--- a/Myhill.thy Thu Jan 27 11:50:58 2011 +0000
+++ b/Myhill.thy Thu Jan 27 12:35:06 2011 +0000
@@ -1,1132 +1,8 @@
theory Myhill
- imports Main List_Prefix Prefix_subtract Prelude
+ imports Myhill_1
begin
-(*
-text {*
- \begin{figure}
- \centering
- \scalebox{0.95}{
- \begin{tikzpicture}[->,>=latex,shorten >=1pt,auto,node distance=1.2cm, semithick]
- \node[state,initial] (n1) {$1$};
- \node[state,accepting] (n2) [right = 10em of n1] {$2$};
-
- \path (n1) edge [bend left] node {$0$} (n2)
- (n1) edge [loop above] node{$1$} (n1)
- (n2) edge [loop above] node{$0$} (n2)
- (n2) edge [bend left] node {$1$} (n1)
- ;
- \end{tikzpicture}}
- \caption{An example automaton (or partition)}\label{fig:example_automata}
- \end{figure}
-*}
-
-*)
-
-section {* Preliminary definitions *}
-
-types lang = "string set"
-
-text {*
- Sequential composition of two languages @{text "L1"} and @{text "L2"}
-*}
-
-
-definition Seq :: "lang \<Rightarrow> lang \<Rightarrow> lang" ("_ ;; _" [100,100] 100)
-where
- "L1 ;; L2 = {s1 @ s2 | s1 s2. s1 \<in> L1 \<and> s2 \<in> L2}"
-
-text {* Transitive closure of language @{text "L"}. *}
-inductive_set
- Star :: "string set \<Rightarrow> string set" ("_\<star>" [101] 102)
- for L :: "string set"
-where
- start[intro]: "[] \<in> L\<star>"
-| step[intro]: "\<lbrakk>s1 \<in> L; s2 \<in> L\<star>\<rbrakk> \<Longrightarrow> s1@s2 \<in> L\<star>"
-
-text {* Some properties of operator @{text ";;"}.*}
-
-lemma seq_union_distrib:
- "(A \<union> B) ;; C = (A ;; C) \<union> (B ;; C)"
-by (auto simp:Seq_def)
-
-lemma seq_intro:
- "\<lbrakk>x \<in> A; y \<in> B\<rbrakk> \<Longrightarrow> x @ y \<in> A ;; B "
-by (auto simp:Seq_def)
-
-lemma seq_assoc:
- "(A ;; B) ;; C = A ;; (B ;; C)"
-apply(auto simp:Seq_def)
-apply blast
-by (metis append_assoc)
-
-lemma star_intro1[rule_format]: "x \<in> lang\<star> \<Longrightarrow> \<forall> y. y \<in> lang\<star> \<longrightarrow> x @ y \<in> lang\<star>"
-by (erule Star.induct, auto)
-
-lemma star_intro2: "y \<in> lang \<Longrightarrow> y \<in> lang\<star>"
-by (drule step[of y lang "[]"], auto simp:start)
-
-lemma star_intro3[rule_format]:
- "x \<in> lang\<star> \<Longrightarrow> \<forall>y . y \<in> lang \<longrightarrow> x @ y \<in> lang\<star>"
-by (erule Star.induct, auto intro:star_intro2)
-
-lemma star_decom:
- "\<lbrakk>x \<in> lang\<star>; x \<noteq> []\<rbrakk> \<Longrightarrow>(\<exists> a b. x = a @ b \<and> a \<noteq> [] \<and> a \<in> lang \<and> b \<in> lang\<star>)"
-by (induct x rule: Star.induct, simp, blast)
-
-lemma star_decom':
- "\<lbrakk>x \<in> lang\<star>; x \<noteq> []\<rbrakk> \<Longrightarrow> \<exists>a b. x = a @ b \<and> a \<in> lang\<star> \<and> b \<in> lang"
-apply (induct x rule:Star.induct, simp)
-apply (case_tac "s2 = []")
-apply (rule_tac x = "[]" in exI, rule_tac x = s1 in exI, simp add:start)
-apply (simp, (erule exE| erule conjE)+)
-by (rule_tac x = "s1 @ a" in exI, rule_tac x = b in exI, simp add:step)
-
-text {* Ardens lemma expressed at the level of language, rather than the level of regular expression. *}
-
-theorem ardens_revised:
- assumes nemp: "[] \<notin> A"
- shows "(X = X ;; A \<union> B) \<longleftrightarrow> (X = B ;; A\<star>)"
-proof
- assume eq: "X = B ;; A\<star>"
- have "A\<star> = {[]} \<union> A\<star> ;; A"
- by (auto simp:Seq_def star_intro3 star_decom')
- then have "B ;; A\<star> = B ;; ({[]} \<union> A\<star> ;; A)"
- unfolding Seq_def by simp
- also have "\<dots> = B \<union> B ;; (A\<star> ;; A)"
- unfolding Seq_def by auto
- also have "\<dots> = B \<union> (B ;; A\<star>) ;; A"
- by (simp only:seq_assoc)
- finally show "X = X ;; A \<union> B"
- using eq by blast
-next
- assume eq': "X = X ;; A \<union> B"
- hence c1': "\<And> x. x \<in> B \<Longrightarrow> x \<in> X"
- and c2': "\<And> x y. \<lbrakk>x \<in> X; y \<in> A\<rbrakk> \<Longrightarrow> x @ y \<in> X"
- using Seq_def by auto
- show "X = B ;; A\<star>"
- proof
- show "B ;; A\<star> \<subseteq> X"
- proof-
- { fix x y
- have "\<lbrakk>y \<in> A\<star>; x \<in> X\<rbrakk> \<Longrightarrow> x @ y \<in> X "
- apply (induct arbitrary:x rule:Star.induct, simp)
- by (auto simp only:append_assoc[THEN sym] dest:c2')
- } thus ?thesis using c1' by (auto simp:Seq_def)
- qed
- next
- show "X \<subseteq> B ;; A\<star>"
- proof-
- { fix x
- have "x \<in> X \<Longrightarrow> x \<in> B ;; A\<star>"
- proof (induct x taking:length rule:measure_induct)
- fix z
- assume hyps:
- "\<forall>y. length y < length z \<longrightarrow> y \<in> X \<longrightarrow> y \<in> B ;; A\<star>"
- and z_in: "z \<in> X"
- show "z \<in> B ;; A\<star>"
- proof (cases "z \<in> B")
- case True thus ?thesis by (auto simp:Seq_def start)
- next
- case False hence "z \<in> X ;; A" using eq' z_in by auto
- then obtain za zb where za_in: "za \<in> X"
- and zab: "z = za @ zb \<and> zb \<in> A" and zbne: "zb \<noteq> []"
- using nemp unfolding Seq_def by blast
- from zbne zab have "length za < length z" by auto
- with za_in hyps have "za \<in> B ;; A\<star>" by blast
- hence "za @ zb \<in> B ;; A\<star>" using zab
- by (clarsimp simp:Seq_def, blast dest:star_intro3)
- thus ?thesis using zab by simp
- qed
- qed
- } thus ?thesis by blast
- qed
- qed
-qed
-
-
-text {* The syntax of regular expressions is defined by the datatype @{text "rexp"}. *}
-datatype rexp =
- NULL
-| EMPTY
-| CHAR char
-| SEQ rexp rexp
-| ALT rexp rexp
-| STAR rexp
-
-
-text {*
- The following @{text "L"} is an overloaded operator, where @{text "L(x)"} evaluates to
- the language represented by the syntactic object @{text "x"}.
-*}
-consts L:: "'a \<Rightarrow> string set"
-
-
-text {*
- The @{text "L(rexp)"} for regular expression @{text "rexp"} is defined by the
- following overloading function @{text "L_rexp"}.
-*}
-overloading L_rexp \<equiv> "L:: rexp \<Rightarrow> string set"
-begin
-fun
- L_rexp :: "rexp \<Rightarrow> string set"
-where
- "L_rexp (NULL) = {}"
- | "L_rexp (EMPTY) = {[]}"
- | "L_rexp (CHAR c) = {[c]}"
- | "L_rexp (SEQ r1 r2) = (L_rexp r1) ;; (L_rexp r2)"
- | "L_rexp (ALT r1 r2) = (L_rexp r1) \<union> (L_rexp r2)"
- | "L_rexp (STAR r) = (L_rexp r)\<star>"
-end
-
-text {*
- To obtain equational system out of finite set of equivalent classes, a fold operation
- on finite set @{text "folds"} is defined. The use of @{text "SOME"} makes @{text "fold"}
- more robust than the @{text "fold"} in Isabelle library. The expression @{text "folds f"}
- makes sense when @{text "f"} is not @{text "associative"} and @{text "commutitive"},
- while @{text "fold f"} does not.
-*}
-
-definition
- folds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"
-where
- "folds f z S \<equiv> SOME x. fold_graph f z S x"
-
-text {*
- The following lemma assures that the arbitrary choice made by the @{text "SOME"} in @{text "folds"}
- does not affect the @{text "L"}-value of the resultant regular expression.
- *}
-lemma folds_alt_simp [simp]:
- "finite rs \<Longrightarrow> L (folds ALT NULL rs) = \<Union> (L ` rs)"
-apply (rule set_eq_intro, simp add:folds_def)
-apply (rule someI2_ex, erule finite_imp_fold_graph)
-by (erule fold_graph.induct, auto)
-
-(* Just a technical lemma. *)
-lemma [simp]:
- shows "(x, y) \<in> {(x, y). P x y} \<longleftrightarrow> P x y"
-by simp
-
-text {*
- @{text "\<approx>L"} is an equivalent class defined by language @{text "Lang"}.
-*}
-
-definition
- str_eq_rel ("\<approx>_" [100] 100)
-where
- "\<approx>Lang \<equiv> {(x, y). (\<forall>z. x @ z \<in> Lang \<longleftrightarrow> y @ z \<in> Lang)}"
-
-text {*
- Among equivlant clases of @{text "\<approx>Lang"}, the set @{text "finals(Lang)"} singles out
- those which contains strings from @{text "Lang"}.
-*}
-
-definition
- "finals Lang \<equiv> {\<approx>Lang `` {x} | x . x \<in> Lang}"
-
-text {*
- The following lemma show the relationshipt between @{text "finals(Lang)"} and @{text "Lang"}.
-*}
-lemma lang_is_union_of_finals:
- "Lang = \<Union> finals(Lang)"
-proof
- show "Lang \<subseteq> \<Union> (finals Lang)"
- proof
- fix x
- assume "x \<in> Lang"
- thus "x \<in> \<Union> (finals Lang)"
- apply (simp add:finals_def, rule_tac x = "(\<approx>Lang) `` {x}" in exI)
- by (auto simp:Image_def str_eq_rel_def)
- qed
-next
- show "\<Union> (finals Lang) \<subseteq> Lang"
- apply (clarsimp simp:finals_def str_eq_rel_def)
- by (drule_tac x = "[]" in spec, auto)
-qed
-
-
-
-
-section {* Direction @{text "finite partition \<Rightarrow> regular language"}*}
-
-text {*
- The relationship between equivalent classes can be described by an
- equational system.
- For example, in equational system \eqref{example_eqns}, $X_0, X_1$ are equivalent
- classes. The first equation says every string in $X_0$ is obtained either by
- appending one $b$ to a string in $X_0$ or by appending one $a$ to a string in
- $X_1$ or just be an empty string (represented by the regular expression $\lambda$). Similary,
- the second equation tells how the strings inside $X_1$ are composed.
- \begin{equation}\label{example_eqns}
- \begin{aligned}
- X_0 & = X_0 b + X_1 a + \lambda \\
- X_1 & = X_0 a + X_1 b
- \end{aligned}
- \end{equation}
- The summands on the right hand side is represented by the following data type
- @{text "rhs_item"}, mnemonic for 'right hand side item'.
- Generally, there are two kinds of right hand side items, one kind corresponds to
- pure regular expressions, like the $\lambda$ in \eqref{example_eqns}, the other kind corresponds to
- transitions from one one equivalent class to another, like the $X_0 b, X_1 a$ etc.
- *}
-
-datatype rhs_item =
- Lam "rexp" (* Lambda *)
- | Trn "(string set)" "rexp" (* Transition *)
-
-text {*
- In this formalization, pure regular expressions like $\lambda$ is
- repsented by @{text "Lam(EMPTY)"}, while transitions like $X_0 a$ is represented by $Trn~X_0~(CHAR~a)$.
- *}
-
-text {*
- The functions @{text "the_r"} and @{text "the_Trn"} are used to extract
- subcomponents from right hand side items.
- *}
-
-fun the_r :: "rhs_item \<Rightarrow> rexp"
-where "the_r (Lam r) = r"
-
-fun the_Trn:: "rhs_item \<Rightarrow> (string set \<times> rexp)"
-where "the_Trn (Trn Y r) = (Y, r)"
-
-text {*
- Every right hand side item @{text "itm"} defines a string set given
- @{text "L(itm)"}, defined as:
-*}
-overloading L_rhs_e \<equiv> "L:: rhs_item \<Rightarrow> string set"
-begin
- fun L_rhs_e:: "rhs_item \<Rightarrow> string set"
- where
- "L_rhs_e (Lam r) = L r" |
- "L_rhs_e (Trn X r) = X ;; L r"
-end
-
-text {*
- The right hand side of every equation is represented by a set of
- items. The string set defined by such a set @{text "itms"} is given
- by @{text "L(itms)"}, defined as:
-*}
-
-overloading L_rhs \<equiv> "L:: rhs_item set \<Rightarrow> string set"
-begin
- fun L_rhs:: "rhs_item set \<Rightarrow> string set"
- where "L_rhs rhs = \<Union> (L ` rhs)"
-end
-
-text {*
- Given a set of equivalent classses @{text "CS"} and one equivalent class @{text "X"} among
- @{text "CS"}, the term @{text "init_rhs CS X"} is used to extract the right hand side of
- the equation describing the formation of @{text "X"}. The definition of @{text "init_rhs"}
- is:
- *}
-
-definition
- "init_rhs CS X \<equiv>
- if ([] \<in> X) then
- {Lam(EMPTY)} \<union> {Trn Y (CHAR c) | Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X}
- else
- {Trn Y (CHAR c)| Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X}"
-
-text {*
- In the definition of @{text "init_rhs"}, the term
- @{text "{Trn Y (CHAR c)| Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X}"} appearing on both branches
- describes the formation of strings in @{text "X"} out of transitions, while
- the term @{text "{Lam(EMPTY)}"} describes the empty string which is intrinsically contained in
- @{text "X"} rather than by transition. This @{text "{Lam(EMPTY)}"} corresponds to
- the $\lambda$ in \eqref{example_eqns}.
-
- With the help of @{text "init_rhs"}, the equitional system descrbing the formation of every
- equivalent class inside @{text "CS"} is given by the following @{text "eqs(CS)"}.
- *}
-
-definition "eqs CS \<equiv> {(X, init_rhs CS X) | X. X \<in> CS}"
-(************ arden's lemma variation ********************)
-
-text {*
- The following @{text "items_of rhs X"} returns all @{text "X"}-items in @{text "rhs"}.
- *}
-definition
- "items_of rhs X \<equiv> {Trn X r | r. (Trn X r) \<in> rhs}"
-
-text {*
- The following @{text "rexp_of rhs X"} combines all regular expressions in @{text "X"}-items
- using @{text "ALT"} to form a single regular expression.
- It will be used later to implement @{text "arden_variate"} and @{text "rhs_subst"}.
- *}
-
-definition
- "rexp_of rhs X \<equiv> folds ALT NULL ((snd o the_Trn) ` items_of rhs X)"
-
-text {*
- The following @{text "lam_of rhs"} returns all pure regular expression items in @{text "rhs"}.
- *}
-
-definition
- "lam_of rhs \<equiv> {Lam r | r. Lam r \<in> rhs}"
-
-text {*
- The following @{text "rexp_of_lam rhs"} combines pure regular expression items in @{text "rhs"}
- using @{text "ALT"} to form a single regular expression.
- When all variables inside @{text "rhs"} are eliminated, @{text "rexp_of_lam rhs"}
- is used to compute compute the regular expression corresponds to @{text "rhs"}.
- *}
-
-definition
- "rexp_of_lam rhs \<equiv> folds ALT NULL (the_r ` lam_of rhs)"
-
-text {*
- The following @{text "attach_rexp rexp' itm"} attach
- the regular expression @{text "rexp'"} to
- the right of right hand side item @{text "itm"}.
- *}
-
-fun attach_rexp :: "rexp \<Rightarrow> rhs_item \<Rightarrow> rhs_item"
-where
- "attach_rexp rexp' (Lam rexp) = Lam (SEQ rexp rexp')"
-| "attach_rexp rexp' (Trn X rexp) = Trn X (SEQ rexp rexp')"
-
-text {*
- The following @{text "append_rhs_rexp rhs rexp"} attaches
- @{text "rexp"} to every item in @{text "rhs"}.
- *}
-
-definition
- "append_rhs_rexp rhs rexp \<equiv> (attach_rexp rexp) ` rhs"
-
-text {*
- With the help of the two functions immediately above, Ardens'
- transformation on right hand side @{text "rhs"} is implemented
- by the following function @{text "arden_variate X rhs"}.
- After this transformation, the recursive occurent of @{text "X"}
- in @{text "rhs"} will be eliminated, while the
- string set defined by @{text "rhs"} is kept unchanged.
- *}
-definition
- "arden_variate X rhs \<equiv>
- append_rhs_rexp (rhs - items_of rhs X) (STAR (rexp_of rhs X))"
-
-
-(*********** substitution of ES *************)
-
-text {*
- Suppose the equation defining @{text "X"} is $X = xrhs$,
- the purpose of @{text "rhs_subst"} is to substitute all occurences of @{text "X"} in
- @{text "rhs"} by @{text "xrhs"}.
- A litte thought may reveal that the final result
- should be: first append $(a_1 | a_2 | \ldots | a_n)$ to every item of @{text "xrhs"} and then
- union the result with all non-@{text "X"}-items of @{text "rhs"}.
- *}
-definition
- "rhs_subst rhs X xrhs \<equiv>
- (rhs - (items_of rhs X)) \<union> (append_rhs_rexp xrhs (rexp_of rhs X))"
-
-text {*
- Suppose the equation defining @{text "X"} is $X = xrhs$, the follwing
- @{text "eqs_subst ES X xrhs"} substitute @{text "xrhs"} into every equation
- of the equational system @{text "ES"}.
- *}
-
-definition
- "eqs_subst ES X xrhs \<equiv> {(Y, rhs_subst yrhs X xrhs) | Y yrhs. (Y, yrhs) \<in> ES}"
-
-text {*
- The computation of regular expressions for equivalent classes is accomplished
- using a iteration principle given by the following lemma.
- *}
-
-lemma wf_iter [rule_format]:
- fixes f
- assumes step: "\<And> e. \<lbrakk>P e; \<not> Q e\<rbrakk> \<Longrightarrow> (\<exists> e'. P e' \<and> (f(e'), f(e)) \<in> less_than)"
- shows pe: "P e \<longrightarrow> (\<exists> e'. P e' \<and> Q e')"
-proof(induct e rule: wf_induct
- [OF wf_inv_image[OF wf_less_than, where f = "f"]], clarify)
- fix x
- assume h [rule_format]:
- "\<forall>y. (y, x) \<in> inv_image less_than f \<longrightarrow> P y \<longrightarrow> (\<exists>e'. P e' \<and> Q e')"
- and px: "P x"
- show "\<exists>e'. P e' \<and> Q e'"
- proof(cases "Q x")
- assume "Q x" with px show ?thesis by blast
- next
- assume nq: "\<not> Q x"
- from step [OF px nq]
- obtain e' where pe': "P e'" and ltf: "(f e', f x) \<in> less_than" by auto
- show ?thesis
- proof(rule h)
- from ltf show "(e', x) \<in> inv_image less_than f"
- by (simp add:inv_image_def)
- next
- from pe' show "P e'" .
- qed
- qed
-qed
-
-text {*
- The @{text "P"} in lemma @{text "wf_iter"} is an invaiant kept throughout the iteration procedure.
- The particular invariant used to solve our problem is defined by function @{text "Inv(ES)"},
- an invariant over equal system @{text "ES"}.
- Every definition starting next till @{text "Inv"} stipulates a property to be satisfied by @{text "ES"}.
-*}
-
-text {*
- Every variable is defined at most onece in @{text "ES"}.
- *}
-definition
- "distinct_equas ES \<equiv>
- \<forall> X rhs rhs'. (X, rhs) \<in> ES \<and> (X, rhs') \<in> ES \<longrightarrow> rhs = rhs'"
-text {*
- Every equation in @{text "ES"} (represented by @{text "(X, rhs)"}) is valid, i.e. @{text "(X = L rhs)"}.
- *}
-definition
- "valid_eqns ES \<equiv> \<forall> X rhs. (X, rhs) \<in> ES \<longrightarrow> (X = L rhs)"
-
-text {*
- The following @{text "rhs_nonempty rhs"} requires regular expressions occuring in transitional
- items of @{text "rhs"} does not contain empty string. This is necessary for
- the application of Arden's transformation to @{text "rhs"}.
- *}
-definition
- "rhs_nonempty rhs \<equiv> (\<forall> Y r. Trn Y r \<in> rhs \<longrightarrow> [] \<notin> L r)"
-
-text {*
- The following @{text "ardenable ES"} requires that Arden's transformation is applicable
- to every equation of equational system @{text "ES"}.
- *}
-definition
- "ardenable ES \<equiv> \<forall> X rhs. (X, rhs) \<in> ES \<longrightarrow> rhs_nonempty rhs"
-
-(* The following non_empty seems useless. *)
-definition
- "non_empty ES \<equiv> \<forall> X rhs. (X, rhs) \<in> ES \<longrightarrow> X \<noteq> {}"
-
-text {*
- The following @{text "finite_rhs ES"} requires every equation in @{text "rhs"} be finite.
- *}
-definition
- "finite_rhs ES \<equiv> \<forall> X rhs. (X, rhs) \<in> ES \<longrightarrow> finite rhs"
-
-text {*
- The following @{text "classes_of rhs"} returns all variables (or equivalent classes)
- occuring in @{text "rhs"}.
- *}
-definition
- "classes_of rhs \<equiv> {X. \<exists> r. Trn X r \<in> rhs}"
-
-text {*
- The following @{text "lefts_of ES"} returns all variables
- defined by equational system @{text "ES"}.
- *}
-definition
- "lefts_of ES \<equiv> {Y | Y yrhs. (Y, yrhs) \<in> ES}"
-
-text {*
- The following @{text "self_contained ES"} requires that every
- variable occuring on the right hand side of equations is already defined by some
- equation in @{text "ES"}.
- *}
-definition
- "self_contained ES \<equiv> \<forall> (X, xrhs) \<in> ES. classes_of xrhs \<subseteq> lefts_of ES"
-
-
-text {*
- The invariant @{text "Inv(ES)"} is a conjunction of all the previously defined constaints.
- *}
-definition
- "Inv ES \<equiv> valid_eqns ES \<and> finite ES \<and> distinct_equas ES \<and> ardenable ES \<and>
- non_empty ES \<and> finite_rhs ES \<and> self_contained ES"
-
-subsection {* The proof of this direction *}
-
-subsubsection {* Basic properties *}
-
-text {*
- The following are some basic properties of the above definitions.
-*}
-
-lemma L_rhs_union_distrib:
- " L (A::rhs_item set) \<union> L B = L (A \<union> B)"
-by simp
-
-lemma finite_snd_Trn:
- assumes finite:"finite rhs"
- shows "finite {r\<^isub>2. Trn Y r\<^isub>2 \<in> rhs}" (is "finite ?B")
-proof-
- def rhs' \<equiv> "{e \<in> rhs. \<exists> r. e = Trn Y r}"
- have "?B = (snd o the_Trn) ` rhs'" using rhs'_def by (auto simp:image_def)
- moreover have "finite rhs'" using finite rhs'_def by auto
- ultimately show ?thesis by simp
-qed
-
-lemma rexp_of_empty:
- assumes finite:"finite rhs"
- and nonempty:"rhs_nonempty rhs"
- shows "[] \<notin> L (rexp_of rhs X)"
-using finite nonempty rhs_nonempty_def
-by (drule_tac finite_snd_Trn[where Y = X], auto simp:rexp_of_def items_of_def)
-
-lemma [intro!]:
- "P (Trn X r) \<Longrightarrow> (\<exists>a. (\<exists>r. a = Trn X r \<and> P a))" by auto
-
-lemma finite_items_of:
- "finite rhs \<Longrightarrow> finite (items_of rhs X)"
-by (auto simp:items_of_def intro:finite_subset)
-
-lemma lang_of_rexp_of:
- assumes finite:"finite rhs"
- shows "L (items_of rhs X) = X ;; (L (rexp_of rhs X))"
-proof -
- have "finite ((snd \<circ> the_Trn) ` items_of rhs X)" using finite_items_of[OF finite] by auto
- thus ?thesis
- apply (auto simp:rexp_of_def Seq_def items_of_def)
- apply (rule_tac x = s1 in exI, rule_tac x = s2 in exI, auto)
- by (rule_tac x= "Trn X r" in exI, auto simp:Seq_def)
-qed
-
-lemma rexp_of_lam_eq_lam_set:
- assumes finite: "finite rhs"
- shows "L (rexp_of_lam rhs) = L (lam_of rhs)"
-proof -
- have "finite (the_r ` {Lam r |r. Lam r \<in> rhs})" using finite
- by (rule_tac finite_imageI, auto intro:finite_subset)
- thus ?thesis by (auto simp:rexp_of_lam_def lam_of_def)
-qed
-
-lemma [simp]:
- " L (attach_rexp r xb) = L xb ;; L r"
-apply (cases xb, auto simp:Seq_def)
-by (rule_tac x = "s1 @ s1a" in exI, rule_tac x = s2a in exI,auto simp:Seq_def)
-
-lemma lang_of_append_rhs:
- "L (append_rhs_rexp rhs r) = L rhs ;; L r"
-apply (auto simp:append_rhs_rexp_def image_def)
-apply (auto simp:Seq_def)
-apply (rule_tac x = "L xb ;; L r" in exI, auto simp add:Seq_def)
-by (rule_tac x = "attach_rexp r xb" in exI, auto simp:Seq_def)
-
-lemma classes_of_union_distrib:
- "classes_of A \<union> classes_of B = classes_of (A \<union> B)"
-by (auto simp add:classes_of_def)
-
-lemma lefts_of_union_distrib:
- "lefts_of A \<union> lefts_of B = lefts_of (A \<union> B)"
-by (auto simp:lefts_of_def)
-
-
-subsubsection {* Intialization *}
-
-text {*
- The following several lemmas until @{text "init_ES_satisfy_Inv"} shows that
- the initial equational system satisfies invariant @{text "Inv"}.
- *}
-
-lemma defined_by_str:
- "\<lbrakk>s \<in> X; X \<in> UNIV // (\<approx>Lang)\<rbrakk> \<Longrightarrow> X = (\<approx>Lang) `` {s}"
-by (auto simp:quotient_def Image_def str_eq_rel_def)
-
-lemma every_eqclass_has_transition:
- assumes has_str: "s @ [c] \<in> X"
- and in_CS: "X \<in> UNIV // (\<approx>Lang)"
- obtains Y where "Y \<in> UNIV // (\<approx>Lang)" and "Y ;; {[c]} \<subseteq> X" and "s \<in> Y"
-proof -
- def Y \<equiv> "(\<approx>Lang) `` {s}"
- have "Y \<in> UNIV // (\<approx>Lang)"
- unfolding Y_def quotient_def by auto
- moreover
- have "X = (\<approx>Lang) `` {s @ [c]}"
- using has_str in_CS defined_by_str by blast
- then have "Y ;; {[c]} \<subseteq> X"
- unfolding Y_def Image_def Seq_def
- unfolding str_eq_rel_def
- by clarsimp
- moreover
- have "s \<in> Y" unfolding Y_def
- unfolding Image_def str_eq_rel_def by simp
- ultimately show thesis by (blast intro: that)
-qed
-
-lemma l_eq_r_in_eqs:
- assumes X_in_eqs: "(X, xrhs) \<in> (eqs (UNIV // (\<approx>Lang)))"
- shows "X = L xrhs"
-proof
- show "X \<subseteq> L xrhs"
- proof
- fix x
- assume "(1)": "x \<in> X"
- show "x \<in> L xrhs"
- proof (cases "x = []")
- assume empty: "x = []"
- thus ?thesis using X_in_eqs "(1)"
- by (auto simp:eqs_def init_rhs_def)
- next
- assume not_empty: "x \<noteq> []"
- then obtain clist c where decom: "x = clist @ [c]"
- by (case_tac x rule:rev_cases, auto)
- have "X \<in> UNIV // (\<approx>Lang)" using X_in_eqs by (auto simp:eqs_def)
- then obtain Y
- where "Y \<in> UNIV // (\<approx>Lang)"
- and "Y ;; {[c]} \<subseteq> X"
- and "clist \<in> Y"
- using decom "(1)" every_eqclass_has_transition by blast
- hence
- "x \<in> L {Trn Y (CHAR c)| Y c. Y \<in> UNIV // (\<approx>Lang) \<and> Y ;; {[c]} \<subseteq> X}"
- using "(1)" decom
- by (simp, rule_tac x = "Trn Y (CHAR c)" in exI, simp add:Seq_def)
- thus ?thesis using X_in_eqs "(1)"
- by (simp add:eqs_def init_rhs_def)
- qed
- qed
-next
- show "L xrhs \<subseteq> X" using X_in_eqs
- by (auto simp:eqs_def init_rhs_def)
-qed
-
-lemma finite_init_rhs:
- assumes finite: "finite CS"
- shows "finite (init_rhs CS X)"
-proof-
- have "finite {Trn Y (CHAR c) |Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X}" (is "finite ?A")
- proof -
- def S \<equiv> "{(Y, c)| Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X}"
- def h \<equiv> "\<lambda> (Y, c). Trn Y (CHAR c)"
- have "finite (CS \<times> (UNIV::char set))" using finite by auto
- hence "finite S" using S_def
- by (rule_tac B = "CS \<times> UNIV" in finite_subset, auto)
- moreover have "?A = h ` S" by (auto simp: S_def h_def image_def)
- ultimately show ?thesis
- by auto
- qed
- thus ?thesis by (simp add:init_rhs_def)
-qed
-
-lemma init_ES_satisfy_Inv:
- assumes finite_CS: "finite (UNIV // (\<approx>Lang))"
- shows "Inv (eqs (UNIV // (\<approx>Lang)))"
-proof -
- have "finite (eqs (UNIV // (\<approx>Lang)))" using finite_CS
- by (simp add:eqs_def)
- moreover have "distinct_equas (eqs (UNIV // (\<approx>Lang)))"
- by (simp add:distinct_equas_def eqs_def)
- moreover have "ardenable (eqs (UNIV // (\<approx>Lang)))"
- by (auto simp add:ardenable_def eqs_def init_rhs_def rhs_nonempty_def del:L_rhs.simps)
- moreover have "valid_eqns (eqs (UNIV // (\<approx>Lang)))"
- using l_eq_r_in_eqs by (simp add:valid_eqns_def)
- moreover have "non_empty (eqs (UNIV // (\<approx>Lang)))"
- by (auto simp:non_empty_def eqs_def quotient_def Image_def str_eq_rel_def)
- moreover have "finite_rhs (eqs (UNIV // (\<approx>Lang)))"
- using finite_init_rhs[OF finite_CS]
- by (auto simp:finite_rhs_def eqs_def)
- moreover have "self_contained (eqs (UNIV // (\<approx>Lang)))"
- by (auto simp:self_contained_def eqs_def init_rhs_def classes_of_def lefts_of_def)
- ultimately show ?thesis by (simp add:Inv_def)
-qed
-
-subsubsection {*
- Interation step
- *}
-
-text {*
- From this point until @{text "iteration_step"}, it is proved
- that there exists iteration steps which keep @{text "Inv(ES)"} while
- decreasing the size of @{text "ES"}.
- *}
-lemma arden_variate_keeps_eq:
- assumes l_eq_r: "X = L rhs"
- and not_empty: "[] \<notin> L (rexp_of rhs X)"
- and finite: "finite rhs"
- shows "X = L (arden_variate X rhs)"
-proof -
- def A \<equiv> "L (rexp_of rhs X)"
- def b \<equiv> "rhs - items_of rhs X"
- def B \<equiv> "L b"
- have "X = B ;; A\<star>"
- proof-
- have "rhs = items_of rhs X \<union> b" by (auto simp:b_def items_of_def)
- hence "L rhs = L(items_of rhs X \<union> b)" by simp
- hence "L rhs = L(items_of rhs X) \<union> B" by (simp only:L_rhs_union_distrib B_def)
- with lang_of_rexp_of
- have "L rhs = X ;; A \<union> B " using finite by (simp only:B_def b_def A_def)
- thus ?thesis
- using l_eq_r not_empty
- apply (drule_tac B = B and X = X in ardens_revised)
- by (auto simp:A_def simp del:L_rhs.simps)
- qed
- moreover have "L (arden_variate X rhs) = (B ;; A\<star>)" (is "?L = ?R")
- by (simp only:arden_variate_def L_rhs_union_distrib lang_of_append_rhs
- B_def A_def b_def L_rexp.simps seq_union_distrib)
- ultimately show ?thesis by simp
-qed
-
-lemma append_keeps_finite:
- "finite rhs \<Longrightarrow> finite (append_rhs_rexp rhs r)"
-by (auto simp:append_rhs_rexp_def)
-
-lemma arden_variate_keeps_finite:
- "finite rhs \<Longrightarrow> finite (arden_variate X rhs)"
-by (auto simp:arden_variate_def append_keeps_finite)
-
-lemma append_keeps_nonempty:
- "rhs_nonempty rhs \<Longrightarrow> rhs_nonempty (append_rhs_rexp rhs r)"
-apply (auto simp:rhs_nonempty_def append_rhs_rexp_def)
-by (case_tac x, auto simp:Seq_def)
-
-lemma nonempty_set_sub:
- "rhs_nonempty rhs \<Longrightarrow> rhs_nonempty (rhs - A)"
-by (auto simp:rhs_nonempty_def)
-
-lemma nonempty_set_union:
- "\<lbrakk>rhs_nonempty rhs; rhs_nonempty rhs'\<rbrakk> \<Longrightarrow> rhs_nonempty (rhs \<union> rhs')"
-by (auto simp:rhs_nonempty_def)
-
-lemma arden_variate_keeps_nonempty:
- "rhs_nonempty rhs \<Longrightarrow> rhs_nonempty (arden_variate X rhs)"
-by (simp only:arden_variate_def append_keeps_nonempty nonempty_set_sub)
-
-
-lemma rhs_subst_keeps_nonempty:
- "\<lbrakk>rhs_nonempty rhs; rhs_nonempty xrhs\<rbrakk> \<Longrightarrow> rhs_nonempty (rhs_subst rhs X xrhs)"
-by (simp only:rhs_subst_def append_keeps_nonempty nonempty_set_union nonempty_set_sub)
-
-lemma rhs_subst_keeps_eq:
- assumes substor: "X = L xrhs"
- and finite: "finite rhs"
- shows "L (rhs_subst rhs X xrhs) = L rhs" (is "?Left = ?Right")
-proof-
- def A \<equiv> "L (rhs - items_of rhs X)"
- have "?Left = A \<union> L (append_rhs_rexp xrhs (rexp_of rhs X))"
- by (simp only:rhs_subst_def L_rhs_union_distrib A_def)
- moreover have "?Right = A \<union> L (items_of rhs X)"
- proof-
- have "rhs = (rhs - items_of rhs X) \<union> (items_of rhs X)" by (auto simp:items_of_def)
- thus ?thesis by (simp only:L_rhs_union_distrib A_def)
- qed
- moreover have "L (append_rhs_rexp xrhs (rexp_of rhs X)) = L (items_of rhs X)"
- using finite substor by (simp only:lang_of_append_rhs lang_of_rexp_of)
- ultimately show ?thesis by simp
-qed
-
-lemma rhs_subst_keeps_finite_rhs:
- "\<lbrakk>finite rhs; finite yrhs\<rbrakk> \<Longrightarrow> finite (rhs_subst rhs Y yrhs)"
-by (auto simp:rhs_subst_def append_keeps_finite)
-
-lemma eqs_subst_keeps_finite:
- assumes finite:"finite (ES:: (string set \<times> rhs_item set) set)"
- shows "finite (eqs_subst ES Y yrhs)"
-proof -
- have "finite {(Ya, rhs_subst yrhsa Y yrhs) |Ya yrhsa. (Ya, yrhsa) \<in> ES}"
- (is "finite ?A")
- proof-
- def eqns' \<equiv> "{((Ya::string set), yrhsa)| Ya yrhsa. (Ya, yrhsa) \<in> ES}"
- def h \<equiv> "\<lambda> ((Ya::string set), yrhsa). (Ya, rhs_subst yrhsa Y yrhs)"
- have "finite (h ` eqns')" using finite h_def eqns'_def by auto
- moreover have "?A = h ` eqns'" by (auto simp:h_def eqns'_def)
- ultimately show ?thesis by auto
- qed
- thus ?thesis by (simp add:eqs_subst_def)
-qed
-
-lemma eqs_subst_keeps_finite_rhs:
- "\<lbrakk>finite_rhs ES; finite yrhs\<rbrakk> \<Longrightarrow> finite_rhs (eqs_subst ES Y yrhs)"
-by (auto intro:rhs_subst_keeps_finite_rhs simp add:eqs_subst_def finite_rhs_def)
-
-lemma append_rhs_keeps_cls:
- "classes_of (append_rhs_rexp rhs r) = classes_of rhs"
-apply (auto simp:classes_of_def append_rhs_rexp_def)
-apply (case_tac xa, auto simp:image_def)
-by (rule_tac x = "SEQ ra r" in exI, rule_tac x = "Trn x ra" in bexI, simp+)
-
-lemma arden_variate_removes_cl:
- "classes_of (arden_variate Y yrhs) = classes_of yrhs - {Y}"
-apply (simp add:arden_variate_def append_rhs_keeps_cls items_of_def)
-by (auto simp:classes_of_def)
-
-lemma lefts_of_keeps_cls:
- "lefts_of (eqs_subst ES Y yrhs) = lefts_of ES"
-by (auto simp:lefts_of_def eqs_subst_def)
-
-lemma rhs_subst_updates_cls:
- "X \<notin> classes_of xrhs \<Longrightarrow>
- classes_of (rhs_subst rhs X xrhs) = classes_of rhs \<union> classes_of xrhs - {X}"
-apply (simp only:rhs_subst_def append_rhs_keeps_cls
- classes_of_union_distrib[THEN sym])
-by (auto simp:classes_of_def items_of_def)
-
-lemma eqs_subst_keeps_self_contained:
- fixes Y
- assumes sc: "self_contained (ES \<union> {(Y, yrhs)})" (is "self_contained ?A")
- shows "self_contained (eqs_subst ES Y (arden_variate Y yrhs))"
- (is "self_contained ?B")
-proof-
- { fix X xrhs'
- assume "(X, xrhs') \<in> ?B"
- then obtain xrhs
- where xrhs_xrhs': "xrhs' = rhs_subst xrhs Y (arden_variate Y yrhs)"
- and X_in: "(X, xrhs) \<in> ES" by (simp add:eqs_subst_def, blast)
- have "classes_of xrhs' \<subseteq> lefts_of ?B"
- proof-
- have "lefts_of ?B = lefts_of ES" by (auto simp add:lefts_of_def eqs_subst_def)
- moreover have "classes_of xrhs' \<subseteq> lefts_of ES"
- proof-
- have "classes_of xrhs' \<subseteq>
- classes_of xrhs \<union> classes_of (arden_variate Y yrhs) - {Y}"
- proof-
- have "Y \<notin> classes_of (arden_variate Y yrhs)"
- using arden_variate_removes_cl by simp
- thus ?thesis using xrhs_xrhs' by (auto simp:rhs_subst_updates_cls)
- qed
- moreover have "classes_of xrhs \<subseteq> lefts_of ES \<union> {Y}" using X_in sc
- apply (simp only:self_contained_def lefts_of_union_distrib[THEN sym])
- by (drule_tac x = "(X, xrhs)" in bspec, auto simp:lefts_of_def)
- moreover have "classes_of (arden_variate Y yrhs) \<subseteq> lefts_of ES \<union> {Y}"
- using sc
- by (auto simp add:arden_variate_removes_cl self_contained_def lefts_of_def)
- ultimately show ?thesis by auto
- qed
- ultimately show ?thesis by simp
- qed
- } thus ?thesis by (auto simp only:eqs_subst_def self_contained_def)
-qed
-
-lemma eqs_subst_satisfy_Inv:
- assumes Inv_ES: "Inv (ES \<union> {(Y, yrhs)})"
- shows "Inv (eqs_subst ES Y (arden_variate Y yrhs))"
-proof -
- have finite_yrhs: "finite yrhs"
- using Inv_ES by (auto simp:Inv_def finite_rhs_def)
- have nonempty_yrhs: "rhs_nonempty yrhs"
- using Inv_ES by (auto simp:Inv_def ardenable_def)
- have Y_eq_yrhs: "Y = L yrhs"
- using Inv_ES by (simp only:Inv_def valid_eqns_def, blast)
- have "distinct_equas (eqs_subst ES Y (arden_variate Y yrhs))"
- using Inv_ES
- by (auto simp:distinct_equas_def eqs_subst_def Inv_def)
- moreover have "finite (eqs_subst ES Y (arden_variate Y yrhs))"
- using Inv_ES by (simp add:Inv_def eqs_subst_keeps_finite)
- moreover have "finite_rhs (eqs_subst ES Y (arden_variate Y yrhs))"
- proof-
- have "finite_rhs ES" using Inv_ES
- by (simp add:Inv_def finite_rhs_def)
- moreover have "finite (arden_variate Y yrhs)"
- proof -
- have "finite yrhs" using Inv_ES
- by (auto simp:Inv_def finite_rhs_def)
- thus ?thesis using arden_variate_keeps_finite by simp
- qed
- ultimately show ?thesis
- by (simp add:eqs_subst_keeps_finite_rhs)
- qed
- moreover have "ardenable (eqs_subst ES Y (arden_variate Y yrhs))"
- proof -
- { fix X rhs
- assume "(X, rhs) \<in> ES"
- hence "rhs_nonempty rhs" using prems Inv_ES
- by (simp add:Inv_def ardenable_def)
- with nonempty_yrhs
- have "rhs_nonempty (rhs_subst rhs Y (arden_variate Y yrhs))"
- by (simp add:nonempty_yrhs
- rhs_subst_keeps_nonempty arden_variate_keeps_nonempty)
- } thus ?thesis by (auto simp add:ardenable_def eqs_subst_def)
- qed
- moreover have "valid_eqns (eqs_subst ES Y (arden_variate Y yrhs))"
- proof-
- have "Y = L (arden_variate Y yrhs)"
- using Y_eq_yrhs Inv_ES finite_yrhs nonempty_yrhs
- by (rule_tac arden_variate_keeps_eq, (simp add:rexp_of_empty)+)
- thus ?thesis using Inv_ES
- by (clarsimp simp add:valid_eqns_def
- eqs_subst_def rhs_subst_keeps_eq Inv_def finite_rhs_def
- simp del:L_rhs.simps)
- qed
- moreover have
- non_empty_subst: "non_empty (eqs_subst ES Y (arden_variate Y yrhs))"
- using Inv_ES by (auto simp:Inv_def non_empty_def eqs_subst_def)
- moreover
- have self_subst: "self_contained (eqs_subst ES Y (arden_variate Y yrhs))"
- using Inv_ES eqs_subst_keeps_self_contained by (simp add:Inv_def)
- ultimately show ?thesis using Inv_ES by (simp add:Inv_def)
-qed
-
-lemma eqs_subst_card_le:
- assumes finite: "finite (ES::(string set \<times> rhs_item set) set)"
- shows "card (eqs_subst ES Y yrhs) <= card ES"
-proof-
- def f \<equiv> "\<lambda> x. ((fst x)::string set, rhs_subst (snd x) Y yrhs)"
- have "eqs_subst ES Y yrhs = f ` ES"
- apply (auto simp:eqs_subst_def f_def image_def)
- by (rule_tac x = "(Ya, yrhsa)" in bexI, simp+)
- thus ?thesis using finite by (auto intro:card_image_le)
-qed
-
-lemma eqs_subst_cls_remains:
- "(X, xrhs) \<in> ES \<Longrightarrow> \<exists> xrhs'. (X, xrhs') \<in> (eqs_subst ES Y yrhs)"
-by (auto simp:eqs_subst_def)
-
-lemma card_noteq_1_has_more:
- assumes card:"card S \<noteq> 1"
- and e_in: "e \<in> S"
- and finite: "finite S"
- obtains e' where "e' \<in> S \<and> e \<noteq> e'"
-proof-
- have "card (S - {e}) > 0"
- proof -
- have "card S > 1" using card e_in finite
- by (case_tac "card S", auto)
- thus ?thesis using finite e_in by auto
- qed
- hence "S - {e} \<noteq> {}" using finite by (rule_tac notI, simp)
- thus "(\<And>e'. e' \<in> S \<and> e \<noteq> e' \<Longrightarrow> thesis) \<Longrightarrow> thesis" by auto
-qed
-
-lemma iteration_step:
- assumes Inv_ES: "Inv ES"
- and X_in_ES: "(X, xrhs) \<in> ES"
- and not_T: "card ES \<noteq> 1"
- shows "\<exists> ES'. (Inv ES' \<and> (\<exists> xrhs'.(X, xrhs') \<in> ES')) \<and>
- (card ES', card ES) \<in> less_than" (is "\<exists> ES'. ?P ES'")
-proof -
- have finite_ES: "finite ES" using Inv_ES by (simp add:Inv_def)
- then obtain Y yrhs
- where Y_in_ES: "(Y, yrhs) \<in> ES" and not_eq: "(X, xrhs) \<noteq> (Y, yrhs)"
- using not_T X_in_ES by (drule_tac card_noteq_1_has_more, auto)
- def ES' == "ES - {(Y, yrhs)}"
- let ?ES'' = "eqs_subst ES' Y (arden_variate Y yrhs)"
- have "?P ?ES''"
- proof -
- have "Inv ?ES''" using Y_in_ES Inv_ES
- by (rule_tac eqs_subst_satisfy_Inv, simp add:ES'_def insert_absorb)
- moreover have "\<exists>xrhs'. (X, xrhs') \<in> ?ES''" using not_eq X_in_ES
- by (rule_tac ES = ES' in eqs_subst_cls_remains, auto simp add:ES'_def)
- moreover have "(card ?ES'', card ES) \<in> less_than"
- proof -
- have "finite ES'" using finite_ES ES'_def by auto
- moreover have "card ES' < card ES" using finite_ES Y_in_ES
- by (auto simp:ES'_def card_gt_0_iff intro:diff_Suc_less)
- ultimately show ?thesis
- by (auto dest:eqs_subst_card_le elim:le_less_trans)
- qed
- ultimately show ?thesis by simp
- qed
- thus ?thesis by blast
-qed
-
-subsubsection {*
- Conclusion of the proof
- *}
-
-text {*
- From this point until @{text "hard_direction"}, the hard direction is proved
- through a simple application of the iteration principle.
-*}
-
-lemma iteration_conc:
- assumes history: "Inv ES"
- and X_in_ES: "\<exists> xrhs. (X, xrhs) \<in> ES"
- shows
- "\<exists> ES'. (Inv ES' \<and> (\<exists> xrhs'. (X, xrhs') \<in> ES')) \<and> card ES' = 1"
- (is "\<exists> ES'. ?P ES'")
-proof (cases "card ES = 1")
- case True
- thus ?thesis using history X_in_ES
- by blast
-next
- case False
- thus ?thesis using history iteration_step X_in_ES
- by (rule_tac f = card in wf_iter, auto)
-qed
-
-lemma last_cl_exists_rexp:
- assumes ES_single: "ES = {(X, xrhs)}"
- and Inv_ES: "Inv ES"
- shows "\<exists> (r::rexp). L r = X" (is "\<exists> r. ?P r")
-proof-
- let ?A = "arden_variate X xrhs"
- have "?P (rexp_of_lam ?A)"
- proof -
- have "L (rexp_of_lam ?A) = L (lam_of ?A)"
- proof(rule rexp_of_lam_eq_lam_set)
- show "finite (arden_variate X xrhs)" using Inv_ES ES_single
- by (rule_tac arden_variate_keeps_finite,
- auto simp add:Inv_def finite_rhs_def)
- qed
- also have "\<dots> = L ?A"
- proof-
- have "lam_of ?A = ?A"
- proof-
- have "classes_of ?A = {}" using Inv_ES ES_single
- by (simp add:arden_variate_removes_cl
- self_contained_def Inv_def lefts_of_def)
- thus ?thesis
- by (auto simp only:lam_of_def classes_of_def, case_tac x, auto)
- qed
- thus ?thesis by simp
- qed
- also have "\<dots> = X"
- proof(rule arden_variate_keeps_eq [THEN sym])
- show "X = L xrhs" using Inv_ES ES_single
- by (auto simp only:Inv_def valid_eqns_def)
- next
- from Inv_ES ES_single show "[] \<notin> L (rexp_of xrhs X)"
- by(simp add:Inv_def ardenable_def rexp_of_empty finite_rhs_def)
- next
- from Inv_ES ES_single show "finite xrhs"
- by (simp add:Inv_def finite_rhs_def)
- qed
- finally show ?thesis by simp
- qed
- thus ?thesis by auto
-qed
-
-lemma every_eqcl_has_reg:
- assumes finite_CS: "finite (UNIV // (\<approx>Lang))"
- and X_in_CS: "X \<in> (UNIV // (\<approx>Lang))"
- shows "\<exists> (reg::rexp). L reg = X" (is "\<exists> r. ?E r")
-proof -
- from X_in_CS have "\<exists> xrhs. (X, xrhs) \<in> (eqs (UNIV // (\<approx>Lang)))"
- by (auto simp:eqs_def init_rhs_def)
- then obtain ES xrhs where Inv_ES: "Inv ES"
- and X_in_ES: "(X, xrhs) \<in> ES"
- and card_ES: "card ES = 1"
- using finite_CS X_in_CS init_ES_satisfy_Inv iteration_conc
- by blast
- hence ES_single_equa: "ES = {(X, xrhs)}"
- by (auto simp:Inv_def dest!:card_Suc_Diff1 simp:card_eq_0_iff)
- thus ?thesis using Inv_ES
- by (rule last_cl_exists_rexp)
-qed
-
-lemma finals_in_partitions:
- "finals Lang \<subseteq> (UNIV // (\<approx>Lang))"
- by (auto simp:finals_def quotient_def)
-
-theorem hard_direction:
- assumes finite_CS: "finite (UNIV // (\<approx>Lang))"
- shows "\<exists> (reg::rexp). Lang = L reg"
-proof -
- have "\<forall> X \<in> (UNIV // (\<approx>Lang)). \<exists> (reg::rexp). X = L reg"
- using finite_CS every_eqcl_has_reg by blast
- then obtain f
- where f_prop: "\<forall> X \<in> (UNIV // (\<approx>Lang)). X = L ((f X)::rexp)"
- by (auto dest:bchoice)
- def rs \<equiv> "f ` (finals Lang)"
- have "Lang = \<Union> (finals Lang)" using lang_is_union_of_finals by auto
- also have "\<dots> = L (folds ALT NULL rs)"
- proof -
- have "finite rs"
- proof -
- have "finite (finals Lang)"
- using finite_CS finals_in_partitions[of "Lang"]
- by (erule_tac finite_subset, simp)
- thus ?thesis using rs_def by auto
- qed
- thus ?thesis
- using f_prop rs_def finals_in_partitions[of "Lang"] by auto
- qed
- finally show ?thesis by blast
-qed
-
-
-
-section {* Direction: @{text "regular language \<Rightarrow> finite partitions"} *}
+section {* Direction: @{text "regular language \<Rightarrow>finite partition"} *}
subsection {* The scheme for this direction *}
@@ -1142,83 +18,148 @@
"x \<approx>Lang y \<equiv> (x, y) \<in> (\<approx>Lang)"
text {*
- The very basic scheme to show the finiteness of the partion generated by a
- language @{text "Lang"} is by attaching tags to every string. The set of
- tags are carefully choosen to make it finite. If it can be proved that
- strings with the same tag are equivlent with respect @{text "Lang"}, then
- the partition given rise by @{text "Lang"} must be finite. The reason for
- this is a lemma in standard library (@{text "finite_imageD"}), which says:
- if the image of an injective function on a set @{text "A"} is finite, then
- @{text "A"} is finite. It can be shown that the function obtained by
- lifting @{text "tag"} to the level of equivalence classes (i.e. @{text "((op
- `) tag)"}) is injective (by lemma @{text "tag_image_injI"}) and the image of
- this function is finite (with the help of lemma @{text
- "finite_tag_imageI"}). This argument is formalized by the following lemma
- @{text "tag_finite_imageD"}.
+ The very basic scheme to show the finiteness of the partion generated by a language @{text "Lang"}
+ is by attaching a tag to every string. The set of tags are carfully choosen to be finite so that
+ the range of tagging function is finite. If it can be proved that strings with the same tag
+ are equivlent with respect @{text "Lang"}, then the partition given rise by @{text "Lang"} must be finite.
+ The detailed argjument for this is formalized by the following lemma @{text "tag_finite_imageD"}.
+ The basic idea is using lemma @{thm [source] "finite_imageD"}
+ from standard library:
+ \[
+ @{thm "finite_imageD" [no_vars]}
+ \]
+ which says: if the image of injective function @{text "f"} over set @{text "A"} is
+ finite, then @{text "A"} must be finte.
+ *}
- {\bf
- Theorems @{text "tag_image_injI"} and @{text
- "finite_tag_imageI"} do not exist. Can this comment be deleted?
- \marginpar{\bf COMMENT}
- }
-*}
+(*
+
+(* I am trying to reduce the following proof to even simpler principles. But not yet succeed. *)
+definition
+ f_eq_rel ("\<cong>_")
+where
+ "\<cong>(f::'a \<Rightarrow> 'b) = {(x, y) | x y. f x = f y}"
+
+thm finite.induct
+
+lemma finite_range_image: "finite (range f) \<Longrightarrow> finite (f ` A)"
+ by (rule_tac B = "{y. \<exists>x. y = f x}" in finite_subset, auto simp:image_def)
+
+lemma "equiv UNIV (\<cong>f)"
+ by (auto simp:equiv_def f_eq_rel_def refl_on_def sym_def trans_def)
+
+lemma
+ assumes rng_fnt: "finite (range tag)"
+ shows "finite (UNIV // (\<cong>tag))"
+proof -
+ let "?f" = "op ` tag" and ?A = "(UNIV // (\<cong>tag))"
+ show ?thesis
+ proof (rule_tac f = "?f" and A = ?A in finite_imageD)
+ -- {*
+ The finiteness of @{text "f"}-image is a simple consequence of assumption @{text "rng_fnt"}:
+ *}
+ show "finite (?f ` ?A)"
+ proof -
+ have "\<forall> X. ?f X \<in> (Pow (range tag))" by (auto simp:image_def Pow_def)
+ moreover from rng_fnt have "finite (Pow (range tag))" by simp
+ ultimately have "finite (range ?f)"
+ by (auto simp only:image_def intro:finite_subset)
+ from finite_range_image [OF this] show ?thesis .
+ qed
+ next
+ -- {*
+ The injectivity of @{text "f"}-image is a consequence of the definition of @{text "\<cong>tag"}
+ *}
+ show "inj_on ?f ?A"
+ proof-
+ { fix X Y
+ assume X_in: "X \<in> ?A"
+ and Y_in: "Y \<in> ?A"
+ and tag_eq: "?f X = ?f Y"
+ have "X = Y"
+ proof -
+ from X_in Y_in tag_eq
+ obtain x y where x_in: "x \<in> X" and y_in: "y \<in> Y" and eq_tg: "tag x = tag y"
+ unfolding quotient_def Image_def str_eq_rel_def str_eq_def image_def f_eq_rel_def
+ apply simp by blast
+ with X_in Y_in show ?thesis
+ by (auto simp:quotient_def str_eq_rel_def str_eq_def f_eq_rel_def)
+ qed
+ } thus ?thesis unfolding inj_on_def by auto
+ qed
+ qed
+qed
+
+*)
+
+lemma finite_range_image: "finite (range f) \<Longrightarrow> finite (f ` A)"
+ by (rule_tac B = "{y. \<exists>x. y = f x}" in finite_subset, auto simp:image_def)
lemma tag_finite_imageD:
- fixes L1::"lang"
- assumes str_inj: "\<And> m n. tag m = tag n \<Longrightarrow> m \<approx>L1 n"
- and range: "finite (range tag)"
- shows "finite (UNIV // \<approx>L1)"
-proof (rule_tac f = "(op `) tag" in finite_imageD)
- show "finite (op ` tag ` UNIV // \<approx>L1)" using range
- apply (rule_tac B = "Pow (tag ` UNIV)" in finite_subset)
- by (auto simp add:image_def Pow_def)
-next
- show "inj_on (op ` tag) (UNIV // \<approx>L1)"
- proof-
- { fix X Y
- assume X_in: "X \<in> UNIV // \<approx>L1"
- and Y_in: "Y \<in> UNIV // \<approx>L1"
- and tag_eq: "tag ` X = tag ` Y"
- then obtain x y where "x \<in> X" and "y \<in> Y" and "tag x = tag y"
- unfolding quotient_def Image_def str_eq_rel_def str_eq_def image_def
- apply simp by blast
- with X_in Y_in str_inj[of x y]
- have "X = Y" by (auto simp:quotient_def str_eq_rel_def str_eq_def)
- } thus ?thesis unfolding inj_on_def by auto
+ fixes tag
+ assumes rng_fnt: "finite (range tag)"
+ -- {* Suppose the rang of tagging fucntion @{text "tag"} is finite. *}
+ and same_tag_eqvt: "\<And> m n. tag m = tag (n::string) \<Longrightarrow> m \<approx>lang n"
+ -- {* And strings with same tag are equivalent *}
+ shows "finite (UNIV // (\<approx>lang))"
+ -- {* Then the partition generated by @{text "(\<approx>lang)"} is finite. *}
+proof -
+ -- {* The particular @{text "f"} and @{text "A"} used in @{thm [source] "finite_imageD"} are:*}
+ let "?f" = "op ` tag" and ?A = "(UNIV // \<approx>lang)"
+ show ?thesis
+ proof (rule_tac f = "?f" and A = ?A in finite_imageD)
+ -- {*
+ The finiteness of @{text "f"}-image is a simple consequence of assumption @{text "rng_fnt"}:
+ *}
+ show "finite (?f ` ?A)"
+ proof -
+ have "\<forall> X. ?f X \<in> (Pow (range tag))" by (auto simp:image_def Pow_def)
+ moreover from rng_fnt have "finite (Pow (range tag))" by simp
+ ultimately have "finite (range ?f)"
+ by (auto simp only:image_def intro:finite_subset)
+ from finite_range_image [OF this] show ?thesis .
+ qed
+ next
+ -- {*
+ The injectivity of @{text "f"} is the consequence of assumption @{text "same_tag_eqvt"}:
+ *}
+ show "inj_on ?f ?A"
+ proof-
+ { fix X Y
+ assume X_in: "X \<in> ?A"
+ and Y_in: "Y \<in> ?A"
+ and tag_eq: "?f X = ?f Y"
+ have "X = Y"
+ proof -
+ from X_in Y_in tag_eq
+ obtain x y where x_in: "x \<in> X" and y_in: "y \<in> Y" and eq_tg: "tag x = tag y"
+ unfolding quotient_def Image_def str_eq_rel_def str_eq_def image_def
+ apply simp by blast
+ from same_tag_eqvt [OF eq_tg] have "x \<approx>lang y" .
+ with X_in Y_in x_in y_in
+ show ?thesis by (auto simp:quotient_def str_eq_rel_def str_eq_def)
+ qed
+ } thus ?thesis unfolding inj_on_def by auto
+ qed
qed
qed
subsection {* Lemmas for basic cases *}
text {*
- The the final result of this direction is in @{text "rexp_imp_finite"},
- which is an induction on the structure of regular expressions. There is one
- case for each regular expression operator. For basic operators such as
- @{const NULL}, @{const EMPTY}, @{const CHAR}, the finiteness of their
- language partition can be established directly with no need of tagging.
- This section contains several technical lemma for these base cases.
-
- The inductive cases involve operators @{const ALT}, @{const SEQ} and @{const
- STAR}. Tagging functions need to be defined individually for each of
- them. There will be one dedicated section for each of these cases, and each
- section goes virtually the same way: gives definition of the tagging
- function and prove that strings with the same tag are equivalent.
-*}
+ The the final result of this direction is in @{text "easier_direction"}, which
+ is an induction on the structure of regular expressions. There is one case
+ for each regular expression operator. For basic operators such as @{text "NULL, EMPTY, CHAR c"},
+ the finiteness of their language partition can be established directly with no need
+ of taggiing. This section contains several technical lemma for these base cases.
-subsection {* The case for @{const "NULL"} *}
-
-lemma quot_null_eq:
- shows "(UNIV // \<approx>{}) = ({UNIV}::lang set)"
- unfolding quotient_def Image_def str_eq_rel_def by auto
-
-lemma quot_null_finiteI [intro]:
- shows "finite ((UNIV // \<approx>{})::lang set)"
-unfolding quot_null_eq by simp
-
-
-subsection {* The case for @{const "EMPTY"} *}
-
+ The inductive cases involve operators @{text "ALT, SEQ"} and @{text "STAR"}.
+ Tagging functions need to be defined individually for each of them. There will be one
+ dedicated section for each of these cases, and each section goes virtually the same way:
+ gives definition of the tagging function and prove that strings
+ with the same tag are equivalent.
+ *}
lemma quot_empty_subset:
"UNIV // (\<approx>{[]}) \<subseteq> {{[]}, UNIV - {[]}}"
@@ -1227,25 +168,18 @@
assume "x \<in> UNIV // \<approx>{[]}"
then obtain y where h: "x = {z. (y, z) \<in> \<approx>{[]}}"
unfolding quotient_def Image_def by blast
- show "x \<in> {{[]}, UNIV - {[]}}"
+ show "x \<in> {{[]}, UNIV - {[]}}"
proof (cases "y = []")
case True with h
- have "x = {[]}" by (auto simp: str_eq_rel_def)
+ have "x = {[]}" by (auto simp:str_eq_rel_def)
thus ?thesis by simp
next
case False with h
- have "x = UNIV - {[]}" by (auto simp: str_eq_rel_def)
+ have "x = UNIV - {[]}" by (auto simp:str_eq_rel_def)
thus ?thesis by simp
qed
qed
-lemma quot_empty_finiteI [intro]:
- shows "finite (UNIV // (\<approx>{[]}))"
-by (rule finite_subset[OF quot_empty_subset]) (simp)
-
-
-subsection {* The case for @{const "CHAR"} *}
-
lemma quot_char_subset:
"UNIV // (\<approx>{[c]}) \<subseteq> {{[]},{[c]}, UNIV - {[], [c]}}"
proof
@@ -1271,17 +205,17 @@
qed
qed
-lemma quot_char_finiteI [intro]:
- shows "finite (UNIV // (\<approx>{[c]}))"
-by (rule finite_subset[OF quot_char_subset]) (simp)
-
-
-subsection {* The case for @{const "SEQ"}*}
+subsection {* The case for @{text "SEQ"}*}
definition
- tag_str_SEQ :: "lang \<Rightarrow> lang \<Rightarrow> string \<Rightarrow> (lang \<times> lang set)"
-where
- "tag_str_SEQ L1 L2 = (\<lambda>x. (\<approx>L1 `` {x}, {(\<approx>L2 `` {x - xa}) | xa. xa \<le> x \<and> xa \<in> L1}))"
+ "tag_str_SEQ L\<^isub>1 L\<^isub>2 x \<equiv>
+ ((\<approx>L\<^isub>1) `` {x}, {(\<approx>L\<^isub>2) `` {x - xa}| xa. xa \<le> x \<and> xa \<in> L\<^isub>1})"
+
+lemma tag_str_seq_range_finite:
+ "\<lbrakk>finite (UNIV // \<approx>L\<^isub>1); finite (UNIV // \<approx>L\<^isub>2)\<rbrakk>
+ \<Longrightarrow> finite (range (tag_str_SEQ L\<^isub>1 L\<^isub>2))"
+apply (rule_tac B = "(UNIV // \<approx>L\<^isub>1) \<times> (Pow (UNIV // \<approx>L\<^isub>2))" in finite_subset)
+by (auto simp:tag_str_SEQ_def Image_def quotient_def split:if_splits)
lemma append_seq_elim:
assumes "x @ y \<in> L\<^isub>1 ;; L\<^isub>2"
@@ -1351,66 +285,40 @@
by (auto simp add: str_eq_def str_eq_rel_def)
qed
-lemma quot_seq_finiteI [intro]:
- fixes L1 L2::"lang"
- assumes fin1: "finite (UNIV // \<approx>L1)"
- and fin2: "finite (UNIV // \<approx>L2)"
- shows "finite (UNIV // \<approx>(L1 ;; L2))"
-proof (rule_tac tag = "tag_str_SEQ L1 L2" in tag_finite_imageD)
- show "\<And>x y. tag_str_SEQ L1 L2 x = tag_str_SEQ L1 L2 y \<Longrightarrow> x \<approx>(L1 ;; L2) y"
- by (rule tag_str_SEQ_injI)
-next
- have *: "finite ((UNIV // \<approx>L1) \<times> (Pow (UNIV // \<approx>L2)))"
- using fin1 fin2 by auto
- show "finite (range (tag_str_SEQ L1 L2))"
- unfolding tag_str_SEQ_def
- apply(rule finite_subset[OF _ *])
- unfolding quotient_def
- by auto
-qed
+lemma quot_seq_finiteI:
+ "\<lbrakk>finite (UNIV // \<approx>L\<^isub>1); finite (UNIV // \<approx>L\<^isub>2)\<rbrakk>
+ \<Longrightarrow> finite (UNIV // \<approx>(L\<^isub>1 ;; L\<^isub>2))"
+ apply (rule_tac tag = "tag_str_SEQ L\<^isub>1 L\<^isub>2" in tag_finite_imageD)
+ by (auto intro:tag_str_SEQ_injI elim:tag_str_seq_range_finite)
-
-subsection {* The case for @{const "ALT"} *}
+subsection {* The case for @{text "ALT"} *}
definition
- tag_str_ALT :: "lang \<Rightarrow> lang \<Rightarrow> string \<Rightarrow> (lang \<times> lang)"
-where
- "tag_str_ALT L1 L2 = (\<lambda>x. (\<approx>L1 `` {x}, \<approx>L2 `` {x}))"
-
+ "tag_str_ALT L\<^isub>1 L\<^isub>2 (x::string) \<equiv> ((\<approx>L\<^isub>1) `` {x}, (\<approx>L\<^isub>2) `` {x})"
-lemma quot_union_finiteI [intro]:
- fixes L1 L2::"lang"
- assumes finite1: "finite (UNIV // \<approx>L1)"
- and finite2: "finite (UNIV // \<approx>L2)"
- shows "finite (UNIV // \<approx>(L1 \<union> L2))"
-proof (rule_tac tag = "tag_str_ALT L1 L2" in tag_finite_imageD)
- show "\<And>x y. tag_str_ALT L1 L2 x = tag_str_ALT L1 L2 y \<Longrightarrow> x \<approx>(L1 \<union> L2) y"
- unfolding tag_str_ALT_def
- unfolding str_eq_def
- unfolding Image_def
- unfolding str_eq_rel_def
- by auto
+lemma quot_union_finiteI:
+ assumes finite1: "finite (UNIV // \<approx>(L\<^isub>1::string set))"
+ and finite2: "finite (UNIV // \<approx>L\<^isub>2)"
+ shows "finite (UNIV // \<approx>(L\<^isub>1 \<union> L\<^isub>2))"
+proof (rule_tac tag = "tag_str_ALT L\<^isub>1 L\<^isub>2" in tag_finite_imageD)
+ show "\<And>m n. tag_str_ALT L\<^isub>1 L\<^isub>2 m = tag_str_ALT L\<^isub>1 L\<^isub>2 n \<Longrightarrow> m \<approx>(L\<^isub>1 \<union> L\<^isub>2) n"
+ unfolding tag_str_ALT_def str_eq_def Image_def str_eq_rel_def by auto
next
- have *: "finite ((UNIV // \<approx>L1) \<times> (UNIV // \<approx>L2))"
- using finite1 finite2 by auto
- show "finite (range (tag_str_ALT L1 L2))"
- unfolding tag_str_ALT_def
- apply(rule finite_subset[OF _ *])
- unfolding quotient_def
- by auto
+ show "finite (range (tag_str_ALT L\<^isub>1 L\<^isub>2))" using finite1 finite2
+ apply (rule_tac B = "(UNIV // \<approx>L\<^isub>1) \<times> (UNIV // \<approx>L\<^isub>2)" in finite_subset)
+ by (auto simp:tag_str_ALT_def Image_def quotient_def)
qed
-
-subsection {* The case for @{const "STAR"} *}
+subsection {*
+ The case for @{text "STAR"}
+ *}
text {*
This turned out to be the trickiest case.
*} (* I will make some illustrations for it. *)
definition
- tag_str_STAR :: "lang \<Rightarrow> string \<Rightarrow> lang set"
-where
- "tag_str_STAR L1 = (\<lambda>x. {\<approx>L1 `` {x - xa} | xa. xa < x \<and> xa \<in> L1\<star>})"
+ "tag_str_STAR L\<^isub>1 x \<equiv> {(\<approx>L\<^isub>1) `` {x - xa} | xa. xa < x \<and> xa \<in> L\<^isub>1\<star>}"
lemma finite_set_has_max: "\<lbrakk>finite A; A \<noteq> {}\<rbrakk> \<Longrightarrow>
(\<exists> max \<in> A. \<forall> a \<in> A. f a <= (f max :: nat))"
@@ -1442,6 +350,13 @@
apply (subgoal_tac "{xa. xa < xs @ [x]} = {xa. xa < xs} \<union> {xs}")
by (auto simp:strict_prefix_def)
+
+lemma tag_str_star_range_finite:
+ "finite (UNIV // \<approx>L\<^isub>1) \<Longrightarrow> finite (range (tag_str_STAR L\<^isub>1))"
+apply (rule_tac B = "Pow (UNIV // \<approx>L\<^isub>1)" in finite_subset)
+by (auto simp:tag_str_STAR_def Image_def
+ quotient_def split:if_splits)
+
lemma tag_str_STAR_injI:
"tag_str_STAR L\<^isub>1 m = tag_str_STAR L\<^isub>1 n \<Longrightarrow> m \<approx>(L\<^isub>1\<star>) n"
proof-
@@ -1526,32 +441,51 @@
by (auto simp add:str_eq_def str_eq_rel_def)
qed
+lemma quot_star_finiteI:
+ "finite (UNIV // \<approx>L\<^isub>1) \<Longrightarrow> finite (UNIV // \<approx>(L\<^isub>1\<star>))"
+ apply (rule_tac tag = "tag_str_STAR L\<^isub>1" in tag_finite_imageD)
+ by (auto intro:tag_str_STAR_injI elim:tag_str_star_range_finite)
-lemma quot_star_finiteI [intro]:
- fixes L1::"lang"
- assumes finite1: "finite (UNIV // \<approx>L1)"
- shows "finite (UNIV // \<approx>(L1\<star>))"
-proof (rule_tac tag = "tag_str_STAR L1" in tag_finite_imageD)
- show "\<And>x y. tag_str_STAR L1 x = tag_str_STAR L1 y \<Longrightarrow> x \<approx>(L1\<star>) y"
- by (rule tag_str_STAR_injI)
+subsection {*
+ The main lemma
+ *}
+
+lemma easier_directio\<nu>:
+ "Lang = L (r::rexp) \<Longrightarrow> finite (UNIV // (\<approx>Lang))"
+proof (induct arbitrary:Lang rule:rexp.induct)
+ case NULL
+ have "UNIV // (\<approx>{}) \<subseteq> {UNIV} "
+ by (auto simp:quotient_def str_eq_rel_def str_eq_def)
+ with prems show "?case" by (auto intro:finite_subset)
+next
+ case EMPTY
+ have "UNIV // (\<approx>{[]}) \<subseteq> {{[]}, UNIV - {[]}}"
+ by (rule quot_empty_subset)
+ with prems show ?case by (auto intro:finite_subset)
next
- have *: "finite (Pow (UNIV // \<approx>L1))"
- using finite1 by auto
- show "finite (range (tag_str_STAR L1))"
- unfolding tag_str_STAR_def
- apply(rule finite_subset[OF _ *])
- unfolding quotient_def
- by auto
-qed
-
-
-subsection {* The main lemma *}
-
-lemma rexp_imp_finite:
- fixes r::"rexp"
- shows "finite (UNIV // \<approx>(L r))"
-by (induct r) (auto)
-
+ case (CHAR c)
+ have "UNIV // (\<approx>{[c]}) \<subseteq> {{[]},{[c]}, UNIV - {[], [c]}}"
+ by (rule quot_char_subset)
+ with prems show ?case by (auto intro:finite_subset)
+next
+ case (SEQ r\<^isub>1 r\<^isub>2)
+ have "\<lbrakk>finite (UNIV // \<approx>(L r\<^isub>1)); finite (UNIV // \<approx>(L r\<^isub>2))\<rbrakk>
+ \<Longrightarrow> finite (UNIV // \<approx>(L r\<^isub>1 ;; L r\<^isub>2))"
+ by (erule quot_seq_finiteI, simp)
+ with prems show ?case by simp
+next
+ case (ALT r\<^isub>1 r\<^isub>2)
+ have "\<lbrakk>finite (UNIV // \<approx>(L r\<^isub>1)); finite (UNIV // \<approx>(L r\<^isub>2))\<rbrakk>
+ \<Longrightarrow> finite (UNIV // \<approx>(L r\<^isub>1 \<union> L r\<^isub>2))"
+ by (erule quot_union_finiteI, simp)
+ with prems show ?case by simp
+next
+ case (STAR r)
+ have "finite (UNIV // \<approx>(L r))
+ \<Longrightarrow> finite (UNIV // \<approx>((L r)\<star>))"
+ by (erule quot_star_finiteI)
+ with prems show ?case by simp
+qed
end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Myhill_1.thy Thu Jan 27 12:35:06 2011 +0000
@@ -0,0 +1,1118 @@
+theory Myhill_1
+ imports Main List_Prefix Prefix_subtract
+begin
+
+(*
+text {*
+ \begin{figure}
+ \centering
+ \scalebox{0.95}{
+ \begin{tikzpicture}[->,>=latex,shorten >=1pt,auto,node distance=1.2cm, semithick]
+ \node[state,initial] (n1) {$1$};
+ \node[state,accepting] (n2) [right = 10em of n1] {$2$};
+
+ \path (n1) edge [bend left] node {$0$} (n2)
+ (n1) edge [loop above] node{$1$} (n1)
+ (n2) edge [loop above] node{$0$} (n2)
+ (n2) edge [bend left] node {$1$} (n1)
+ ;
+ \end{tikzpicture}}
+ \caption{An example automaton (or partition)}\label{fig:example_automata}
+ \end{figure}
+*}
+
+*)
+
+
+section {* Preliminary definitions *}
+
+text {* Sequential composition of two languages @{text "L1"} and @{text "L2"} *}
+definition Seq :: "string set \<Rightarrow> string set \<Rightarrow> string set" ("_ ;; _" [100,100] 100)
+where
+ "L1 ;; L2 = {s1 @ s2 | s1 s2. s1 \<in> L1 \<and> s2 \<in> L2}"
+
+text {* Transitive closure of language @{text "L"}. *}
+inductive_set
+ Star :: "string set \<Rightarrow> string set" ("_\<star>" [101] 102)
+ for L :: "string set"
+where
+ start[intro]: "[] \<in> L\<star>"
+| step[intro]: "\<lbrakk>s1 \<in> L; s2 \<in> L\<star>\<rbrakk> \<Longrightarrow> s1@s2 \<in> L\<star>"
+
+text {* Some properties of operator @{text ";;"}.*}
+
+lemma seq_union_distrib:
+ "(A \<union> B) ;; C = (A ;; C) \<union> (B ;; C)"
+by (auto simp:Seq_def)
+
+lemma seq_intro:
+ "\<lbrakk>x \<in> A; y \<in> B\<rbrakk> \<Longrightarrow> x @ y \<in> A ;; B "
+by (auto simp:Seq_def)
+
+lemma seq_assoc:
+ "(A ;; B) ;; C = A ;; (B ;; C)"
+apply(auto simp:Seq_def)
+apply blast
+by (metis append_assoc)
+
+lemma star_intro1[rule_format]: "x \<in> lang\<star> \<Longrightarrow> \<forall> y. y \<in> lang\<star> \<longrightarrow> x @ y \<in> lang\<star>"
+by (erule Star.induct, auto)
+
+lemma star_intro2: "y \<in> lang \<Longrightarrow> y \<in> lang\<star>"
+by (drule step[of y lang "[]"], auto simp:start)
+
+lemma star_intro3[rule_format]:
+ "x \<in> lang\<star> \<Longrightarrow> \<forall>y . y \<in> lang \<longrightarrow> x @ y \<in> lang\<star>"
+by (erule Star.induct, auto intro:star_intro2)
+
+lemma star_decom:
+ "\<lbrakk>x \<in> lang\<star>; x \<noteq> []\<rbrakk> \<Longrightarrow>(\<exists> a b. x = a @ b \<and> a \<noteq> [] \<and> a \<in> lang \<and> b \<in> lang\<star>)"
+by (induct x rule: Star.induct, simp, blast)
+
+lemma star_decom':
+ "\<lbrakk>x \<in> lang\<star>; x \<noteq> []\<rbrakk> \<Longrightarrow> \<exists>a b. x = a @ b \<and> a \<in> lang\<star> \<and> b \<in> lang"
+apply (induct x rule:Star.induct, simp)
+apply (case_tac "s2 = []")
+apply (rule_tac x = "[]" in exI, rule_tac x = s1 in exI, simp add:start)
+apply (simp, (erule exE| erule conjE)+)
+by (rule_tac x = "s1 @ a" in exI, rule_tac x = b in exI, simp add:step)
+
+text {* Ardens lemma expressed at the level of language, rather than the level of regular expression. *}
+
+theorem ardens_revised:
+ assumes nemp: "[] \<notin> A"
+ shows "(X = X ;; A \<union> B) \<longleftrightarrow> (X = B ;; A\<star>)"
+proof
+ assume eq: "X = B ;; A\<star>"
+ have "A\<star> = {[]} \<union> A\<star> ;; A"
+ by (auto simp:Seq_def star_intro3 star_decom')
+ then have "B ;; A\<star> = B ;; ({[]} \<union> A\<star> ;; A)"
+ unfolding Seq_def by simp
+ also have "\<dots> = B \<union> B ;; (A\<star> ;; A)"
+ unfolding Seq_def by auto
+ also have "\<dots> = B \<union> (B ;; A\<star>) ;; A"
+ by (simp only:seq_assoc)
+ finally show "X = X ;; A \<union> B"
+ using eq by blast
+next
+ assume eq': "X = X ;; A \<union> B"
+ hence c1': "\<And> x. x \<in> B \<Longrightarrow> x \<in> X"
+ and c2': "\<And> x y. \<lbrakk>x \<in> X; y \<in> A\<rbrakk> \<Longrightarrow> x @ y \<in> X"
+ using Seq_def by auto
+ show "X = B ;; A\<star>"
+ proof
+ show "B ;; A\<star> \<subseteq> X"
+ proof-
+ { fix x y
+ have "\<lbrakk>y \<in> A\<star>; x \<in> X\<rbrakk> \<Longrightarrow> x @ y \<in> X "
+ apply (induct arbitrary:x rule:Star.induct, simp)
+ by (auto simp only:append_assoc[THEN sym] dest:c2')
+ } thus ?thesis using c1' by (auto simp:Seq_def)
+ qed
+ next
+ show "X \<subseteq> B ;; A\<star>"
+ proof-
+ { fix x
+ have "x \<in> X \<Longrightarrow> x \<in> B ;; A\<star>"
+ proof (induct x taking:length rule:measure_induct)
+ fix z
+ assume hyps:
+ "\<forall>y. length y < length z \<longrightarrow> y \<in> X \<longrightarrow> y \<in> B ;; A\<star>"
+ and z_in: "z \<in> X"
+ show "z \<in> B ;; A\<star>"
+ proof (cases "z \<in> B")
+ case True thus ?thesis by (auto simp:Seq_def start)
+ next
+ case False hence "z \<in> X ;; A" using eq' z_in by auto
+ then obtain za zb where za_in: "za \<in> X"
+ and zab: "z = za @ zb \<and> zb \<in> A" and zbne: "zb \<noteq> []"
+ using nemp unfolding Seq_def by blast
+ from zbne zab have "length za < length z" by auto
+ with za_in hyps have "za \<in> B ;; A\<star>" by blast
+ hence "za @ zb \<in> B ;; A\<star>" using zab
+ by (clarsimp simp:Seq_def, blast dest:star_intro3)
+ thus ?thesis using zab by simp
+ qed
+ qed
+ } thus ?thesis by blast
+ qed
+ qed
+qed
+
+
+text {* The syntax of regular expressions is defined by the datatype @{text "rexp"}. *}
+datatype rexp =
+ NULL
+| EMPTY
+| CHAR char
+| SEQ rexp rexp
+| ALT rexp rexp
+| STAR rexp
+
+
+text {*
+ The following @{text "L"} is an overloaded operator, where @{text "L(x)"} evaluates to
+ the language represented by the syntactic object @{text "x"}.
+*}
+consts L:: "'a \<Rightarrow> string set"
+
+
+text {*
+ The @{text "L(rexp)"} for regular expression @{text "rexp"} is defined by the
+ following overloading function @{text "L_rexp"}.
+*}
+overloading L_rexp \<equiv> "L:: rexp \<Rightarrow> string set"
+begin
+fun
+ L_rexp :: "rexp \<Rightarrow> string set"
+where
+ "L_rexp (NULL) = {}"
+ | "L_rexp (EMPTY) = {[]}"
+ | "L_rexp (CHAR c) = {[c]}"
+ | "L_rexp (SEQ r1 r2) = (L_rexp r1) ;; (L_rexp r2)"
+ | "L_rexp (ALT r1 r2) = (L_rexp r1) \<union> (L_rexp r2)"
+ | "L_rexp (STAR r) = (L_rexp r)\<star>"
+end
+
+text {*
+ To obtain equational system out of finite set of equivalent classes, a fold operation
+ on finite set @{text "folds"} is defined. The use of @{text "SOME"} makes @{text "fold"}
+ more robust than the @{text "fold"} in Isabelle library. The expression @{text "folds f"}
+ makes sense when @{text "f"} is not @{text "associative"} and @{text "commutitive"},
+ while @{text "fold f"} does not.
+*}
+
+definition
+ folds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"
+where
+ "folds f z S \<equiv> SOME x. fold_graph f z S x"
+
+text {*
+ The following lemma assures that the arbitrary choice made by the @{text "SOME"} in @{text "folds"}
+ does not affect the @{text "L"}-value of the resultant regular expression.
+ *}
+lemma folds_alt_simp [simp]:
+ "finite rs \<Longrightarrow> L (folds ALT NULL rs) = \<Union> (L ` rs)"
+apply (rule set_ext, simp add:folds_def)
+apply (rule someI2_ex, erule finite_imp_fold_graph)
+by (erule fold_graph.induct, auto)
+
+(* Just a technical lemma. *)
+lemma [simp]:
+ shows "(x, y) \<in> {(x, y). P x y} \<longleftrightarrow> P x y"
+by simp
+
+text {*
+ @{text "\<approx>L"} is an equivalent class defined by language @{text "Lang"}.
+*}
+definition
+ str_eq_rel ("\<approx>_")
+where
+ "\<approx>Lang \<equiv> {(x, y). (\<forall>z. x @ z \<in> Lang \<longleftrightarrow> y @ z \<in> Lang)}"
+
+text {*
+ Among equivlant clases of @{text "\<approx>Lang"}, the set @{text "finals(Lang)"} singles out
+ those which contains strings from @{text "Lang"}.
+*}
+
+definition
+ "finals Lang \<equiv> {\<approx>Lang `` {x} | x . x \<in> Lang}"
+
+text {*
+ The following lemma show the relationshipt between @{text "finals(Lang)"} and @{text "Lang"}.
+*}
+lemma lang_is_union_of_finals:
+ "Lang = \<Union> finals(Lang)"
+proof
+ show "Lang \<subseteq> \<Union> (finals Lang)"
+ proof
+ fix x
+ assume "x \<in> Lang"
+ thus "x \<in> \<Union> (finals Lang)"
+ apply (simp add:finals_def, rule_tac x = "(\<approx>Lang) `` {x}" in exI)
+ by (auto simp:Image_def str_eq_rel_def)
+ qed
+next
+ show "\<Union> (finals Lang) \<subseteq> Lang"
+ apply (clarsimp simp:finals_def str_eq_rel_def)
+ by (drule_tac x = "[]" in spec, auto)
+qed
+
+section {* Direction @{text "finite partition \<Rightarrow> regular language"}*}
+
+text {*
+ The relationship between equivalent classes can be described by an
+ equational system.
+ For example, in equational system \eqref{example_eqns}, $X_0, X_1$ are equivalent
+ classes. The first equation says every string in $X_0$ is obtained either by
+ appending one $b$ to a string in $X_0$ or by appending one $a$ to a string in
+ $X_1$ or just be an empty string (represented by the regular expression $\lambda$). Similary,
+ the second equation tells how the strings inside $X_1$ are composed.
+ \begin{equation}\label{example_eqns}
+ \begin{aligned}
+ X_0 & = X_0 b + X_1 a + \lambda \\
+ X_1 & = X_0 a + X_1 b
+ \end{aligned}
+ \end{equation}
+ The summands on the right hand side is represented by the following data type
+ @{text "rhs_item"}, mnemonic for 'right hand side item'.
+ Generally, there are two kinds of right hand side items, one kind corresponds to
+ pure regular expressions, like the $\lambda$ in \eqref{example_eqns}, the other kind corresponds to
+ transitions from one one equivalent class to another, like the $X_0 b, X_1 a$ etc.
+ *}
+
+datatype rhs_item =
+ Lam "rexp" (* Lambda *)
+ | Trn "(string set)" "rexp" (* Transition *)
+
+text {*
+ In this formalization, pure regular expressions like $\lambda$ is
+ repsented by @{text "Lam(EMPTY)"}, while transitions like $X_0 a$ is represented by $Trn~X_0~(CHAR~a)$.
+ *}
+
+text {*
+ The functions @{text "the_r"} and @{text "the_Trn"} are used to extract
+ subcomponents from right hand side items.
+ *}
+
+fun the_r :: "rhs_item \<Rightarrow> rexp"
+where "the_r (Lam r) = r"
+
+fun the_Trn:: "rhs_item \<Rightarrow> (string set \<times> rexp)"
+where "the_Trn (Trn Y r) = (Y, r)"
+
+text {*
+ Every right hand side item @{text "itm"} defines a string set given
+ @{text "L(itm)"}, defined as:
+*}
+overloading L_rhs_e \<equiv> "L:: rhs_item \<Rightarrow> string set"
+begin
+ fun L_rhs_e:: "rhs_item \<Rightarrow> string set"
+ where
+ "L_rhs_e (Lam r) = L r" |
+ "L_rhs_e (Trn X r) = X ;; L r"
+end
+
+text {*
+ The right hand side of every equation is represented by a set of
+ items. The string set defined by such a set @{text "itms"} is given
+ by @{text "L(itms)"}, defined as:
+*}
+
+overloading L_rhs \<equiv> "L:: rhs_item set \<Rightarrow> string set"
+begin
+ fun L_rhs:: "rhs_item set \<Rightarrow> string set"
+ where "L_rhs rhs = \<Union> (L ` rhs)"
+end
+
+text {*
+ Given a set of equivalent classses @{text "CS"} and one equivalent class @{text "X"} among
+ @{text "CS"}, the term @{text "init_rhs CS X"} is used to extract the right hand side of
+ the equation describing the formation of @{text "X"}. The definition of @{text "init_rhs"}
+ is:
+ *}
+
+definition
+ "init_rhs CS X \<equiv>
+ if ([] \<in> X) then
+ {Lam(EMPTY)} \<union> {Trn Y (CHAR c) | Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X}
+ else
+ {Trn Y (CHAR c)| Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X}"
+
+text {*
+ In the definition of @{text "init_rhs"}, the term
+ @{text "{Trn Y (CHAR c)| Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X}"} appearing on both branches
+ describes the formation of strings in @{text "X"} out of transitions, while
+ the term @{text "{Lam(EMPTY)}"} describes the empty string which is intrinsically contained in
+ @{text "X"} rather than by transition. This @{text "{Lam(EMPTY)}"} corresponds to
+ the $\lambda$ in \eqref{example_eqns}.
+
+ With the help of @{text "init_rhs"}, the equitional system descrbing the formation of every
+ equivalent class inside @{text "CS"} is given by the following @{text "eqs(CS)"}.
+ *}
+
+definition "eqs CS \<equiv> {(X, init_rhs CS X) | X. X \<in> CS}"
+(************ arden's lemma variation ********************)
+
+text {*
+ The following @{text "items_of rhs X"} returns all @{text "X"}-items in @{text "rhs"}.
+ *}
+definition
+ "items_of rhs X \<equiv> {Trn X r | r. (Trn X r) \<in> rhs}"
+
+text {*
+ The following @{text "rexp_of rhs X"} combines all regular expressions in @{text "X"}-items
+ using @{text "ALT"} to form a single regular expression.
+ It will be used later to implement @{text "arden_variate"} and @{text "rhs_subst"}.
+ *}
+
+definition
+ "rexp_of rhs X \<equiv> folds ALT NULL ((snd o the_Trn) ` items_of rhs X)"
+
+text {*
+ The following @{text "lam_of rhs"} returns all pure regular expression items in @{text "rhs"}.
+ *}
+
+definition
+ "lam_of rhs \<equiv> {Lam r | r. Lam r \<in> rhs}"
+
+text {*
+ The following @{text "rexp_of_lam rhs"} combines pure regular expression items in @{text "rhs"}
+ using @{text "ALT"} to form a single regular expression.
+ When all variables inside @{text "rhs"} are eliminated, @{text "rexp_of_lam rhs"}
+ is used to compute compute the regular expression corresponds to @{text "rhs"}.
+ *}
+
+definition
+ "rexp_of_lam rhs \<equiv> folds ALT NULL (the_r ` lam_of rhs)"
+
+text {*
+ The following @{text "attach_rexp rexp' itm"} attach
+ the regular expression @{text "rexp'"} to
+ the right of right hand side item @{text "itm"}.
+ *}
+
+fun attach_rexp :: "rexp \<Rightarrow> rhs_item \<Rightarrow> rhs_item"
+where
+ "attach_rexp rexp' (Lam rexp) = Lam (SEQ rexp rexp')"
+| "attach_rexp rexp' (Trn X rexp) = Trn X (SEQ rexp rexp')"
+
+text {*
+ The following @{text "append_rhs_rexp rhs rexp"} attaches
+ @{text "rexp"} to every item in @{text "rhs"}.
+ *}
+
+definition
+ "append_rhs_rexp rhs rexp \<equiv> (attach_rexp rexp) ` rhs"
+
+text {*
+ With the help of the two functions immediately above, Ardens'
+ transformation on right hand side @{text "rhs"} is implemented
+ by the following function @{text "arden_variate X rhs"}.
+ After this transformation, the recursive occurent of @{text "X"}
+ in @{text "rhs"} will be eliminated, while the
+ string set defined by @{text "rhs"} is kept unchanged.
+ *}
+definition
+ "arden_variate X rhs \<equiv>
+ append_rhs_rexp (rhs - items_of rhs X) (STAR (rexp_of rhs X))"
+
+
+(*********** substitution of ES *************)
+
+text {*
+ Suppose the equation defining @{text "X"} is $X = xrhs$,
+ the purpose of @{text "rhs_subst"} is to substitute all occurences of @{text "X"} in
+ @{text "rhs"} by @{text "xrhs"}.
+ A litte thought may reveal that the final result
+ should be: first append $(a_1 | a_2 | \ldots | a_n)$ to every item of @{text "xrhs"} and then
+ union the result with all non-@{text "X"}-items of @{text "rhs"}.
+ *}
+definition
+ "rhs_subst rhs X xrhs \<equiv>
+ (rhs - (items_of rhs X)) \<union> (append_rhs_rexp xrhs (rexp_of rhs X))"
+
+text {*
+ Suppose the equation defining @{text "X"} is $X = xrhs$, the follwing
+ @{text "eqs_subst ES X xrhs"} substitute @{text "xrhs"} into every equation
+ of the equational system @{text "ES"}.
+ *}
+
+definition
+ "eqs_subst ES X xrhs \<equiv> {(Y, rhs_subst yrhs X xrhs) | Y yrhs. (Y, yrhs) \<in> ES}"
+
+text {*
+ The computation of regular expressions for equivalent classes is accomplished
+ using a iteration principle given by the following lemma.
+ *}
+
+lemma wf_iter [rule_format]:
+ fixes f
+ assumes step: "\<And> e. \<lbrakk>P e; \<not> Q e\<rbrakk> \<Longrightarrow> (\<exists> e'. P e' \<and> (f(e'), f(e)) \<in> less_than)"
+ shows pe: "P e \<longrightarrow> (\<exists> e'. P e' \<and> Q e')"
+proof(induct e rule: wf_induct
+ [OF wf_inv_image[OF wf_less_than, where f = "f"]], clarify)
+ fix x
+ assume h [rule_format]:
+ "\<forall>y. (y, x) \<in> inv_image less_than f \<longrightarrow> P y \<longrightarrow> (\<exists>e'. P e' \<and> Q e')"
+ and px: "P x"
+ show "\<exists>e'. P e' \<and> Q e'"
+ proof(cases "Q x")
+ assume "Q x" with px show ?thesis by blast
+ next
+ assume nq: "\<not> Q x"
+ from step [OF px nq]
+ obtain e' where pe': "P e'" and ltf: "(f e', f x) \<in> less_than" by auto
+ show ?thesis
+ proof(rule h)
+ from ltf show "(e', x) \<in> inv_image less_than f"
+ by (simp add:inv_image_def)
+ next
+ from pe' show "P e'" .
+ qed
+ qed
+qed
+
+text {*
+ The @{text "P"} in lemma @{text "wf_iter"} is an invaiant kept throughout the iteration procedure.
+ The particular invariant used to solve our problem is defined by function @{text "Inv(ES)"},
+ an invariant over equal system @{text "ES"}.
+ Every definition starting next till @{text "Inv"} stipulates a property to be satisfied by @{text "ES"}.
+*}
+
+text {*
+ Every variable is defined at most onece in @{text "ES"}.
+ *}
+definition
+ "distinct_equas ES \<equiv>
+ \<forall> X rhs rhs'. (X, rhs) \<in> ES \<and> (X, rhs') \<in> ES \<longrightarrow> rhs = rhs'"
+text {*
+ Every equation in @{text "ES"} (represented by @{text "(X, rhs)"}) is valid, i.e. @{text "(X = L rhs)"}.
+ *}
+definition
+ "valid_eqns ES \<equiv> \<forall> X rhs. (X, rhs) \<in> ES \<longrightarrow> (X = L rhs)"
+
+text {*
+ The following @{text "rhs_nonempty rhs"} requires regular expressions occuring in transitional
+ items of @{text "rhs"} does not contain empty string. This is necessary for
+ the application of Arden's transformation to @{text "rhs"}.
+ *}
+definition
+ "rhs_nonempty rhs \<equiv> (\<forall> Y r. Trn Y r \<in> rhs \<longrightarrow> [] \<notin> L r)"
+
+text {*
+ The following @{text "ardenable ES"} requires that Arden's transformation is applicable
+ to every equation of equational system @{text "ES"}.
+ *}
+definition
+ "ardenable ES \<equiv> \<forall> X rhs. (X, rhs) \<in> ES \<longrightarrow> rhs_nonempty rhs"
+
+(* The following non_empty seems useless. *)
+definition
+ "non_empty ES \<equiv> \<forall> X rhs. (X, rhs) \<in> ES \<longrightarrow> X \<noteq> {}"
+
+text {*
+ The following @{text "finite_rhs ES"} requires every equation in @{text "rhs"} be finite.
+ *}
+definition
+ "finite_rhs ES \<equiv> \<forall> X rhs. (X, rhs) \<in> ES \<longrightarrow> finite rhs"
+
+text {*
+ The following @{text "classes_of rhs"} returns all variables (or equivalent classes)
+ occuring in @{text "rhs"}.
+ *}
+definition
+ "classes_of rhs \<equiv> {X. \<exists> r. Trn X r \<in> rhs}"
+
+text {*
+ The following @{text "lefts_of ES"} returns all variables
+ defined by equational system @{text "ES"}.
+ *}
+definition
+ "lefts_of ES \<equiv> {Y | Y yrhs. (Y, yrhs) \<in> ES}"
+
+text {*
+ The following @{text "self_contained ES"} requires that every
+ variable occuring on the right hand side of equations is already defined by some
+ equation in @{text "ES"}.
+ *}
+definition
+ "self_contained ES \<equiv> \<forall> (X, xrhs) \<in> ES. classes_of xrhs \<subseteq> lefts_of ES"
+
+
+text {*
+ The invariant @{text "Inv(ES)"} is a conjunction of all the previously defined constaints.
+ *}
+definition
+ "Inv ES \<equiv> valid_eqns ES \<and> finite ES \<and> distinct_equas ES \<and> ardenable ES \<and>
+ non_empty ES \<and> finite_rhs ES \<and> self_contained ES"
+
+subsection {* The proof of this direction *}
+
+subsubsection {* Basic properties *}
+
+text {*
+ The following are some basic properties of the above definitions.
+*}
+
+lemma L_rhs_union_distrib:
+ " L (A::rhs_item set) \<union> L B = L (A \<union> B)"
+by simp
+
+lemma finite_snd_Trn:
+ assumes finite:"finite rhs"
+ shows "finite {r\<^isub>2. Trn Y r\<^isub>2 \<in> rhs}" (is "finite ?B")
+proof-
+ def rhs' \<equiv> "{e \<in> rhs. \<exists> r. e = Trn Y r}"
+ have "?B = (snd o the_Trn) ` rhs'" using rhs'_def by (auto simp:image_def)
+ moreover have "finite rhs'" using finite rhs'_def by auto
+ ultimately show ?thesis by simp
+qed
+
+lemma rexp_of_empty:
+ assumes finite:"finite rhs"
+ and nonempty:"rhs_nonempty rhs"
+ shows "[] \<notin> L (rexp_of rhs X)"
+using finite nonempty rhs_nonempty_def
+by (drule_tac finite_snd_Trn[where Y = X], auto simp:rexp_of_def items_of_def)
+
+lemma [intro!]:
+ "P (Trn X r) \<Longrightarrow> (\<exists>a. (\<exists>r. a = Trn X r \<and> P a))" by auto
+
+lemma finite_items_of:
+ "finite rhs \<Longrightarrow> finite (items_of rhs X)"
+by (auto simp:items_of_def intro:finite_subset)
+
+lemma lang_of_rexp_of:
+ assumes finite:"finite rhs"
+ shows "L (items_of rhs X) = X ;; (L (rexp_of rhs X))"
+proof -
+ have "finite ((snd \<circ> the_Trn) ` items_of rhs X)" using finite_items_of[OF finite] by auto
+ thus ?thesis
+ apply (auto simp:rexp_of_def Seq_def items_of_def)
+ apply (rule_tac x = s1 in exI, rule_tac x = s2 in exI, auto)
+ by (rule_tac x= "Trn X r" in exI, auto simp:Seq_def)
+qed
+
+lemma rexp_of_lam_eq_lam_set:
+ assumes finite: "finite rhs"
+ shows "L (rexp_of_lam rhs) = L (lam_of rhs)"
+proof -
+ have "finite (the_r ` {Lam r |r. Lam r \<in> rhs})" using finite
+ by (rule_tac finite_imageI, auto intro:finite_subset)
+ thus ?thesis by (auto simp:rexp_of_lam_def lam_of_def)
+qed
+
+lemma [simp]:
+ " L (attach_rexp r xb) = L xb ;; L r"
+apply (cases xb, auto simp:Seq_def)
+by (rule_tac x = "s1 @ s1a" in exI, rule_tac x = s2a in exI,auto simp:Seq_def)
+
+lemma lang_of_append_rhs:
+ "L (append_rhs_rexp rhs r) = L rhs ;; L r"
+apply (auto simp:append_rhs_rexp_def image_def)
+apply (auto simp:Seq_def)
+apply (rule_tac x = "L xb ;; L r" in exI, auto simp add:Seq_def)
+by (rule_tac x = "attach_rexp r xb" in exI, auto simp:Seq_def)
+
+lemma classes_of_union_distrib:
+ "classes_of A \<union> classes_of B = classes_of (A \<union> B)"
+by (auto simp add:classes_of_def)
+
+lemma lefts_of_union_distrib:
+ "lefts_of A \<union> lefts_of B = lefts_of (A \<union> B)"
+by (auto simp:lefts_of_def)
+
+
+subsubsection {* Intialization *}
+
+text {*
+ The following several lemmas until @{text "init_ES_satisfy_Inv"} shows that
+ the initial equational system satisfies invariant @{text "Inv"}.
+ *}
+
+lemma defined_by_str:
+ "\<lbrakk>s \<in> X; X \<in> UNIV // (\<approx>Lang)\<rbrakk> \<Longrightarrow> X = (\<approx>Lang) `` {s}"
+by (auto simp:quotient_def Image_def str_eq_rel_def)
+
+lemma every_eqclass_has_transition:
+ assumes has_str: "s @ [c] \<in> X"
+ and in_CS: "X \<in> UNIV // (\<approx>Lang)"
+ obtains Y where "Y \<in> UNIV // (\<approx>Lang)" and "Y ;; {[c]} \<subseteq> X" and "s \<in> Y"
+proof -
+ def Y \<equiv> "(\<approx>Lang) `` {s}"
+ have "Y \<in> UNIV // (\<approx>Lang)"
+ unfolding Y_def quotient_def by auto
+ moreover
+ have "X = (\<approx>Lang) `` {s @ [c]}"
+ using has_str in_CS defined_by_str by blast
+ then have "Y ;; {[c]} \<subseteq> X"
+ unfolding Y_def Image_def Seq_def
+ unfolding str_eq_rel_def
+ by clarsimp
+ moreover
+ have "s \<in> Y" unfolding Y_def
+ unfolding Image_def str_eq_rel_def by simp
+ ultimately show thesis by (blast intro: that)
+qed
+
+lemma l_eq_r_in_eqs:
+ assumes X_in_eqs: "(X, xrhs) \<in> (eqs (UNIV // (\<approx>Lang)))"
+ shows "X = L xrhs"
+proof
+ show "X \<subseteq> L xrhs"
+ proof
+ fix x
+ assume "(1)": "x \<in> X"
+ show "x \<in> L xrhs"
+ proof (cases "x = []")
+ assume empty: "x = []"
+ thus ?thesis using X_in_eqs "(1)"
+ by (auto simp:eqs_def init_rhs_def)
+ next
+ assume not_empty: "x \<noteq> []"
+ then obtain clist c where decom: "x = clist @ [c]"
+ by (case_tac x rule:rev_cases, auto)
+ have "X \<in> UNIV // (\<approx>Lang)" using X_in_eqs by (auto simp:eqs_def)
+ then obtain Y
+ where "Y \<in> UNIV // (\<approx>Lang)"
+ and "Y ;; {[c]} \<subseteq> X"
+ and "clist \<in> Y"
+ using decom "(1)" every_eqclass_has_transition by blast
+ hence
+ "x \<in> L {Trn Y (CHAR c)| Y c. Y \<in> UNIV // (\<approx>Lang) \<and> Y ;; {[c]} \<subseteq> X}"
+ using "(1)" decom
+ by (simp, rule_tac x = "Trn Y (CHAR c)" in exI, simp add:Seq_def)
+ thus ?thesis using X_in_eqs "(1)"
+ by (simp add:eqs_def init_rhs_def)
+ qed
+ qed
+next
+ show "L xrhs \<subseteq> X" using X_in_eqs
+ by (auto simp:eqs_def init_rhs_def)
+qed
+
+lemma finite_init_rhs:
+ assumes finite: "finite CS"
+ shows "finite (init_rhs CS X)"
+proof-
+ have "finite {Trn Y (CHAR c) |Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X}" (is "finite ?A")
+ proof -
+ def S \<equiv> "{(Y, c)| Y c. Y \<in> CS \<and> Y ;; {[c]} \<subseteq> X}"
+ def h \<equiv> "\<lambda> (Y, c). Trn Y (CHAR c)"
+ have "finite (CS \<times> (UNIV::char set))" using finite by auto
+ hence "finite S" using S_def
+ by (rule_tac B = "CS \<times> UNIV" in finite_subset, auto)
+ moreover have "?A = h ` S" by (auto simp: S_def h_def image_def)
+ ultimately show ?thesis
+ by auto
+ qed
+ thus ?thesis by (simp add:init_rhs_def)
+qed
+
+lemma init_ES_satisfy_Inv:
+ assumes finite_CS: "finite (UNIV // (\<approx>Lang))"
+ shows "Inv (eqs (UNIV // (\<approx>Lang)))"
+proof -
+ have "finite (eqs (UNIV // (\<approx>Lang)))" using finite_CS
+ by (simp add:eqs_def)
+ moreover have "distinct_equas (eqs (UNIV // (\<approx>Lang)))"
+ by (simp add:distinct_equas_def eqs_def)
+ moreover have "ardenable (eqs (UNIV // (\<approx>Lang)))"
+ by (auto simp add:ardenable_def eqs_def init_rhs_def rhs_nonempty_def del:L_rhs.simps)
+ moreover have "valid_eqns (eqs (UNIV // (\<approx>Lang)))"
+ using l_eq_r_in_eqs by (simp add:valid_eqns_def)
+ moreover have "non_empty (eqs (UNIV // (\<approx>Lang)))"
+ by (auto simp:non_empty_def eqs_def quotient_def Image_def str_eq_rel_def)
+ moreover have "finite_rhs (eqs (UNIV // (\<approx>Lang)))"
+ using finite_init_rhs[OF finite_CS]
+ by (auto simp:finite_rhs_def eqs_def)
+ moreover have "self_contained (eqs (UNIV // (\<approx>Lang)))"
+ by (auto simp:self_contained_def eqs_def init_rhs_def classes_of_def lefts_of_def)
+ ultimately show ?thesis by (simp add:Inv_def)
+qed
+
+subsubsection {*
+ Interation step
+ *}
+
+text {*
+ From this point until @{text "iteration_step"}, it is proved
+ that there exists iteration steps which keep @{text "Inv(ES)"} while
+ decreasing the size of @{text "ES"}.
+ *}
+lemma arden_variate_keeps_eq:
+ assumes l_eq_r: "X = L rhs"
+ and not_empty: "[] \<notin> L (rexp_of rhs X)"
+ and finite: "finite rhs"
+ shows "X = L (arden_variate X rhs)"
+proof -
+ def A \<equiv> "L (rexp_of rhs X)"
+ def b \<equiv> "rhs - items_of rhs X"
+ def B \<equiv> "L b"
+ have "X = B ;; A\<star>"
+ proof-
+ have "rhs = items_of rhs X \<union> b" by (auto simp:b_def items_of_def)
+ hence "L rhs = L(items_of rhs X \<union> b)" by simp
+ hence "L rhs = L(items_of rhs X) \<union> B" by (simp only:L_rhs_union_distrib B_def)
+ with lang_of_rexp_of
+ have "L rhs = X ;; A \<union> B " using finite by (simp only:B_def b_def A_def)
+ thus ?thesis
+ using l_eq_r not_empty
+ apply (drule_tac B = B and X = X in ardens_revised)
+ by (auto simp:A_def simp del:L_rhs.simps)
+ qed
+ moreover have "L (arden_variate X rhs) = (B ;; A\<star>)" (is "?L = ?R")
+ by (simp only:arden_variate_def L_rhs_union_distrib lang_of_append_rhs
+ B_def A_def b_def L_rexp.simps seq_union_distrib)
+ ultimately show ?thesis by simp
+qed
+
+lemma append_keeps_finite:
+ "finite rhs \<Longrightarrow> finite (append_rhs_rexp rhs r)"
+by (auto simp:append_rhs_rexp_def)
+
+lemma arden_variate_keeps_finite:
+ "finite rhs \<Longrightarrow> finite (arden_variate X rhs)"
+by (auto simp:arden_variate_def append_keeps_finite)
+
+lemma append_keeps_nonempty:
+ "rhs_nonempty rhs \<Longrightarrow> rhs_nonempty (append_rhs_rexp rhs r)"
+apply (auto simp:rhs_nonempty_def append_rhs_rexp_def)
+by (case_tac x, auto simp:Seq_def)
+
+lemma nonempty_set_sub:
+ "rhs_nonempty rhs \<Longrightarrow> rhs_nonempty (rhs - A)"
+by (auto simp:rhs_nonempty_def)
+
+lemma nonempty_set_union:
+ "\<lbrakk>rhs_nonempty rhs; rhs_nonempty rhs'\<rbrakk> \<Longrightarrow> rhs_nonempty (rhs \<union> rhs')"
+by (auto simp:rhs_nonempty_def)
+
+lemma arden_variate_keeps_nonempty:
+ "rhs_nonempty rhs \<Longrightarrow> rhs_nonempty (arden_variate X rhs)"
+by (simp only:arden_variate_def append_keeps_nonempty nonempty_set_sub)
+
+
+lemma rhs_subst_keeps_nonempty:
+ "\<lbrakk>rhs_nonempty rhs; rhs_nonempty xrhs\<rbrakk> \<Longrightarrow> rhs_nonempty (rhs_subst rhs X xrhs)"
+by (simp only:rhs_subst_def append_keeps_nonempty nonempty_set_union nonempty_set_sub)
+
+lemma rhs_subst_keeps_eq:
+ assumes substor: "X = L xrhs"
+ and finite: "finite rhs"
+ shows "L (rhs_subst rhs X xrhs) = L rhs" (is "?Left = ?Right")
+proof-
+ def A \<equiv> "L (rhs - items_of rhs X)"
+ have "?Left = A \<union> L (append_rhs_rexp xrhs (rexp_of rhs X))"
+ by (simp only:rhs_subst_def L_rhs_union_distrib A_def)
+ moreover have "?Right = A \<union> L (items_of rhs X)"
+ proof-
+ have "rhs = (rhs - items_of rhs X) \<union> (items_of rhs X)" by (auto simp:items_of_def)
+ thus ?thesis by (simp only:L_rhs_union_distrib A_def)
+ qed
+ moreover have "L (append_rhs_rexp xrhs (rexp_of rhs X)) = L (items_of rhs X)"
+ using finite substor by (simp only:lang_of_append_rhs lang_of_rexp_of)
+ ultimately show ?thesis by simp
+qed
+
+lemma rhs_subst_keeps_finite_rhs:
+ "\<lbrakk>finite rhs; finite yrhs\<rbrakk> \<Longrightarrow> finite (rhs_subst rhs Y yrhs)"
+by (auto simp:rhs_subst_def append_keeps_finite)
+
+lemma eqs_subst_keeps_finite:
+ assumes finite:"finite (ES:: (string set \<times> rhs_item set) set)"
+ shows "finite (eqs_subst ES Y yrhs)"
+proof -
+ have "finite {(Ya, rhs_subst yrhsa Y yrhs) |Ya yrhsa. (Ya, yrhsa) \<in> ES}"
+ (is "finite ?A")
+ proof-
+ def eqns' \<equiv> "{((Ya::string set), yrhsa)| Ya yrhsa. (Ya, yrhsa) \<in> ES}"
+ def h \<equiv> "\<lambda> ((Ya::string set), yrhsa). (Ya, rhs_subst yrhsa Y yrhs)"
+ have "finite (h ` eqns')" using finite h_def eqns'_def by auto
+ moreover have "?A = h ` eqns'" by (auto simp:h_def eqns'_def)
+ ultimately show ?thesis by auto
+ qed
+ thus ?thesis by (simp add:eqs_subst_def)
+qed
+
+lemma eqs_subst_keeps_finite_rhs:
+ "\<lbrakk>finite_rhs ES; finite yrhs\<rbrakk> \<Longrightarrow> finite_rhs (eqs_subst ES Y yrhs)"
+by (auto intro:rhs_subst_keeps_finite_rhs simp add:eqs_subst_def finite_rhs_def)
+
+lemma append_rhs_keeps_cls:
+ "classes_of (append_rhs_rexp rhs r) = classes_of rhs"
+apply (auto simp:classes_of_def append_rhs_rexp_def)
+apply (case_tac xa, auto simp:image_def)
+by (rule_tac x = "SEQ ra r" in exI, rule_tac x = "Trn x ra" in bexI, simp+)
+
+lemma arden_variate_removes_cl:
+ "classes_of (arden_variate Y yrhs) = classes_of yrhs - {Y}"
+apply (simp add:arden_variate_def append_rhs_keeps_cls items_of_def)
+by (auto simp:classes_of_def)
+
+lemma lefts_of_keeps_cls:
+ "lefts_of (eqs_subst ES Y yrhs) = lefts_of ES"
+by (auto simp:lefts_of_def eqs_subst_def)
+
+lemma rhs_subst_updates_cls:
+ "X \<notin> classes_of xrhs \<Longrightarrow>
+ classes_of (rhs_subst rhs X xrhs) = classes_of rhs \<union> classes_of xrhs - {X}"
+apply (simp only:rhs_subst_def append_rhs_keeps_cls
+ classes_of_union_distrib[THEN sym])
+by (auto simp:classes_of_def items_of_def)
+
+lemma eqs_subst_keeps_self_contained:
+ fixes Y
+ assumes sc: "self_contained (ES \<union> {(Y, yrhs)})" (is "self_contained ?A")
+ shows "self_contained (eqs_subst ES Y (arden_variate Y yrhs))"
+ (is "self_contained ?B")
+proof-
+ { fix X xrhs'
+ assume "(X, xrhs') \<in> ?B"
+ then obtain xrhs
+ where xrhs_xrhs': "xrhs' = rhs_subst xrhs Y (arden_variate Y yrhs)"
+ and X_in: "(X, xrhs) \<in> ES" by (simp add:eqs_subst_def, blast)
+ have "classes_of xrhs' \<subseteq> lefts_of ?B"
+ proof-
+ have "lefts_of ?B = lefts_of ES" by (auto simp add:lefts_of_def eqs_subst_def)
+ moreover have "classes_of xrhs' \<subseteq> lefts_of ES"
+ proof-
+ have "classes_of xrhs' \<subseteq>
+ classes_of xrhs \<union> classes_of (arden_variate Y yrhs) - {Y}"
+ proof-
+ have "Y \<notin> classes_of (arden_variate Y yrhs)"
+ using arden_variate_removes_cl by simp
+ thus ?thesis using xrhs_xrhs' by (auto simp:rhs_subst_updates_cls)
+ qed
+ moreover have "classes_of xrhs \<subseteq> lefts_of ES \<union> {Y}" using X_in sc
+ apply (simp only:self_contained_def lefts_of_union_distrib[THEN sym])
+ by (drule_tac x = "(X, xrhs)" in bspec, auto simp:lefts_of_def)
+ moreover have "classes_of (arden_variate Y yrhs) \<subseteq> lefts_of ES \<union> {Y}"
+ using sc
+ by (auto simp add:arden_variate_removes_cl self_contained_def lefts_of_def)
+ ultimately show ?thesis by auto
+ qed
+ ultimately show ?thesis by simp
+ qed
+ } thus ?thesis by (auto simp only:eqs_subst_def self_contained_def)
+qed
+
+lemma eqs_subst_satisfy_Inv:
+ assumes Inv_ES: "Inv (ES \<union> {(Y, yrhs)})"
+ shows "Inv (eqs_subst ES Y (arden_variate Y yrhs))"
+proof -
+ have finite_yrhs: "finite yrhs"
+ using Inv_ES by (auto simp:Inv_def finite_rhs_def)
+ have nonempty_yrhs: "rhs_nonempty yrhs"
+ using Inv_ES by (auto simp:Inv_def ardenable_def)
+ have Y_eq_yrhs: "Y = L yrhs"
+ using Inv_ES by (simp only:Inv_def valid_eqns_def, blast)
+ have "distinct_equas (eqs_subst ES Y (arden_variate Y yrhs))"
+ using Inv_ES
+ by (auto simp:distinct_equas_def eqs_subst_def Inv_def)
+ moreover have "finite (eqs_subst ES Y (arden_variate Y yrhs))"
+ using Inv_ES by (simp add:Inv_def eqs_subst_keeps_finite)
+ moreover have "finite_rhs (eqs_subst ES Y (arden_variate Y yrhs))"
+ proof-
+ have "finite_rhs ES" using Inv_ES
+ by (simp add:Inv_def finite_rhs_def)
+ moreover have "finite (arden_variate Y yrhs)"
+ proof -
+ have "finite yrhs" using Inv_ES
+ by (auto simp:Inv_def finite_rhs_def)
+ thus ?thesis using arden_variate_keeps_finite by simp
+ qed
+ ultimately show ?thesis
+ by (simp add:eqs_subst_keeps_finite_rhs)
+ qed
+ moreover have "ardenable (eqs_subst ES Y (arden_variate Y yrhs))"
+ proof -
+ { fix X rhs
+ assume "(X, rhs) \<in> ES"
+ hence "rhs_nonempty rhs" using prems Inv_ES
+ by (simp add:Inv_def ardenable_def)
+ with nonempty_yrhs
+ have "rhs_nonempty (rhs_subst rhs Y (arden_variate Y yrhs))"
+ by (simp add:nonempty_yrhs
+ rhs_subst_keeps_nonempty arden_variate_keeps_nonempty)
+ } thus ?thesis by (auto simp add:ardenable_def eqs_subst_def)
+ qed
+ moreover have "valid_eqns (eqs_subst ES Y (arden_variate Y yrhs))"
+ proof-
+ have "Y = L (arden_variate Y yrhs)"
+ using Y_eq_yrhs Inv_ES finite_yrhs nonempty_yrhs
+ by (rule_tac arden_variate_keeps_eq, (simp add:rexp_of_empty)+)
+ thus ?thesis using Inv_ES
+ by (clarsimp simp add:valid_eqns_def
+ eqs_subst_def rhs_subst_keeps_eq Inv_def finite_rhs_def
+ simp del:L_rhs.simps)
+ qed
+ moreover have
+ non_empty_subst: "non_empty (eqs_subst ES Y (arden_variate Y yrhs))"
+ using Inv_ES by (auto simp:Inv_def non_empty_def eqs_subst_def)
+ moreover
+ have self_subst: "self_contained (eqs_subst ES Y (arden_variate Y yrhs))"
+ using Inv_ES eqs_subst_keeps_self_contained by (simp add:Inv_def)
+ ultimately show ?thesis using Inv_ES by (simp add:Inv_def)
+qed
+
+lemma eqs_subst_card_le:
+ assumes finite: "finite (ES::(string set \<times> rhs_item set) set)"
+ shows "card (eqs_subst ES Y yrhs) <= card ES"
+proof-
+ def f \<equiv> "\<lambda> x. ((fst x)::string set, rhs_subst (snd x) Y yrhs)"
+ have "eqs_subst ES Y yrhs = f ` ES"
+ apply (auto simp:eqs_subst_def f_def image_def)
+ by (rule_tac x = "(Ya, yrhsa)" in bexI, simp+)
+ thus ?thesis using finite by (auto intro:card_image_le)
+qed
+
+lemma eqs_subst_cls_remains:
+ "(X, xrhs) \<in> ES \<Longrightarrow> \<exists> xrhs'. (X, xrhs') \<in> (eqs_subst ES Y yrhs)"
+by (auto simp:eqs_subst_def)
+
+lemma card_noteq_1_has_more:
+ assumes card:"card S \<noteq> 1"
+ and e_in: "e \<in> S"
+ and finite: "finite S"
+ obtains e' where "e' \<in> S \<and> e \<noteq> e'"
+proof-
+ have "card (S - {e}) > 0"
+ proof -
+ have "card S > 1" using card e_in finite
+ by (case_tac "card S", auto)
+ thus ?thesis using finite e_in by auto
+ qed
+ hence "S - {e} \<noteq> {}" using finite by (rule_tac notI, simp)
+ thus "(\<And>e'. e' \<in> S \<and> e \<noteq> e' \<Longrightarrow> thesis) \<Longrightarrow> thesis" by auto
+qed
+
+lemma iteration_step:
+ assumes Inv_ES: "Inv ES"
+ and X_in_ES: "(X, xrhs) \<in> ES"
+ and not_T: "card ES \<noteq> 1"
+ shows "\<exists> ES'. (Inv ES' \<and> (\<exists> xrhs'.(X, xrhs') \<in> ES')) \<and>
+ (card ES', card ES) \<in> less_than" (is "\<exists> ES'. ?P ES'")
+proof -
+ have finite_ES: "finite ES" using Inv_ES by (simp add:Inv_def)
+ then obtain Y yrhs
+ where Y_in_ES: "(Y, yrhs) \<in> ES" and not_eq: "(X, xrhs) \<noteq> (Y, yrhs)"
+ using not_T X_in_ES by (drule_tac card_noteq_1_has_more, auto)
+ def ES' == "ES - {(Y, yrhs)}"
+ let ?ES'' = "eqs_subst ES' Y (arden_variate Y yrhs)"
+ have "?P ?ES''"
+ proof -
+ have "Inv ?ES''" using Y_in_ES Inv_ES
+ by (rule_tac eqs_subst_satisfy_Inv, simp add:ES'_def insert_absorb)
+ moreover have "\<exists>xrhs'. (X, xrhs') \<in> ?ES''" using not_eq X_in_ES
+ by (rule_tac ES = ES' in eqs_subst_cls_remains, auto simp add:ES'_def)
+ moreover have "(card ?ES'', card ES) \<in> less_than"
+ proof -
+ have "finite ES'" using finite_ES ES'_def by auto
+ moreover have "card ES' < card ES" using finite_ES Y_in_ES
+ by (auto simp:ES'_def card_gt_0_iff intro:diff_Suc_less)
+ ultimately show ?thesis
+ by (auto dest:eqs_subst_card_le elim:le_less_trans)
+ qed
+ ultimately show ?thesis by simp
+ qed
+ thus ?thesis by blast
+qed
+
+subsubsection {*
+ Conclusion of the proof
+ *}
+
+text {*
+ From this point until @{text "hard_direction"}, the hard direction is proved
+ through a simple application of the iteration principle.
+*}
+
+lemma iteration_conc:
+ assumes history: "Inv ES"
+ and X_in_ES: "\<exists> xrhs. (X, xrhs) \<in> ES"
+ shows
+ "\<exists> ES'. (Inv ES' \<and> (\<exists> xrhs'. (X, xrhs') \<in> ES')) \<and> card ES' = 1"
+ (is "\<exists> ES'. ?P ES'")
+proof (cases "card ES = 1")
+ case True
+ thus ?thesis using history X_in_ES
+ by blast
+next
+ case False
+ thus ?thesis using history iteration_step X_in_ES
+ by (rule_tac f = card in wf_iter, auto)
+qed
+
+lemma last_cl_exists_rexp:
+ assumes ES_single: "ES = {(X, xrhs)}"
+ and Inv_ES: "Inv ES"
+ shows "\<exists> (r::rexp). L r = X" (is "\<exists> r. ?P r")
+proof-
+ let ?A = "arden_variate X xrhs"
+ have "?P (rexp_of_lam ?A)"
+ proof -
+ have "L (rexp_of_lam ?A) = L (lam_of ?A)"
+ proof(rule rexp_of_lam_eq_lam_set)
+ show "finite (arden_variate X xrhs)" using Inv_ES ES_single
+ by (rule_tac arden_variate_keeps_finite,
+ auto simp add:Inv_def finite_rhs_def)
+ qed
+ also have "\<dots> = L ?A"
+ proof-
+ have "lam_of ?A = ?A"
+ proof-
+ have "classes_of ?A = {}" using Inv_ES ES_single
+ by (simp add:arden_variate_removes_cl
+ self_contained_def Inv_def lefts_of_def)
+ thus ?thesis
+ by (auto simp only:lam_of_def classes_of_def, case_tac x, auto)
+ qed
+ thus ?thesis by simp
+ qed
+ also have "\<dots> = X"
+ proof(rule arden_variate_keeps_eq [THEN sym])
+ show "X = L xrhs" using Inv_ES ES_single
+ by (auto simp only:Inv_def valid_eqns_def)
+ next
+ from Inv_ES ES_single show "[] \<notin> L (rexp_of xrhs X)"
+ by(simp add:Inv_def ardenable_def rexp_of_empty finite_rhs_def)
+ next
+ from Inv_ES ES_single show "finite xrhs"
+ by (simp add:Inv_def finite_rhs_def)
+ qed
+ finally show ?thesis by simp
+ qed
+ thus ?thesis by auto
+qed
+
+lemma every_eqcl_has_reg:
+ assumes finite_CS: "finite (UNIV // (\<approx>Lang))"
+ and X_in_CS: "X \<in> (UNIV // (\<approx>Lang))"
+ shows "\<exists> (reg::rexp). L reg = X" (is "\<exists> r. ?E r")
+proof -
+ from X_in_CS have "\<exists> xrhs. (X, xrhs) \<in> (eqs (UNIV // (\<approx>Lang)))"
+ by (auto simp:eqs_def init_rhs_def)
+ then obtain ES xrhs where Inv_ES: "Inv ES"
+ and X_in_ES: "(X, xrhs) \<in> ES"
+ and card_ES: "card ES = 1"
+ using finite_CS X_in_CS init_ES_satisfy_Inv iteration_conc
+ by blast
+ hence ES_single_equa: "ES = {(X, xrhs)}"
+ by (auto simp:Inv_def dest!:card_Suc_Diff1 simp:card_eq_0_iff)
+ thus ?thesis using Inv_ES
+ by (rule last_cl_exists_rexp)
+qed
+
+lemma finals_in_partitions:
+ "finals Lang \<subseteq> (UNIV // (\<approx>Lang))"
+ by (auto simp:finals_def quotient_def)
+
+theorem hard_direction:
+ assumes finite_CS: "finite (UNIV // (\<approx>Lang))"
+ shows "\<exists> (reg::rexp). Lang = L reg"
+proof -
+ have "\<forall> X \<in> (UNIV // (\<approx>Lang)). \<exists> (reg::rexp). X = L reg"
+ using finite_CS every_eqcl_has_reg by blast
+ then obtain f
+ where f_prop: "\<forall> X \<in> (UNIV // (\<approx>Lang)). X = L ((f X)::rexp)"
+ by (auto dest:bchoice)
+ def rs \<equiv> "f ` (finals Lang)"
+ have "Lang = \<Union> (finals Lang)" using lang_is_union_of_finals by auto
+ also have "\<dots> = L (folds ALT NULL rs)"
+ proof -
+ have "finite rs"
+ proof -
+ have "finite (finals Lang)"
+ using finite_CS finals_in_partitions[of "Lang"]
+ by (erule_tac finite_subset, simp)
+ thus ?thesis using rs_def by auto
+ qed
+ thus ?thesis
+ using f_prop rs_def finals_in_partitions[of "Lang"] by auto
+ qed
+ finally show ?thesis by blast
+qed
+
+end
\ No newline at end of file
--- a/pres/ROOT.ML Thu Jan 27 11:50:58 2011 +0000
+++ b/pres/ROOT.ML Thu Jan 27 12:35:06 2011 +0000
@@ -3,4 +3,5 @@
use_thys ["That_Theory1", "That_Theory2", "That_Theory3"];
*)
-use_thy "ListP";;
+no_document use_thy "ListP";
+no_document use_thys ["../Prefix_subtract", "../Myhill_1"];
\ No newline at end of file
--- a/tphols-2011/ROOT.ML Thu Jan 27 11:50:58 2011 +0000
+++ b/tphols-2011/ROOT.ML Thu Jan 27 12:35:06 2011 +0000
@@ -2,7 +2,4 @@
no_document use_thys ["This_Theory1", "This_Theory2"];
use_thys ["That_Theory1", "That_Theory2", "That_Theory3"];
*)
-
-no_document use_thys ["../Prefix_subtract", "../Prelude"];
-
-use_thys ["../Myhill"];
+ use_thys ["../Myhill"];