc:
"⋀L R. start(conc L R) = True#start L"

lemma final_conc:
"⋀L R. fin(conc L R) p = ((fin R (start R) ∧ (∃s. p = True#s ∧ fin L s)) ∨
(∃s. p = False#s ∧ fin R s))"
apply blast
done

lemma accepts_conc:
"accepts (conc L R) w = (∃u v. w = u@v ∧ accepts L u ∧ accepts R v)"
apply (simp add: accepts_conv_steps True_steps_conc final_conc start_conc)
apply (rule iffI)
apply (clarify)
apply (erule disjE)
apply (clarify)
apply (erule disjE)
apply (rule_tac x = "w" in exI)
apply simp
apply blast
apply blast
apply (erule disjE)
apply blast
apply (clarify)
apply (rule_tac x = "u" in exI)
apply simp
apply blast
apply (clarify)
apply (case_tac "v")
apply simp
apply blast
apply simp
apply blast
done

(******************************************************)
(*                     epsilon                        *)
(******************************************************)

lemma step_epsilon[simp]: "step epsilon a = {}"

lemma steps_epsilon: "((p,q) : steps epsilon w) = (w=[] ∧ p=q)"
by (induct "w") auto

lemma accepts_epsilon[iff]: "accepts epsilon w = (w = [])"
done

(******************************************************)
(*                       plus                         *)
(******************************************************)

lemma start_plus[simp]: "⋀A. start (plus A) = start A"

lemma fin_plus[iff]: "⋀A. fin (plus A) = fin A"

lemma step_plusI:
"⋀A. (p,q) : step A a ⟹ (p,q) : step (plus A) a"

lemma steps_plusI: "⋀p. (p,q) : steps A w ⟹ (p,q) ∈ steps (plus A) w"
apply (induct "w")
apply simp
apply simp
apply (blast intro: step_plusI)
done

lemma step_plus_conv[iff]:
"⋀A. (p,r): step (plus A) a =
( (p,r): step A a | fin A p ∧ (start A,r) : step A a )"

lemma fin_steps_plusI:
"[| (start A,q) : steps A u; u ≠ []; fin A p |]
==> (p,q) : steps (plus A) u"
apply (case_tac "u")
apply blast
apply simp
apply (blast intro: steps_plusI)
done

(* reverse list induction! Complicates matters for conc? *)
lemma start_steps_plusD[rule_format]:
"∀r. (start A,r) ∈ steps (plus A) w ⟶
(∃us v. w = concat us @ v ∧
(∀u∈set us. accepts A u) ∧
(start A,r) ∈ steps A v)"
apply (induct w rule: rev_induct)
apply simp
apply (rule_tac x = "[]" in exI)
apply simp
apply simp
apply (clarify)
apply (erule allE, erule impE, assumption)
apply (clarify)
apply (erule disjE)
apply (rule_tac x = "us" in exI)
apply (simp)
apply blast
apply (rule_tac x = "us@[v]" in exI)
apply blast
done

lemma steps_star_cycle[rule_format]:
"us ≠ [] ⟶ (∀u ∈ set us. accepts A u) ⟶ accepts (plus A) (concat us)"
apply (induct us rule: rev_induct)
apply simp
apply (rename_tac u us)
apply simp
apply (clarify)
apply (case_tac "us = []")
apply (simp)
apply (blast intro: steps_plusI fin_steps_plusI)
apply (clarify)
apply (case_tac "u = []")
apply (simp)
apply (blast intro: steps_plusI fin_steps_plusI)
apply (blast intro: steps_plusI fin_steps_plusI)
done

lemma accepts_plus[iff]:
"accepts (plus A) w =
(∃us. us ≠ [] ∧ w = concat us ∧ (∀u ∈ set us. accepts A u))"
apply (rule iffI)
apply (clarify)
apply (drule start_steps_plusD)
apply (clarify)
apply (rule_tac x = "us@[v]" in exI)
apply blast
apply (blast intro: steps_star_cycle)
done

(******************************************************)
(*                       star                         *)
(******************************************************)

lemma accepts_star:
"accepts (star A) w = (∃us. (∀u ∈ set us. accepts A u) ∧ w = concat us)"
apply(unfold star_def)
apply (rule iffI)
apply (clarify)
apply (erule disjE)
apply (rule_tac x = "[]" in exI)
apply simp
apply blast
apply force
done

(***** Correctness of r2n *****)

lemma accepts_rexp2na:
"⋀w. accepts (rexp2na r) w = (w : lang r)"
apply (induct "r")