src/HOL/BNF_LFP.thy
author blanchet
Wed Apr 23 10:23:26 2014 +0200 (2014-04-23)
changeset 56638 092a306bcc3d
parent 56346 42533f8f4729
child 56639 c9d6b581bd3b
permissions -rw-r--r--
generate size instances for new-style datatypes
     1 (*  Title:      HOL/BNF_LFP.thy
     2     Author:     Dmitriy Traytel, TU Muenchen
     3     Author:     Lorenz Panny, TU Muenchen
     4     Author:     Jasmin Blanchette, TU Muenchen
     5     Copyright   2012, 2013
     6 
     7 Least fixed point operation on bounded natural functors.
     8 *)
     9 
    10 header {* Least Fixed Point Operation on Bounded Natural Functors *}
    11 
    12 theory BNF_LFP
    13 imports BNF_FP_Base
    14 keywords
    15   "datatype_new" :: thy_decl and
    16   "datatype_compat" :: thy_decl
    17 begin
    18 
    19 lemma subset_emptyI: "(\<And>x. x \<in> A \<Longrightarrow> False) \<Longrightarrow> A \<subseteq> {}"
    20 by blast
    21 
    22 lemma image_Collect_subsetI: "(\<And>x. P x \<Longrightarrow> f x \<in> B) \<Longrightarrow> f ` {x. P x} \<subseteq> B"
    23 by blast
    24 
    25 lemma Collect_restrict: "{x. x \<in> X \<and> P x} \<subseteq> X"
    26 by auto
    27 
    28 lemma prop_restrict: "\<lbrakk>x \<in> Z; Z \<subseteq> {x. x \<in> X \<and> P x}\<rbrakk> \<Longrightarrow> P x"
    29 by auto
    30 
    31 lemma underS_I: "\<lbrakk>i \<noteq> j; (i, j) \<in> R\<rbrakk> \<Longrightarrow> i \<in> underS R j"
    32 unfolding underS_def by simp
    33 
    34 lemma underS_E: "i \<in> underS R j \<Longrightarrow> i \<noteq> j \<and> (i, j) \<in> R"
    35 unfolding underS_def by simp
    36 
    37 lemma underS_Field: "i \<in> underS R j \<Longrightarrow> i \<in> Field R"
    38 unfolding underS_def Field_def by auto
    39 
    40 lemma FieldI2: "(i, j) \<in> R \<Longrightarrow> j \<in> Field R"
    41 unfolding Field_def by auto
    42 
    43 lemma fst_convol': "fst (<f, g> x) = f x"
    44 using fst_convol unfolding convol_def by simp
    45 
    46 lemma snd_convol': "snd (<f, g> x) = g x"
    47 using snd_convol unfolding convol_def by simp
    48 
    49 lemma convol_expand_snd: "fst o f = g \<Longrightarrow>  <g, snd o f> = f"
    50 unfolding convol_def by auto
    51 
    52 lemma convol_expand_snd':
    53   assumes "(fst o f = g)"
    54   shows "h = snd o f \<longleftrightarrow> <g, h> = f"
    55 proof -
    56   from assms have *: "<g, snd o f> = f" by (rule convol_expand_snd)
    57   then have "h = snd o f \<longleftrightarrow> h = snd o <g, snd o f>" by simp
    58   moreover have "\<dots> \<longleftrightarrow> h = snd o f" by (simp add: snd_convol)
    59   moreover have "\<dots> \<longleftrightarrow> <g, h> = f" by (subst (2) *[symmetric]) (auto simp: convol_def fun_eq_iff)
    60   ultimately show ?thesis by simp
    61 qed
    62 lemma bij_betwE: "bij_betw f A B \<Longrightarrow> \<forall>a\<in>A. f a \<in> B"
    63 unfolding bij_betw_def by auto
    64 
    65 lemma bij_betw_imageE: "bij_betw f A B \<Longrightarrow> f ` A = B"
    66 unfolding bij_betw_def by auto
    67 
    68 lemma f_the_inv_into_f_bij_betw: "bij_betw f A B \<Longrightarrow>
    69   (bij_betw f A B \<Longrightarrow> x \<in> B) \<Longrightarrow> f (the_inv_into A f x) = x"
    70   unfolding bij_betw_def by (blast intro: f_the_inv_into_f)
    71 
    72 lemma ex_bij_betw: "|A| \<le>o (r :: 'b rel) \<Longrightarrow> \<exists>f B :: 'b set. bij_betw f B A"
    73   by (subst (asm) internalize_card_of_ordLeq)
    74     (auto dest!: iffD2[OF card_of_ordIso ordIso_symmetric])
    75 
    76 lemma bij_betwI':
    77   "\<lbrakk>\<And>x y. \<lbrakk>x \<in> X; y \<in> X\<rbrakk> \<Longrightarrow> (f x = f y) = (x = y);
    78     \<And>x. x \<in> X \<Longrightarrow> f x \<in> Y;
    79     \<And>y. y \<in> Y \<Longrightarrow> \<exists>x \<in> X. y = f x\<rbrakk> \<Longrightarrow> bij_betw f X Y"
    80 unfolding bij_betw_def inj_on_def by blast
    81 
    82 lemma surj_fun_eq:
    83   assumes surj_on: "f ` X = UNIV" and eq_on: "\<forall>x \<in> X. (g1 o f) x = (g2 o f) x"
    84   shows "g1 = g2"
    85 proof (rule ext)
    86   fix y
    87   from surj_on obtain x where "x \<in> X" and "y = f x" by blast
    88   thus "g1 y = g2 y" using eq_on by simp
    89 qed
    90 
    91 lemma Card_order_wo_rel: "Card_order r \<Longrightarrow> wo_rel r"
    92 unfolding wo_rel_def card_order_on_def by blast
    93 
    94 lemma Cinfinite_limit: "\<lbrakk>x \<in> Field r; Cinfinite r\<rbrakk> \<Longrightarrow>
    95   \<exists>y \<in> Field r. x \<noteq> y \<and> (x, y) \<in> r"
    96 unfolding cinfinite_def by (auto simp add: infinite_Card_order_limit)
    97 
    98 lemma Card_order_trans:
    99   "\<lbrakk>Card_order r; x \<noteq> y; (x, y) \<in> r; y \<noteq> z; (y, z) \<in> r\<rbrakk> \<Longrightarrow> x \<noteq> z \<and> (x, z) \<in> r"
   100 unfolding card_order_on_def well_order_on_def linear_order_on_def
   101   partial_order_on_def preorder_on_def trans_def antisym_def by blast
   102 
   103 lemma Cinfinite_limit2:
   104  assumes x1: "x1 \<in> Field r" and x2: "x2 \<in> Field r" and r: "Cinfinite r"
   105  shows "\<exists>y \<in> Field r. (x1 \<noteq> y \<and> (x1, y) \<in> r) \<and> (x2 \<noteq> y \<and> (x2, y) \<in> r)"
   106 proof -
   107   from r have trans: "trans r" and total: "Total r" and antisym: "antisym r"
   108     unfolding card_order_on_def well_order_on_def linear_order_on_def
   109       partial_order_on_def preorder_on_def by auto
   110   obtain y1 where y1: "y1 \<in> Field r" "x1 \<noteq> y1" "(x1, y1) \<in> r"
   111     using Cinfinite_limit[OF x1 r] by blast
   112   obtain y2 where y2: "y2 \<in> Field r" "x2 \<noteq> y2" "(x2, y2) \<in> r"
   113     using Cinfinite_limit[OF x2 r] by blast
   114   show ?thesis
   115   proof (cases "y1 = y2")
   116     case True with y1 y2 show ?thesis by blast
   117   next
   118     case False
   119     with y1(1) y2(1) total have "(y1, y2) \<in> r \<or> (y2, y1) \<in> r"
   120       unfolding total_on_def by auto
   121     thus ?thesis
   122     proof
   123       assume *: "(y1, y2) \<in> r"
   124       with trans y1(3) have "(x1, y2) \<in> r" unfolding trans_def by blast
   125       with False y1 y2 * antisym show ?thesis by (cases "x1 = y2") (auto simp: antisym_def)
   126     next
   127       assume *: "(y2, y1) \<in> r"
   128       with trans y2(3) have "(x2, y1) \<in> r" unfolding trans_def by blast
   129       with False y1 y2 * antisym show ?thesis by (cases "x2 = y1") (auto simp: antisym_def)
   130     qed
   131   qed
   132 qed
   133 
   134 lemma Cinfinite_limit_finite: "\<lbrakk>finite X; X \<subseteq> Field r; Cinfinite r\<rbrakk>
   135  \<Longrightarrow> \<exists>y \<in> Field r. \<forall>x \<in> X. (x \<noteq> y \<and> (x, y) \<in> r)"
   136 proof (induct X rule: finite_induct)
   137   case empty thus ?case unfolding cinfinite_def using ex_in_conv[of "Field r"] finite.emptyI by auto
   138 next
   139   case (insert x X)
   140   then obtain y where y: "y \<in> Field r" "\<forall>x \<in> X. (x \<noteq> y \<and> (x, y) \<in> r)" by blast
   141   then obtain z where z: "z \<in> Field r" "x \<noteq> z \<and> (x, z) \<in> r" "y \<noteq> z \<and> (y, z) \<in> r"
   142     using Cinfinite_limit2[OF _ y(1) insert(5), of x] insert(4) by blast
   143   show ?case
   144     apply (intro bexI ballI)
   145     apply (erule insertE)
   146     apply hypsubst
   147     apply (rule z(2))
   148     using Card_order_trans[OF insert(5)[THEN conjunct2]] y(2) z(3)
   149     apply blast
   150     apply (rule z(1))
   151     done
   152 qed
   153 
   154 lemma insert_subsetI: "\<lbrakk>x \<in> A; X \<subseteq> A\<rbrakk> \<Longrightarrow> insert x X \<subseteq> A"
   155 by auto
   156 
   157 (*helps resolution*)
   158 lemma well_order_induct_imp:
   159   "wo_rel r \<Longrightarrow> (\<And>x. \<forall>y. y \<noteq> x \<and> (y, x) \<in> r \<longrightarrow> y \<in> Field r \<longrightarrow> P y \<Longrightarrow> x \<in> Field r \<longrightarrow> P x) \<Longrightarrow>
   160      x \<in> Field r \<longrightarrow> P x"
   161 by (erule wo_rel.well_order_induct)
   162 
   163 lemma meta_spec2:
   164   assumes "(\<And>x y. PROP P x y)"
   165   shows "PROP P x y"
   166 by (rule assms)
   167 
   168 lemma nchotomy_relcomppE:
   169   assumes "\<And>y. \<exists>x. y = f x" "(r OO s) a c" "\<And>b. r a (f b) \<Longrightarrow> s (f b) c \<Longrightarrow> P"
   170   shows P
   171 proof (rule relcompp.cases[OF assms(2)], hypsubst)
   172   fix b assume "r a b" "s b c"
   173   moreover from assms(1) obtain b' where "b = f b'" by blast
   174   ultimately show P by (blast intro: assms(3))
   175 qed
   176 
   177 lemma vimage2p_rel_fun: "rel_fun (vimage2p f g R) R f g"
   178   unfolding rel_fun_def vimage2p_def by auto
   179 
   180 lemma predicate2D_vimage2p: "\<lbrakk>R \<le> vimage2p f g S; R x y\<rbrakk> \<Longrightarrow> S (f x) (g y)"
   181   unfolding vimage2p_def by auto
   182 
   183 lemma id_transfer: "rel_fun A A id id"
   184   unfolding rel_fun_def by simp
   185 
   186 lemma ssubst_Pair_rhs: "\<lbrakk>(r, s) \<in> R; s' = s\<rbrakk> \<Longrightarrow> (r, s') \<in> R"
   187   by (rule ssubst)
   188 
   189 lemma snd_o_convol: "(snd \<circ> (\<lambda>x. (f x, g x))) = g"
   190   by (rule ext) simp
   191 
   192 lemma inj_on_convol_id: "inj_on (\<lambda>x. (x, f x)) X"
   193   unfolding inj_on_def by simp
   194 
   195 ML_file "Tools/BNF/bnf_lfp_util.ML"
   196 ML_file "Tools/BNF/bnf_lfp_tactics.ML"
   197 ML_file "Tools/BNF/bnf_lfp.ML"
   198 ML_file "Tools/BNF/bnf_lfp_compat.ML"
   199 ML_file "Tools/BNF/bnf_lfp_rec_sugar_more.ML"
   200 ML_file "Tools/BNF/bnf_lfp_size.ML"
   201 
   202 hide_fact (open) id_transfer
   203 
   204 datatype_new x = X nat
   205 thm x.size
   206 
   207 datatype_new 'a l = N | C 'a "'a l"
   208 thm l.size
   209 thm l.size_map
   210 thm size_l_def size_l_overloaded_def
   211 
   212 datatype_new
   213   'a tl = TN | TC "'a mt" "'a tl" and
   214   'a mt = MT 'a "'a tl"
   215 
   216 thm size_tl_def size_tl_overloaded_def
   217 thm size_mt_def size_mt_overloaded_def
   218 
   219 datatype_new 'a t = T 'a "'a t l"
   220 thm t.size
   221 
   222 lemma size_l_cong: "(ALL x : set_l t. f x = g x) --> size_l f t = size_l g t"
   223   apply (induct_tac t)
   224   apply (simp only: l.size simp_thms)
   225   apply (simp add: l.set l.size simp_thms)
   226   done
   227 
   228 lemma t_size_map_t: "size_t g (map_t f t) = size_t (g \<circ> f) t"
   229   apply (rule t.induct)
   230   apply (simp_all only: t.map t.size comp_def l.size_map)
   231   apply (auto intro: size_l_cong)
   232   apply (subst size_l_cong[rule_format], assumption)
   233   apply (rule refl)
   234   done
   235 
   236 
   237 thm t.size
   238 
   239 lemmas size_t_def' =
   240   size_t_def[THEN meta_eq_to_obj_eq, THEN fun_cong, THEN fun_cong]
   241 
   242 thm trans[OF size_t_def' t.rec(1), unfolded l.size_map snd_o_convol, folded size_t_def']
   243 
   244 lemma "size_t f (T x ts) = f x + size_l (size_t f) ts + Suc 0"
   245   unfolding size_t_def t.rec l.size_map snd_o_convol
   246   by rule
   247 
   248 
   249 lemma "       (\<And>x2aa. x2aa \<in> set_l x2a \<Longrightarrow>
   250                 size_t f1 (map_t g1 x2aa) = size_t (f1 \<circ> g1) x2aa) \<Longrightarrow>
   251        f1 (g1 x1a) +
   252        size_l snd (map_l (\<lambda>t. (t, size_t f1 t)) (map_l (map_t g1) x2a)) +
   253        Suc 0 =
   254        f1 (g1 x1a) + size_l snd (map_l (\<lambda>t. (t, size_t (\<lambda>x1. f1 (g1 x1)) t)) x2a) +
   255        Suc 0"
   256 apply (simp only: l.size_map comp_def snd_conv t.size_map snd_o_convol)
   257 
   258 thm size_t_def size_t_overloaded_def
   259 
   260 xdatatype_new ('a, 'b, 'c) x = XN 'c | XC 'a "('a, 'b, 'c) x"
   261 thm size_x_def size_x_overloaded_def
   262 
   263 end