src/HOL/BNF_LFP.thy
author blanchet
Mon Jan 20 20:42:43 2014 +0100 (2014-01-20)
changeset 55084 8ee9aabb2bca
parent 55066 4e5ddf3162ac
child 55530 3dfb724db099
permissions -rw-r--r--
rationalized lemmas
     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_new_compat" :: thy_decl and
    17   "primrec_new" :: thy_decl
    18 begin
    19 
    20 lemma subset_emptyI: "(\<And>x. x \<in> A \<Longrightarrow> False) \<Longrightarrow> A \<subseteq> {}"
    21 by blast
    22 
    23 lemma image_Collect_subsetI:
    24   "(\<And>x. P x \<Longrightarrow> f x \<in> B) \<Longrightarrow> f ` {x. P x} \<subseteq> B"
    25 by blast
    26 
    27 lemma Collect_restrict: "{x. x \<in> X \<and> P x} \<subseteq> X"
    28 by auto
    29 
    30 lemma prop_restrict: "\<lbrakk>x \<in> Z; Z \<subseteq> {x. x \<in> X \<and> P x}\<rbrakk> \<Longrightarrow> P x"
    31 by auto
    32 
    33 lemma underS_I: "\<lbrakk>i \<noteq> j; (i, j) \<in> R\<rbrakk> \<Longrightarrow> i \<in> underS R j"
    34 unfolding underS_def by simp
    35 
    36 lemma underS_E: "i \<in> underS R j \<Longrightarrow> i \<noteq> j \<and> (i, j) \<in> R"
    37 unfolding underS_def by simp
    38 
    39 lemma underS_Field: "i \<in> underS R j \<Longrightarrow> i \<in> Field R"
    40 unfolding underS_def Field_def by auto
    41 
    42 lemma FieldI2: "(i, j) \<in> R \<Longrightarrow> j \<in> Field R"
    43 unfolding Field_def by auto
    44 
    45 lemma fst_convol': "fst (<f, g> x) = f x"
    46 using fst_convol unfolding convol_def by simp
    47 
    48 lemma snd_convol': "snd (<f, g> x) = g x"
    49 using snd_convol unfolding convol_def by simp
    50 
    51 lemma convol_expand_snd: "fst o f = g \<Longrightarrow>  <g, snd o f> = f"
    52 unfolding convol_def by auto
    53 
    54 lemma convol_expand_snd': "(fst o f = g) \<Longrightarrow> (h = snd o f) \<longleftrightarrow> (<g, h> = f)"
    55   by (metis convol_expand_snd snd_convol)
    56 
    57 definition inver where
    58   "inver g f A = (ALL a : A. g (f a) = a)"
    59 
    60 lemma bij_betw_iff_ex:
    61   "bij_betw f A B = (EX g. g ` B = A \<and> inver g f A \<and> inver f g B)" (is "?L = ?R")
    62 proof (rule iffI)
    63   assume ?L
    64   hence f: "f ` A = B" and inj_f: "inj_on f A" unfolding bij_betw_def by auto
    65   let ?phi = "% b a. a : A \<and> f a = b"
    66   have "ALL b : B. EX a. ?phi b a" using f by blast
    67   then obtain g where g: "ALL b : B. g b : A \<and> f (g b) = b"
    68     using bchoice[of B ?phi] by blast
    69   hence gg: "ALL b : f ` A. g b : A \<and> f (g b) = b" using f by blast
    70   have gf: "inver g f A" unfolding inver_def
    71     by (metis (no_types) gg imageI[of _ A f] the_inv_into_f_f[OF inj_f])
    72   moreover have "g ` B \<le> A \<and> inver f g B" using g unfolding inver_def by blast
    73   moreover have "A \<le> g ` B"
    74   proof safe
    75     fix a assume a: "a : A"
    76     hence "f a : B" using f by auto
    77     moreover have "a = g (f a)" using a gf unfolding inver_def by auto
    78     ultimately show "a : g ` B" by blast
    79   qed
    80   ultimately show ?R by blast
    81 next
    82   assume ?R
    83   then obtain g where g: "g ` B = A \<and> inver g f A \<and> inver f g B" by blast
    84   show ?L unfolding bij_betw_def
    85   proof safe
    86     show "inj_on f A" unfolding inj_on_def
    87     proof safe
    88       fix a1 a2 assume a: "a1 : A"  "a2 : A" and "f a1 = f a2"
    89       hence "g (f a1) = g (f a2)" by simp
    90       thus "a1 = a2" using a g unfolding inver_def by simp
    91     qed
    92   next
    93     fix a assume "a : A"
    94     then obtain b where b: "b : B" and a: "a = g b" using g by blast
    95     hence "b = f (g b)" using g unfolding inver_def by auto
    96     thus "f a : B" unfolding a using b by simp
    97   next
    98     fix b assume "b : B"
    99     hence "g b : A \<and> b = f (g b)" using g unfolding inver_def by auto
   100     thus "b : f ` A" by auto
   101   qed
   102 qed
   103 
   104 lemma bij_betw_ex_weakE:
   105   "\<lbrakk>bij_betw f A B\<rbrakk> \<Longrightarrow> \<exists>g. g ` B \<subseteq> A \<and> inver g f A \<and> inver f g B"
   106 by (auto simp only: bij_betw_iff_ex)
   107 
   108 lemma inver_surj: "\<lbrakk>g ` B \<subseteq> A; f ` A \<subseteq> B; inver g f A\<rbrakk> \<Longrightarrow> g ` B = A"
   109 unfolding inver_def by auto (rule rev_image_eqI, auto)
   110 
   111 lemma inver_mono: "\<lbrakk>A \<subseteq> B; inver f g B\<rbrakk> \<Longrightarrow> inver f g A"
   112 unfolding inver_def by auto
   113 
   114 lemma inver_pointfree: "inver f g A = (\<forall>a \<in> A. (f o g) a = a)"
   115 unfolding inver_def by simp
   116 
   117 lemma bij_betwE: "bij_betw f A B \<Longrightarrow> \<forall>a\<in>A. f a \<in> B"
   118 unfolding bij_betw_def by auto
   119 
   120 lemma bij_betw_imageE: "bij_betw f A B \<Longrightarrow> f ` A = B"
   121 unfolding bij_betw_def by auto
   122 
   123 lemma inverE: "\<lbrakk>inver f f' A; x \<in> A\<rbrakk> \<Longrightarrow> f (f' x) = x"
   124 unfolding inver_def by auto
   125 
   126 lemma bij_betw_inver1: "bij_betw f A B \<Longrightarrow> inver (inv_into A f) f A"
   127 unfolding bij_betw_def inver_def by auto
   128 
   129 lemma bij_betw_inver2: "bij_betw f A B \<Longrightarrow> inver f (inv_into A f) B"
   130 unfolding bij_betw_def inver_def by auto
   131 
   132 lemma bij_betwI: "\<lbrakk>bij_betw g B A; inver g f A; inver f g B\<rbrakk> \<Longrightarrow> bij_betw f A B"
   133 by (drule bij_betw_imageE, unfold bij_betw_iff_ex) blast
   134 
   135 lemma bij_betwI':
   136   "\<lbrakk>\<And>x y. \<lbrakk>x \<in> X; y \<in> X\<rbrakk> \<Longrightarrow> (f x = f y) = (x = y);
   137     \<And>x. x \<in> X \<Longrightarrow> f x \<in> Y;
   138     \<And>y. y \<in> Y \<Longrightarrow> \<exists>x \<in> X. y = f x\<rbrakk> \<Longrightarrow> bij_betw f X Y"
   139 unfolding bij_betw_def inj_on_def by blast
   140 
   141 lemma surj_fun_eq:
   142   assumes surj_on: "f ` X = UNIV" and eq_on: "\<forall>x \<in> X. (g1 o f) x = (g2 o f) x"
   143   shows "g1 = g2"
   144 proof (rule ext)
   145   fix y
   146   from surj_on obtain x where "x \<in> X" and "y = f x" by blast
   147   thus "g1 y = g2 y" using eq_on by simp
   148 qed
   149 
   150 lemma Card_order_wo_rel: "Card_order r \<Longrightarrow> wo_rel r"
   151 unfolding wo_rel_def card_order_on_def by blast
   152 
   153 lemma Cinfinite_limit: "\<lbrakk>x \<in> Field r; Cinfinite r\<rbrakk> \<Longrightarrow>
   154   \<exists>y \<in> Field r. x \<noteq> y \<and> (x, y) \<in> r"
   155 unfolding cinfinite_def by (auto simp add: infinite_Card_order_limit)
   156 
   157 lemma Card_order_trans:
   158   "\<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"
   159 unfolding card_order_on_def well_order_on_def linear_order_on_def
   160   partial_order_on_def preorder_on_def trans_def antisym_def by blast
   161 
   162 lemma Cinfinite_limit2:
   163  assumes x1: "x1 \<in> Field r" and x2: "x2 \<in> Field r" and r: "Cinfinite r"
   164  shows "\<exists>y \<in> Field r. (x1 \<noteq> y \<and> (x1, y) \<in> r) \<and> (x2 \<noteq> y \<and> (x2, y) \<in> r)"
   165 proof -
   166   from r have trans: "trans r" and total: "Total r" and antisym: "antisym r"
   167     unfolding card_order_on_def well_order_on_def linear_order_on_def
   168       partial_order_on_def preorder_on_def by auto
   169   obtain y1 where y1: "y1 \<in> Field r" "x1 \<noteq> y1" "(x1, y1) \<in> r"
   170     using Cinfinite_limit[OF x1 r] by blast
   171   obtain y2 where y2: "y2 \<in> Field r" "x2 \<noteq> y2" "(x2, y2) \<in> r"
   172     using Cinfinite_limit[OF x2 r] by blast
   173   show ?thesis
   174   proof (cases "y1 = y2")
   175     case True with y1 y2 show ?thesis by blast
   176   next
   177     case False
   178     with y1(1) y2(1) total have "(y1, y2) \<in> r \<or> (y2, y1) \<in> r"
   179       unfolding total_on_def by auto
   180     thus ?thesis
   181     proof
   182       assume *: "(y1, y2) \<in> r"
   183       with trans y1(3) have "(x1, y2) \<in> r" unfolding trans_def by blast
   184       with False y1 y2 * antisym show ?thesis by (cases "x1 = y2") (auto simp: antisym_def)
   185     next
   186       assume *: "(y2, y1) \<in> r"
   187       with trans y2(3) have "(x2, y1) \<in> r" unfolding trans_def by blast
   188       with False y1 y2 * antisym show ?thesis by (cases "x2 = y1") (auto simp: antisym_def)
   189     qed
   190   qed
   191 qed
   192 
   193 lemma Cinfinite_limit_finite: "\<lbrakk>finite X; X \<subseteq> Field r; Cinfinite r\<rbrakk>
   194  \<Longrightarrow> \<exists>y \<in> Field r. \<forall>x \<in> X. (x \<noteq> y \<and> (x, y) \<in> r)"
   195 proof (induct X rule: finite_induct)
   196   case empty thus ?case unfolding cinfinite_def using ex_in_conv[of "Field r"] finite.emptyI by auto
   197 next
   198   case (insert x X)
   199   then obtain y where y: "y \<in> Field r" "\<forall>x \<in> X. (x \<noteq> y \<and> (x, y) \<in> r)" by blast
   200   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"
   201     using Cinfinite_limit2[OF _ y(1) insert(5), of x] insert(4) by blast
   202   show ?case
   203     apply (intro bexI ballI)
   204     apply (erule insertE)
   205     apply hypsubst
   206     apply (rule z(2))
   207     using Card_order_trans[OF insert(5)[THEN conjunct2]] y(2) z(3)
   208     apply blast
   209     apply (rule z(1))
   210     done
   211 qed
   212 
   213 lemma insert_subsetI: "\<lbrakk>x \<in> A; X \<subseteq> A\<rbrakk> \<Longrightarrow> insert x X \<subseteq> A"
   214 by auto
   215 
   216 (*helps resolution*)
   217 lemma well_order_induct_imp:
   218   "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>
   219      x \<in> Field r \<longrightarrow> P x"
   220 by (erule wo_rel.well_order_induct)
   221 
   222 lemma meta_spec2:
   223   assumes "(\<And>x y. PROP P x y)"
   224   shows "PROP P x y"
   225 by (rule assms)
   226 
   227 lemma nchotomy_relcomppE:
   228   "\<lbrakk>\<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\<rbrakk> \<Longrightarrow> P"
   229   by (metis relcompp.cases)
   230 
   231 lemma vimage2p_fun_rel: "fun_rel (vimage2p f g R) R f g"
   232   unfolding fun_rel_def vimage2p_def by auto
   233 
   234 lemma predicate2D_vimage2p: "\<lbrakk>R \<le> vimage2p f g S; R x y\<rbrakk> \<Longrightarrow> S (f x) (g y)"
   235   unfolding vimage2p_def by auto
   236 
   237 lemma id_transfer: "fun_rel A A id id"
   238 unfolding fun_rel_def by simp
   239 
   240 ML_file "Tools/BNF/bnf_lfp_rec_sugar.ML"
   241 ML_file "Tools/BNF/bnf_lfp_util.ML"
   242 ML_file "Tools/BNF/bnf_lfp_tactics.ML"
   243 ML_file "Tools/BNF/bnf_lfp.ML"
   244 ML_file "Tools/BNF/bnf_lfp_compat.ML"
   245 
   246 hide_fact (open) id_transfer
   247 
   248 end