src/HOL/BNF/BNF_Def.thy
author traytel
Mon Jul 22 21:12:15 2013 +0200 (2013-07-22)
changeset 52719 480a3479fa47
parent 52660 7f7311d04727
child 52730 6bf02eb4ddf7
permissions -rw-r--r--
transfer rule for map (not yet registered as a transfer rule)
     1 (*  Title:      HOL/BNF/BNF_Def.thy
     2     Author:     Dmitriy Traytel, TU Muenchen
     3     Copyright   2012
     4 
     5 Definition of bounded natural functors.
     6 *)
     7 
     8 header {* Definition of Bounded Natural Functors *}
     9 
    10 theory BNF_Def
    11 imports BNF_Util
    12 keywords
    13   "print_bnfs" :: diag and
    14   "bnf" :: thy_goal
    15 begin
    16 
    17 lemma collect_o: "collect F o g = collect ((\<lambda>f. f o g) ` F)"
    18 by (rule ext) (auto simp only: o_apply collect_def)
    19 
    20 lemma conversep_mono:
    21 "R1 ^--1 \<le> R2 ^--1 \<longleftrightarrow> R1 \<le> R2"
    22 unfolding conversep.simps by auto
    23 
    24 lemma converse_shift:
    25 "R1 \<subseteq> R2 ^-1 \<Longrightarrow> R1 ^-1 \<subseteq> R2"
    26 unfolding converse_def by auto
    27 
    28 lemma conversep_shift:
    29 "R1 \<le> R2 ^--1 \<Longrightarrow> R1 ^--1 \<le> R2"
    30 unfolding conversep.simps by auto
    31 
    32 definition convol ("<_ , _>") where
    33 "<f , g> \<equiv> %a. (f a, g a)"
    34 
    35 lemma fst_convol:
    36 "fst o <f , g> = f"
    37 apply(rule ext)
    38 unfolding convol_def by simp
    39 
    40 lemma snd_convol:
    41 "snd o <f , g> = g"
    42 apply(rule ext)
    43 unfolding convol_def by simp
    44 
    45 lemma convol_mem_GrpI:
    46 "\<lbrakk>g x = g' x; x \<in> A\<rbrakk> \<Longrightarrow> <id , g> x \<in> (Collect (split (Grp A g)))"
    47 unfolding convol_def Grp_def by auto
    48 
    49 definition csquare where
    50 "csquare A f1 f2 p1 p2 \<longleftrightarrow> (\<forall> a \<in> A. f1 (p1 a) = f2 (p2 a))"
    51 
    52 (* The pullback of sets *)
    53 definition thePull where
    54 "thePull B1 B2 f1 f2 = {(b1,b2). b1 \<in> B1 \<and> b2 \<in> B2 \<and> f1 b1 = f2 b2}"
    55 
    56 lemma wpull_thePull:
    57 "wpull (thePull B1 B2 f1 f2) B1 B2 f1 f2 fst snd"
    58 unfolding wpull_def thePull_def by auto
    59 
    60 lemma wppull_thePull:
    61 assumes "wppull A B1 B2 f1 f2 e1 e2 p1 p2"
    62 shows
    63 "\<exists> j. \<forall> a' \<in> thePull B1 B2 f1 f2.
    64    j a' \<in> A \<and>
    65    e1 (p1 (j a')) = e1 (fst a') \<and> e2 (p2 (j a')) = e2 (snd a')"
    66 (is "\<exists> j. \<forall> a' \<in> ?A'. ?phi a' (j a')")
    67 proof(rule bchoice[of ?A' ?phi], default)
    68   fix a' assume a': "a' \<in> ?A'"
    69   hence "fst a' \<in> B1" unfolding thePull_def by auto
    70   moreover
    71   from a' have "snd a' \<in> B2" unfolding thePull_def by auto
    72   moreover have "f1 (fst a') = f2 (snd a')"
    73   using a' unfolding csquare_def thePull_def by auto
    74   ultimately show "\<exists> ja'. ?phi a' ja'"
    75   using assms unfolding wppull_def by blast
    76 qed
    77 
    78 lemma wpull_wppull:
    79 assumes wp: "wpull A' B1 B2 f1 f2 p1' p2'" and
    80 1: "\<forall> a' \<in> A'. j a' \<in> A \<and> e1 (p1 (j a')) = e1 (p1' a') \<and> e2 (p2 (j a')) = e2 (p2' a')"
    81 shows "wppull A B1 B2 f1 f2 e1 e2 p1 p2"
    82 unfolding wppull_def proof safe
    83   fix b1 b2
    84   assume b1: "b1 \<in> B1" and b2: "b2 \<in> B2" and f: "f1 b1 = f2 b2"
    85   then obtain a' where a': "a' \<in> A'" and b1: "b1 = p1' a'" and b2: "b2 = p2' a'"
    86   using wp unfolding wpull_def by blast
    87   show "\<exists>a\<in>A. e1 (p1 a) = e1 b1 \<and> e2 (p2 a) = e2 b2"
    88   apply (rule bexI[of _ "j a'"]) unfolding b1 b2 using a' 1 by auto
    89 qed
    90 
    91 lemma wppull_id: "\<lbrakk>wpull UNIV UNIV UNIV f1 f2 p1 p2; e1 = id; e2 = id\<rbrakk> \<Longrightarrow>
    92    wppull UNIV UNIV UNIV f1 f2 e1 e2 p1 p2"
    93 by (erule wpull_wppull) auto
    94 
    95 lemma eq_alt: "op = = Grp UNIV id"
    96 unfolding Grp_def by auto
    97 
    98 lemma leq_conversepI: "R = op = \<Longrightarrow> R \<le> R^--1"
    99   by auto
   100 
   101 lemma eq_OOI: "R = op = \<Longrightarrow> R = R OO R"
   102   by auto
   103 
   104 lemma Grp_UNIV_id: "f = id \<Longrightarrow> (Grp UNIV f)^--1 OO Grp UNIV f = Grp UNIV f"
   105 unfolding Grp_def by auto
   106 
   107 lemma Grp_UNIV_idI: "x = y \<Longrightarrow> Grp UNIV id x y"
   108 unfolding Grp_def by auto
   109 
   110 lemma Grp_mono: "A \<le> B \<Longrightarrow> Grp A f \<le> Grp B f"
   111 unfolding Grp_def by auto
   112 
   113 lemma GrpI: "\<lbrakk>f x = y; x \<in> A\<rbrakk> \<Longrightarrow> Grp A f x y"
   114 unfolding Grp_def by auto
   115 
   116 lemma GrpE: "Grp A f x y \<Longrightarrow> (\<lbrakk>f x = y; x \<in> A\<rbrakk> \<Longrightarrow> R) \<Longrightarrow> R"
   117 unfolding Grp_def by auto
   118 
   119 lemma Collect_split_Grp_eqD: "z \<in> Collect (split (Grp A f)) \<Longrightarrow> (f \<circ> fst) z = snd z"
   120 unfolding Grp_def o_def by auto
   121 
   122 lemma Collect_split_Grp_inD: "z \<in> Collect (split (Grp A f)) \<Longrightarrow> fst z \<in> A"
   123 unfolding Grp_def o_def by auto
   124 
   125 lemma wpull_Grp:
   126 "wpull (Collect (split (Grp A f))) A (f ` A) f id fst snd"
   127 unfolding wpull_def Grp_def by auto
   128 
   129 definition "pick_middlep P Q a c = (SOME b. P a b \<and> Q b c)"
   130 
   131 lemma pick_middlep:
   132 "(P OO Q) a c \<Longrightarrow> P a (pick_middlep P Q a c) \<and> Q (pick_middlep P Q a c) c"
   133 unfolding pick_middlep_def apply(rule someI_ex) by auto
   134 
   135 definition fstOp where "fstOp P Q ac = (fst ac, pick_middlep P Q (fst ac) (snd ac))"
   136 definition sndOp where "sndOp P Q ac = (pick_middlep P Q (fst ac) (snd ac), (snd ac))"
   137 
   138 lemma fstOp_in: "ac \<in> Collect (split (P OO Q)) \<Longrightarrow> fstOp P Q ac \<in> Collect (split P)"
   139 unfolding fstOp_def mem_Collect_eq
   140 by (subst (asm) surjective_pairing, unfold prod.cases) (erule pick_middlep[THEN conjunct1])
   141 
   142 lemma fst_fstOp: "fst bc = (fst \<circ> fstOp P Q) bc"
   143 unfolding comp_def fstOp_def by simp
   144 
   145 lemma snd_sndOp: "snd bc = (snd \<circ> sndOp P Q) bc"
   146 unfolding comp_def sndOp_def by simp
   147 
   148 lemma sndOp_in: "ac \<in> Collect (split (P OO Q)) \<Longrightarrow> sndOp P Q ac \<in> Collect (split Q)"
   149 unfolding sndOp_def mem_Collect_eq
   150 by (subst (asm) surjective_pairing, unfold prod.cases) (erule pick_middlep[THEN conjunct2])
   151 
   152 lemma csquare_fstOp_sndOp:
   153 "csquare (Collect (split (P OO Q))) snd fst (fstOp P Q) (sndOp P Q)"
   154 unfolding csquare_def fstOp_def sndOp_def using pick_middlep by simp
   155 
   156 lemma wppull_fstOp_sndOp:
   157 shows "wppull (Collect (split (P OO Q))) (Collect (split P)) (Collect (split Q))
   158   snd fst fst snd (fstOp P Q) (sndOp P Q)"
   159 using pick_middlep unfolding wppull_def fstOp_def sndOp_def relcompp.simps by auto
   160 
   161 lemma snd_fst_flip: "snd xy = (fst o (%(x, y). (y, x))) xy"
   162 by (simp split: prod.split)
   163 
   164 lemma fst_snd_flip: "fst xy = (snd o (%(x, y). (y, x))) xy"
   165 by (simp split: prod.split)
   166 
   167 lemma flip_pred: "A \<subseteq> Collect (split (R ^--1)) \<Longrightarrow> (%(x, y). (y, x)) ` A \<subseteq> Collect (split R)"
   168 by auto
   169 
   170 lemma Collect_split_mono: "A \<le> B \<Longrightarrow> Collect (split A) \<subseteq> Collect (split B)"
   171   by auto
   172 
   173 lemma Collect_split_mono_strong: 
   174   "\<lbrakk>\<forall>a\<in>fst ` A. \<forall>b \<in> snd ` A. P a b \<longrightarrow> Q a b; A \<subseteq> Collect (split P)\<rbrakk> \<Longrightarrow>
   175   A \<subseteq> Collect (split Q)"
   176   by fastforce
   177 
   178 lemma predicate2_eqD: "A = B \<Longrightarrow> A a b \<longleftrightarrow> B a b"
   179 by metis
   180 
   181 lemma sum_case_o_inj:
   182 "sum_case f g \<circ> Inl = f"
   183 "sum_case f g \<circ> Inr = g"
   184 by auto
   185 
   186 lemma card_order_csum_cone_cexp_def:
   187   "card_order r \<Longrightarrow> ( |A1| +c cone) ^c r = |Func UNIV (Inl ` A1 \<union> {Inr ()})|"
   188   unfolding cexp_def cone_def Field_csum Field_card_of by (auto dest: Field_card_order)
   189 
   190 lemma If_the_inv_into_in_Func:
   191   "\<lbrakk>inj_on g C; C \<subseteq> B \<union> {x}\<rbrakk> \<Longrightarrow>
   192   (\<lambda>i. if i \<in> g ` C then the_inv_into C g i else x) \<in> Func UNIV (B \<union> {x})"
   193 unfolding Func_def by (auto dest: the_inv_into_into)
   194 
   195 lemma If_the_inv_into_f_f:
   196   "\<lbrakk>i \<in> C; inj_on g C\<rbrakk> \<Longrightarrow>
   197   ((\<lambda>i. if i \<in> g ` C then the_inv_into C g i else x) o g) i = id i"
   198 unfolding Func_def by (auto elim: the_inv_into_f_f)
   199 
   200 definition vimagep where
   201   "vimagep f g R = (\<lambda>x y. R (f x) (g y))"
   202 
   203 lemma vimagepI: "R (f x) (g y) \<Longrightarrow> vimagep f g R x y"
   204   unfolding vimagep_def by -
   205 
   206 lemma vimagepD: "vimagep f g R x y \<Longrightarrow> R (f x) (g y)"
   207   unfolding vimagep_def by -
   208 
   209 lemma fun_rel_iff_leq_vimagep: "(fun_rel R S) f g = (R \<le> vimagep f g S)"
   210   unfolding fun_rel_def vimagep_def by auto
   211 
   212 lemma convol_image_vimagep: "<f o fst, g o snd> ` Collect (split (vimagep f g R)) \<subseteq> Collect (split R)"
   213   unfolding vimagep_def convol_def by auto
   214 
   215 ML_file "Tools/bnf_def_tactics.ML"
   216 ML_file "Tools/bnf_def.ML"
   217 
   218 
   219 end