src/HOL/BNF_Least_Fixpoint.thy
author wenzelm
Sat Nov 04 15:24:40 2017 +0100 (19 months ago)
changeset 67003 49850a679c2c
parent 66290 88714f2e40e8
child 67091 1393c2340eec
permissions -rw-r--r--
more robust sorted_entries;
     1 (*  Title:      HOL/BNF_Least_Fixpoint.thy
     2     Author:     Dmitriy Traytel, TU Muenchen
     3     Author:     Lorenz Panny, TU Muenchen
     4     Author:     Jasmin Blanchette, TU Muenchen
     5     Copyright   2012, 2013, 2014
     6 
     7 Least fixpoint (datatype) operation on bounded natural functors.
     8 *)
     9 
    10 section \<open>Least Fixpoint (Datatype) Operation on Bounded Natural Functors\<close>
    11 
    12 theory BNF_Least_Fixpoint
    13 imports BNF_Fixpoint_Base
    14 keywords
    15   "datatype" :: 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 bij_betwE: "bij_betw f A B \<Longrightarrow> \<forall>a\<in>A. f a \<in> B"
    41   unfolding bij_betw_def by auto
    42 
    43 lemma f_the_inv_into_f_bij_betw:
    44   "bij_betw f A B \<Longrightarrow> (bij_betw f A B \<Longrightarrow> x \<in> B) \<Longrightarrow> f (the_inv_into A f x) = x"
    45   unfolding bij_betw_def by (blast intro: f_the_inv_into_f)
    46 
    47 lemma ex_bij_betw: "|A| \<le>o (r :: 'b rel) \<Longrightarrow> \<exists>f B :: 'b set. bij_betw f B A"
    48   by (subst (asm) internalize_card_of_ordLeq) (auto dest!: iffD2[OF card_of_ordIso ordIso_symmetric])
    49 
    50 lemma bij_betwI':
    51   "\<lbrakk>\<And>x y. \<lbrakk>x \<in> X; y \<in> X\<rbrakk> \<Longrightarrow> (f x = f y) = (x = y);
    52     \<And>x. x \<in> X \<Longrightarrow> f x \<in> Y;
    53     \<And>y. y \<in> Y \<Longrightarrow> \<exists>x \<in> X. y = f x\<rbrakk> \<Longrightarrow> bij_betw f X Y"
    54   unfolding bij_betw_def inj_on_def by blast
    55 
    56 lemma surj_fun_eq:
    57   assumes surj_on: "f ` X = UNIV" and eq_on: "\<forall>x \<in> X. (g1 o f) x = (g2 o f) x"
    58   shows "g1 = g2"
    59 proof (rule ext)
    60   fix y
    61   from surj_on obtain x where "x \<in> X" and "y = f x" by blast
    62   thus "g1 y = g2 y" using eq_on by simp
    63 qed
    64 
    65 lemma Card_order_wo_rel: "Card_order r \<Longrightarrow> wo_rel r"
    66   unfolding wo_rel_def card_order_on_def by blast
    67 
    68 lemma Cinfinite_limit: "\<lbrakk>x \<in> Field r; Cinfinite r\<rbrakk> \<Longrightarrow> \<exists>y \<in> Field r. x \<noteq> y \<and> (x, y) \<in> r"
    69   unfolding cinfinite_def by (auto simp add: infinite_Card_order_limit)
    70 
    71 lemma Card_order_trans:
    72   "\<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"
    73   unfolding card_order_on_def well_order_on_def linear_order_on_def
    74     partial_order_on_def preorder_on_def trans_def antisym_def by blast
    75 
    76 lemma Cinfinite_limit2:
    77   assumes x1: "x1 \<in> Field r" and x2: "x2 \<in> Field r" and r: "Cinfinite r"
    78   shows "\<exists>y \<in> Field r. (x1 \<noteq> y \<and> (x1, y) \<in> r) \<and> (x2 \<noteq> y \<and> (x2, y) \<in> r)"
    79 proof -
    80   from r have trans: "trans r" and total: "Total r" and antisym: "antisym r"
    81     unfolding card_order_on_def well_order_on_def linear_order_on_def
    82       partial_order_on_def preorder_on_def by auto
    83   obtain y1 where y1: "y1 \<in> Field r" "x1 \<noteq> y1" "(x1, y1) \<in> r"
    84     using Cinfinite_limit[OF x1 r] by blast
    85   obtain y2 where y2: "y2 \<in> Field r" "x2 \<noteq> y2" "(x2, y2) \<in> r"
    86     using Cinfinite_limit[OF x2 r] by blast
    87   show ?thesis
    88   proof (cases "y1 = y2")
    89     case True with y1 y2 show ?thesis by blast
    90   next
    91     case False
    92     with y1(1) y2(1) total have "(y1, y2) \<in> r \<or> (y2, y1) \<in> r"
    93       unfolding total_on_def by auto
    94     thus ?thesis
    95     proof
    96       assume *: "(y1, y2) \<in> r"
    97       with trans y1(3) have "(x1, y2) \<in> r" unfolding trans_def by blast
    98       with False y1 y2 * antisym show ?thesis by (cases "x1 = y2") (auto simp: antisym_def)
    99     next
   100       assume *: "(y2, y1) \<in> r"
   101       with trans y2(3) have "(x2, y1) \<in> r" unfolding trans_def by blast
   102       with False y1 y2 * antisym show ?thesis by (cases "x2 = y1") (auto simp: antisym_def)
   103     qed
   104   qed
   105 qed
   106 
   107 lemma Cinfinite_limit_finite:
   108   "\<lbrakk>finite X; X \<subseteq> Field r; Cinfinite r\<rbrakk> \<Longrightarrow> \<exists>y \<in> Field r. \<forall>x \<in> X. (x \<noteq> y \<and> (x, y) \<in> r)"
   109 proof (induct X rule: finite_induct)
   110   case empty thus ?case unfolding cinfinite_def using ex_in_conv[of "Field r"] finite.emptyI by auto
   111 next
   112   case (insert x X)
   113   then obtain y where y: "y \<in> Field r" "\<forall>x \<in> X. (x \<noteq> y \<and> (x, y) \<in> r)" by blast
   114   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"
   115     using Cinfinite_limit2[OF _ y(1) insert(5), of x] insert(4) by blast
   116   show ?case
   117     apply (intro bexI ballI)
   118     apply (erule insertE)
   119     apply hypsubst
   120     apply (rule z(2))
   121     using Card_order_trans[OF insert(5)[THEN conjunct2]] y(2) z(3)
   122     apply blast
   123     apply (rule z(1))
   124     done
   125 qed
   126 
   127 lemma insert_subsetI: "\<lbrakk>x \<in> A; X \<subseteq> A\<rbrakk> \<Longrightarrow> insert x X \<subseteq> A"
   128   by auto
   129 
   130 lemmas well_order_induct_imp = wo_rel.well_order_induct[of r "\<lambda>x. x \<in> Field r \<longrightarrow> P x" for r P]
   131 
   132 lemma meta_spec2:
   133   assumes "(\<And>x y. PROP P x y)"
   134   shows "PROP P x y"
   135   by (rule assms)
   136 
   137 lemma nchotomy_relcomppE:
   138   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"
   139   shows P
   140 proof (rule relcompp.cases[OF assms(2)], hypsubst)
   141   fix b assume "r a b" "s b c"
   142   moreover from assms(1) obtain b' where "b = f b'" by blast
   143   ultimately show P by (blast intro: assms(3))
   144 qed
   145 
   146 lemma predicate2D_vimage2p: "\<lbrakk>R \<le> vimage2p f g S; R x y\<rbrakk> \<Longrightarrow> S (f x) (g y)"
   147   unfolding vimage2p_def by auto
   148 
   149 lemma ssubst_Pair_rhs: "\<lbrakk>(r, s) \<in> R; s' = s\<rbrakk> \<Longrightarrow> (r, s') \<in> R"
   150   by (rule ssubst)
   151 
   152 lemma all_mem_range1:
   153   "(\<And>y. y \<in> range f \<Longrightarrow> P y) \<equiv> (\<And>x. P (f x)) "
   154   by (rule equal_intr_rule) fast+
   155 
   156 lemma all_mem_range2:
   157   "(\<And>fa y. fa \<in> range f \<Longrightarrow> y \<in> range fa \<Longrightarrow> P y) \<equiv> (\<And>x xa. P (f x xa))"
   158   by (rule equal_intr_rule) fast+
   159 
   160 lemma all_mem_range3:
   161   "(\<And>fa fb y. fa \<in> range f \<Longrightarrow> fb \<in> range fa \<Longrightarrow> y \<in> range fb \<Longrightarrow> P y) \<equiv> (\<And>x xa xb. P (f x xa xb))"
   162   by (rule equal_intr_rule) fast+
   163 
   164 lemma all_mem_range4:
   165   "(\<And>fa fb fc y. fa \<in> range f \<Longrightarrow> fb \<in> range fa \<Longrightarrow> fc \<in> range fb \<Longrightarrow> y \<in> range fc \<Longrightarrow> P y) \<equiv>
   166    (\<And>x xa xb xc. P (f x xa xb xc))"
   167   by (rule equal_intr_rule) fast+
   168 
   169 lemma all_mem_range5:
   170   "(\<And>fa fb fc fd y. fa \<in> range f \<Longrightarrow> fb \<in> range fa \<Longrightarrow> fc \<in> range fb \<Longrightarrow> fd \<in> range fc \<Longrightarrow>
   171      y \<in> range fd \<Longrightarrow> P y) \<equiv>
   172    (\<And>x xa xb xc xd. P (f x xa xb xc xd))"
   173   by (rule equal_intr_rule) fast+
   174 
   175 lemma all_mem_range6:
   176   "(\<And>fa fb fc fd fe ff y. fa \<in> range f \<Longrightarrow> fb \<in> range fa \<Longrightarrow> fc \<in> range fb \<Longrightarrow> fd \<in> range fc \<Longrightarrow>
   177      fe \<in> range fd \<Longrightarrow> ff \<in> range fe \<Longrightarrow> y \<in> range ff \<Longrightarrow> P y) \<equiv>
   178    (\<And>x xa xb xc xd xe xf. P (f x xa xb xc xd xe xf))"
   179   by (rule equal_intr_rule) (fastforce, fast)
   180 
   181 lemma all_mem_range7:
   182   "(\<And>fa fb fc fd fe ff fg y. fa \<in> range f \<Longrightarrow> fb \<in> range fa \<Longrightarrow> fc \<in> range fb \<Longrightarrow> fd \<in> range fc \<Longrightarrow>
   183      fe \<in> range fd \<Longrightarrow> ff \<in> range fe \<Longrightarrow> fg \<in> range ff \<Longrightarrow> y \<in> range fg \<Longrightarrow> P y) \<equiv>
   184    (\<And>x xa xb xc xd xe xf xg. P (f x xa xb xc xd xe xf xg))"
   185   by (rule equal_intr_rule) (fastforce, fast)
   186 
   187 lemma all_mem_range8:
   188   "(\<And>fa fb fc fd fe ff fg fh y. fa \<in> range f \<Longrightarrow> fb \<in> range fa \<Longrightarrow> fc \<in> range fb \<Longrightarrow> fd \<in> range fc \<Longrightarrow>
   189      fe \<in> range fd \<Longrightarrow> ff \<in> range fe \<Longrightarrow> fg \<in> range ff \<Longrightarrow> fh \<in> range fg \<Longrightarrow> y \<in> range fh \<Longrightarrow> P y) \<equiv>
   190    (\<And>x xa xb xc xd xe xf xg xh. P (f x xa xb xc xd xe xf xg xh))"
   191   by (rule equal_intr_rule) (fastforce, fast)
   192 
   193 lemmas all_mem_range = all_mem_range1 all_mem_range2 all_mem_range3 all_mem_range4 all_mem_range5
   194   all_mem_range6 all_mem_range7 all_mem_range8
   195 
   196 lemma pred_fun_True_id: "NO_MATCH id p \<Longrightarrow> pred_fun (\<lambda>x. True) p f = pred_fun (\<lambda>x. True) id (p \<circ> f)"
   197   unfolding fun.pred_map unfolding comp_def id_def ..
   198 
   199 ML_file "Tools/BNF/bnf_lfp_util.ML"
   200 ML_file "Tools/BNF/bnf_lfp_tactics.ML"
   201 ML_file "Tools/BNF/bnf_lfp.ML"
   202 ML_file "Tools/BNF/bnf_lfp_compat.ML"
   203 ML_file "Tools/BNF/bnf_lfp_rec_sugar_more.ML"
   204 ML_file "Tools/BNF/bnf_lfp_size.ML"
   205 
   206 end