src/HOL/Basic_BNF_LFPs.thy
author haftmann
Fri Jun 19 07:53:35 2015 +0200 (2015-06-19)
changeset 60517 f16e4fb20652
parent 59819 dbec7f33093d
child 61076 bdc1e2f0a86a
permissions -rw-r--r--
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
     1 (*  Title:      HOL/Basic_BNF_LFPs.thy
     2     Author:     Jasmin Blanchette, TU Muenchen
     3     Copyright   2014
     4 
     5 Registration of basic types as BNF least fixpoints (datatypes).
     6 *)
     7 
     8 theory Basic_BNF_LFPs
     9 imports BNF_Least_Fixpoint
    10 begin
    11 
    12 definition xtor :: "'a \<Rightarrow> 'a" where
    13   "xtor x = x"
    14 
    15 lemma xtor_map: "f (xtor x) = xtor (f x)"
    16   unfolding xtor_def by (rule refl)
    17 
    18 lemma xtor_set: "f (xtor x) = f x"
    19   unfolding xtor_def by (rule refl)
    20 
    21 lemma xtor_rel: "R (xtor x) (xtor y) = R x y"
    22   unfolding xtor_def by (rule refl)
    23 
    24 lemma xtor_induct: "(\<And>x. P (xtor x)) \<Longrightarrow> P z"
    25   unfolding xtor_def by assumption
    26 
    27 lemma xtor_xtor: "xtor (xtor x) = x"
    28   unfolding xtor_def by (rule refl)
    29 
    30 lemmas xtor_inject = xtor_rel[of "op ="]
    31 
    32 lemma xtor_rel_induct: "(\<And>x y. vimage2p id_bnf id_bnf R x y \<Longrightarrow> IR (xtor x) (xtor y)) \<Longrightarrow> R \<le> IR"
    33   unfolding xtor_def vimage2p_def id_bnf_def by default
    34 
    35 lemma Inl_def_alt: "Inl \<equiv> (\<lambda>a. xtor (id_bnf (Inl a)))"
    36   unfolding xtor_def id_bnf_def by (rule reflexive)
    37 
    38 lemma Inr_def_alt: "Inr \<equiv> (\<lambda>a. xtor (id_bnf (Inr a)))"
    39   unfolding xtor_def id_bnf_def by (rule reflexive)
    40 
    41 lemma Pair_def_alt: "Pair \<equiv> (\<lambda>a b. xtor (id_bnf (a, b)))"
    42   unfolding xtor_def id_bnf_def by (rule reflexive)
    43 
    44 definition ctor_rec :: "'a \<Rightarrow> 'a" where
    45   "ctor_rec x = x"
    46 
    47 lemma ctor_rec: "g = id \<Longrightarrow> ctor_rec f (xtor x) = f ((id_bnf \<circ> g \<circ> id_bnf) x)"
    48   unfolding ctor_rec_def id_bnf_def xtor_def comp_def id_def by hypsubst (rule refl)
    49 
    50 lemma ctor_rec_def_alt: "f = ctor_rec (f \<circ> id_bnf)"
    51   unfolding ctor_rec_def id_bnf_def comp_def by (rule refl)
    52 
    53 lemma ctor_rec_o_map: "ctor_rec f \<circ> g = ctor_rec (f \<circ> (id_bnf \<circ> g \<circ> id_bnf))"
    54   unfolding ctor_rec_def id_bnf_def comp_def by (rule refl)
    55 
    56 lemma eq_fst_iff: "a = fst p \<longleftrightarrow> (\<exists>b. p = (a, b))"
    57   by (cases p) auto
    58 
    59 lemma eq_snd_iff: "b = snd p \<longleftrightarrow> (\<exists>a. p = (a, b))"
    60   by (cases p) auto
    61 
    62 lemma ex_neg_all_pos: "((\<exists>x. P x) \<Longrightarrow> Q) \<equiv> (\<And>x. P x \<Longrightarrow> Q)"
    63   by default blast+
    64 
    65 lemma hypsubst_in_prems: "(\<And>x. y = x \<Longrightarrow> z = f x \<Longrightarrow> P) \<equiv> (z = f y \<Longrightarrow> P)"
    66   by default blast+
    67 
    68 lemma isl_map_sum:
    69   "isl (map_sum f g s) = isl s"
    70   by (cases s) simp_all
    71 
    72 lemma map_sum_sel:
    73   "isl s \<Longrightarrow> projl (map_sum f g s) = f (projl s)"
    74   "\<not> isl s \<Longrightarrow> projr (map_sum f g s) = g (projr s)"
    75   by (case_tac [!] s) simp_all
    76 
    77 lemma set_sum_sel:
    78   "isl s \<Longrightarrow> projl s \<in> setl s"
    79   "\<not> isl s \<Longrightarrow> projr s \<in> setr s"
    80   by (case_tac [!] s) (auto intro: setl.intros setr.intros)
    81 
    82 lemma rel_sum_sel: "rel_sum R1 R2 a b = (isl a = isl b \<and>
    83   (isl a \<longrightarrow> isl b \<longrightarrow> R1 (projl a) (projl b)) \<and>
    84   (\<not> isl a \<longrightarrow> \<not> isl b \<longrightarrow> R2 (projr a) (projr b)))"
    85   by (cases a b rule: sum.exhaust[case_product sum.exhaust]) simp_all
    86 
    87 lemma isl_transfer: "rel_fun (rel_sum A B) (op =) isl isl"
    88   unfolding rel_fun_def rel_sum_sel by simp
    89 
    90 lemma rel_prod_sel: "rel_prod R1 R2 p q = (R1 (fst p) (fst q) \<and> R2 (snd p) (snd q))"
    91   by (force simp: rel_prod.simps elim: rel_prod.cases)
    92 
    93 ML_file "Tools/BNF/bnf_lfp_basic_sugar.ML"
    94 
    95 ML_file "~~/src/HOL/Tools/Old_Datatype/old_size.ML"
    96 
    97 lemma size_bool[code]: "size (b\<Colon>bool) = 0"
    98   by (cases b) auto
    99 
   100 declare prod.size[no_atp]
   101 
   102 lemmas size_nat = size_nat_def
   103 
   104 hide_const (open) xtor ctor_rec
   105 
   106 hide_fact (open)
   107   xtor_def xtor_map xtor_set xtor_rel xtor_induct xtor_xtor xtor_inject ctor_rec_def ctor_rec
   108   ctor_rec_def_alt ctor_rec_o_map xtor_rel_induct Inl_def_alt Inr_def_alt Pair_def_alt
   109 
   110 end