src/HOL/BNF_Composition.thy
author blanchet
Thu Sep 11 18:54:36 2014 +0200 (2014-09-11)
changeset 58306 117ba6cbe414
parent 58282 48e16d74845b
child 58353 c9f374b64d99
permissions -rw-r--r--
renamed 'rep_datatype' to 'old_rep_datatype' (HOL)
     1 (*  Title:      HOL/BNF_Composition.thy
     2     Author:     Dmitriy Traytel, TU Muenchen
     3     Author:     Jasmin Blanchette, TU Muenchen
     4     Copyright   2012, 2013, 2014
     5 
     6 Composition of bounded natural functors.
     7 *)
     8 
     9 header {* Composition of Bounded Natural Functors *}
    10 
    11 theory BNF_Composition
    12 imports BNF_Def
    13 begin
    14 
    15 lemma empty_natural: "(\<lambda>_. {}) o f = image g o (\<lambda>_. {})"
    16   by (rule ext) simp
    17 
    18 lemma Union_natural: "Union o image (image f) = image f o Union"
    19   by (rule ext) (auto simp only: comp_apply)
    20 
    21 lemma in_Union_o_assoc: "x \<in> (Union o gset o gmap) A \<Longrightarrow> x \<in> (Union o (gset o gmap)) A"
    22   by (unfold comp_assoc)
    23 
    24 lemma comp_single_set_bd:
    25   assumes fbd_Card_order: "Card_order fbd" and
    26     fset_bd: "\<And>x. |fset x| \<le>o fbd" and
    27     gset_bd: "\<And>x. |gset x| \<le>o gbd"
    28   shows "|\<Union>(fset ` gset x)| \<le>o gbd *c fbd"
    29   apply simp
    30   apply (rule ordLeq_transitive)
    31   apply (rule card_of_UNION_Sigma)
    32   apply (subst SIGMA_CSUM)
    33   apply (rule ordLeq_transitive)
    34   apply (rule card_of_Csum_Times')
    35   apply (rule fbd_Card_order)
    36   apply (rule ballI)
    37   apply (rule fset_bd)
    38   apply (rule ordLeq_transitive)
    39   apply (rule cprod_mono1)
    40   apply (rule gset_bd)
    41   apply (rule ordIso_imp_ordLeq)
    42   apply (rule ordIso_refl)
    43   apply (rule Card_order_cprod)
    44   done
    45 
    46 lemma csum_dup: "cinfinite r \<Longrightarrow> Card_order r \<Longrightarrow> p +c p' =o r +c r \<Longrightarrow> p +c p' =o r"
    47   apply (erule ordIso_transitive)
    48   apply (frule csum_absorb2')
    49   apply (erule ordLeq_refl)
    50   by simp
    51 
    52 lemma cprod_dup: "cinfinite r \<Longrightarrow> Card_order r \<Longrightarrow> p *c p' =o r *c r \<Longrightarrow> p *c p' =o r"
    53   apply (erule ordIso_transitive)
    54   apply (rule cprod_infinite)
    55   by simp
    56 
    57 lemma Union_image_insert: "\<Union>(f ` insert a B) = f a \<union> \<Union>(f ` B)"
    58   by simp
    59 
    60 lemma Union_image_empty: "A \<union> \<Union>(f ` {}) = A"
    61   by simp
    62 
    63 lemma image_o_collect: "collect ((\<lambda>f. image g o f) ` F) = image g o collect F"
    64   by (rule ext) (auto simp add: collect_def)
    65 
    66 lemma conj_subset_def: "A \<subseteq> {x. P x \<and> Q x} = (A \<subseteq> {x. P x} \<and> A \<subseteq> {x. Q x})"
    67   by blast
    68 
    69 lemma UN_image_subset: "\<Union>(f ` g x) \<subseteq> X = (g x \<subseteq> {x. f x \<subseteq> X})"
    70   by blast
    71 
    72 lemma comp_set_bd_Union_o_collect: "|\<Union>\<Union>((\<lambda>f. f x) ` X)| \<le>o hbd \<Longrightarrow> |(Union \<circ> collect X) x| \<le>o hbd"
    73   by (unfold comp_apply collect_def) simp
    74 
    75 lemma wpull_cong:
    76   "\<lbrakk>A' = A; B1' = B1; B2' = B2; wpull A B1 B2 f1 f2 p1 p2\<rbrakk> \<Longrightarrow> wpull A' B1' B2' f1 f2 p1 p2"
    77   by simp
    78 
    79 lemma Grp_fst_snd: "(Grp (Collect (split R)) fst)^--1 OO Grp (Collect (split R)) snd = R"
    80   unfolding Grp_def fun_eq_iff relcompp.simps by auto
    81 
    82 lemma OO_Grp_cong: "A = B \<Longrightarrow> (Grp A f)^--1 OO Grp A g = (Grp B f)^--1 OO Grp B g"
    83   by (rule arg_cong)
    84 
    85 lemma vimage2p_relcompp_mono: "R OO S \<le> T \<Longrightarrow>
    86   vimage2p f g R OO vimage2p g h S \<le> vimage2p f h T"
    87   unfolding vimage2p_def by auto
    88 
    89 lemma type_copy_map_cong0: "M (g x) = N (h x) \<Longrightarrow> (f o M o g) x = (f o N o h) x"
    90   by auto
    91 
    92 lemma type_copy_set_bd: "(\<And>y. |S y| \<le>o bd) \<Longrightarrow> |(S o Rep) x| \<le>o bd"
    93   by auto
    94 
    95 lemma vimage2p_cong: "R = S \<Longrightarrow> vimage2p f g R = vimage2p f g S"
    96   by simp
    97 
    98 context
    99   fixes Rep Abs
   100   assumes type_copy: "type_definition Rep Abs UNIV"
   101 begin
   102 
   103 lemma type_copy_map_id0: "M = id \<Longrightarrow> Abs o M o Rep = id"
   104   using type_definition.Rep_inverse[OF type_copy] by auto
   105 
   106 lemma type_copy_map_comp0: "M = M1 o M2 \<Longrightarrow> f o M o g = (f o M1 o Rep) o (Abs o M2 o g)"
   107   using type_definition.Abs_inverse[OF type_copy UNIV_I] by auto
   108 
   109 lemma type_copy_set_map0: "S o M = image f o S' \<Longrightarrow> (S o Rep) o (Abs o M o g) = image f o (S' o g)"
   110   using type_definition.Abs_inverse[OF type_copy UNIV_I] by (auto simp: o_def fun_eq_iff)
   111 
   112 lemma type_copy_wit: "x \<in> (S o Rep) (Abs y) \<Longrightarrow> x \<in> S y"
   113   using type_definition.Abs_inverse[OF type_copy UNIV_I] by auto
   114 
   115 lemma type_copy_vimage2p_Grp_Rep: "vimage2p f Rep (Grp (Collect P) h) =
   116     Grp (Collect (\<lambda>x. P (f x))) (Abs o h o f)"
   117   unfolding vimage2p_def Grp_def fun_eq_iff
   118   by (auto simp: type_definition.Abs_inverse[OF type_copy UNIV_I]
   119    type_definition.Rep_inverse[OF type_copy] dest: sym)
   120 
   121 lemma type_copy_vimage2p_Grp_Abs:
   122   "\<And>h. vimage2p g Abs (Grp (Collect P) h) = Grp (Collect (\<lambda>x. P (g x))) (Rep o h o g)"
   123   unfolding vimage2p_def Grp_def fun_eq_iff
   124   by (auto simp: type_definition.Abs_inverse[OF type_copy UNIV_I]
   125    type_definition.Rep_inverse[OF type_copy] dest: sym)
   126 
   127 lemma type_copy_ex_RepI: "(\<exists>b. F b) = (\<exists>b. F (Rep b))"
   128 proof safe
   129   fix b assume "F b"
   130   show "\<exists>b'. F (Rep b')"
   131   proof (rule exI)
   132     from `F b` show "F (Rep (Abs b))" using type_definition.Abs_inverse[OF type_copy] by auto
   133   qed
   134 qed blast
   135 
   136 lemma vimage2p_relcompp_converse:
   137   "vimage2p f g (R^--1 OO S) = (vimage2p Rep f R)^--1 OO vimage2p Rep g S"
   138   unfolding vimage2p_def relcompp.simps conversep.simps fun_eq_iff image_def
   139   by (auto simp: type_copy_ex_RepI)
   140 
   141 end
   142 
   143 bnf DEADID: 'a
   144   map: "id :: 'a \<Rightarrow> 'a"
   145   bd: natLeq
   146   rel: "op = :: 'a \<Rightarrow> 'a \<Rightarrow> bool"
   147   by (auto simp add: Grp_def natLeq_card_order natLeq_cinfinite)
   148 
   149 definition id_bnf :: "'a \<Rightarrow> 'a" where "id_bnf \<equiv> (\<lambda>x. x)"
   150 
   151 lemma id_bnf_apply: "id_bnf x = x"
   152   unfolding id_bnf_def by simp
   153 
   154 bnf ID: 'a
   155   map: "id_bnf :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
   156   sets: "\<lambda>x. {x}"
   157   bd: natLeq
   158   rel: "id_bnf :: ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
   159   unfolding id_bnf_def
   160   apply (auto simp: Grp_def fun_eq_iff relcompp.simps natLeq_card_order natLeq_cinfinite)
   161   apply (rule ordLess_imp_ordLeq[OF finite_ordLess_infinite[OF _ natLeq_Well_order]])
   162   apply (auto simp add: Field_card_of Field_natLeq card_of_well_order_on)[3]
   163   done
   164 
   165 lemma type_definition_id_bnf_UNIV: "type_definition id_bnf id_bnf UNIV"
   166   unfolding id_bnf_def by unfold_locales auto
   167 
   168 ML_file "Tools/BNF/bnf_comp_tactics.ML"
   169 ML_file "Tools/BNF/bnf_comp.ML"
   170 
   171 hide_fact
   172   DEADID.inj_map DEADID.inj_map_strong DEADID.map_comp DEADID.map_cong DEADID.map_cong0
   173   DEADID.map_cong_simp DEADID.map_id DEADID.map_id0 DEADID.map_ident DEADID.map_transfer
   174   DEADID.rel_Grp DEADID.rel_compp DEADID.rel_compp_Grp DEADID.rel_conversep DEADID.rel_eq
   175   DEADID.rel_flip DEADID.rel_map DEADID.rel_mono DEADID.rel_transfer
   176   ID.inj_map ID.inj_map_strong ID.map_comp ID.map_cong ID.map_cong0 ID.map_cong_simp ID.map_id
   177   ID.map_id0 ID.map_ident ID.map_transfer ID.rel_Grp ID.rel_compp ID.rel_compp_Grp ID.rel_conversep
   178   ID.rel_eq ID.rel_flip ID.rel_map ID.rel_mono ID.rel_transfer ID.set_map ID.set_transfer
   179 
   180 hide_const (open) id_bnf
   181 hide_fact (open) id_bnf_def type_definition_id_bnf_UNIV
   182 
   183 end