src/HOL/BNF/BNF_Def.thy
 author blanchet Fri Sep 21 16:45:06 2012 +0200 (2012-09-21) changeset 49510 ba50d204095e parent 49509 src/HOL/Codatatype/BNF_Def.thy@163914705f8d child 49537 fe1deee434b6 permissions -rw-r--r--
renamed "Codatatype" directory "BNF" (and corresponding session) -- this opens the door to no-nonsense session names like "HOL-BNF-LFP"
```     1 (*  Title:      HOL/BNF/BNF_Def.thy
```
```     2     Author:     Dmitriy Traytel, TU Muenchen
```
```     3     Copyright   2012
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```     4
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```     5 Definition of bounded natural functors.
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```     6 *)
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```     7
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```     8 header {* Definition of Bounded Natural Functors *}
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```     9
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```    10 theory BNF_Def
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```    11 imports BNF_Util
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```    12 keywords
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```    13   "print_bnfs" :: diag and
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```    14   "bnf_def" :: thy_goal
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```    15 begin
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```    16
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```    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
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```    20 lemma converse_mono:
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```    21 "R1 ^-1 \<subseteq> R2 ^-1 \<longleftrightarrow> R1 \<subseteq> R2"
```
```    22 unfolding converse_def by auto
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```    23
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```    24 lemma converse_shift:
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```    25 "R1 \<subseteq> R2 ^-1 \<Longrightarrow> R1 ^-1 \<subseteq> R2"
```
```    26 unfolding converse_def by auto
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```    27
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```    28 definition convol ("<_ , _>") where
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```    29 "<f , g> \<equiv> %a. (f a, g a)"
```
```    30
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```    31 lemma fst_convol:
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```    32 "fst o <f , g> = f"
```
```    33 apply(rule ext)
```
```    34 unfolding convol_def by simp
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```    35
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```    36 lemma snd_convol:
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```    37 "snd o <f , g> = g"
```
```    38 apply(rule ext)
```
```    39 unfolding convol_def by simp
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```    40
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```    41 lemma convol_memI:
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```    42 "\<lbrakk>f x = f' x; g x = g' x; P x\<rbrakk> \<Longrightarrow> <f , g> x \<in> {(f' a, g' a) |a. P a}"
```
```    43 unfolding convol_def by auto
```
```    44
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```    45 definition csquare where
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```    46 "csquare A f1 f2 p1 p2 \<longleftrightarrow> (\<forall> a \<in> A. f1 (p1 a) = f2 (p2 a))"
```
```    47
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```    48 (* The pullback of sets *)
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```    49 definition thePull where
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```    50 "thePull B1 B2 f1 f2 = {(b1,b2). b1 \<in> B1 \<and> b2 \<in> B2 \<and> f1 b1 = f2 b2}"
```
```    51
```
```    52 lemma wpull_thePull:
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```    53 "wpull (thePull B1 B2 f1 f2) B1 B2 f1 f2 fst snd"
```
```    54 unfolding wpull_def thePull_def by auto
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```    55
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```    56 lemma wppull_thePull:
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```    57 assumes "wppull A B1 B2 f1 f2 e1 e2 p1 p2"
```
```    58 shows
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```    59 "\<exists> j. \<forall> a' \<in> thePull B1 B2 f1 f2.
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```    60    j a' \<in> A \<and>
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```    61    e1 (p1 (j a')) = e1 (fst a') \<and> e2 (p2 (j a')) = e2 (snd a')"
```
```    62 (is "\<exists> j. \<forall> a' \<in> ?A'. ?phi a' (j a')")
```
```    63 proof(rule bchoice[of ?A' ?phi], default)
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```    64   fix a' assume a': "a' \<in> ?A'"
```
```    65   hence "fst a' \<in> B1" unfolding thePull_def by auto
```
```    66   moreover
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```    67   from a' have "snd a' \<in> B2" unfolding thePull_def by auto
```
```    68   moreover have "f1 (fst a') = f2 (snd a')"
```
```    69   using a' unfolding csquare_def thePull_def by auto
```
```    70   ultimately show "\<exists> ja'. ?phi a' ja'"
```
```    71   using assms unfolding wppull_def by blast
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```    72 qed
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```    73
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```    74 lemma wpull_wppull:
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```    75 assumes wp: "wpull A' B1 B2 f1 f2 p1' p2'" and
```
```    76 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')"
```
```    77 shows "wppull A B1 B2 f1 f2 e1 e2 p1 p2"
```
```    78 unfolding wppull_def proof safe
```
```    79   fix b1 b2
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```    80   assume b1: "b1 \<in> B1" and b2: "b2 \<in> B2" and f: "f1 b1 = f2 b2"
```
```    81   then obtain a' where a': "a' \<in> A'" and b1: "b1 = p1' a'" and b2: "b2 = p2' a'"
```
```    82   using wp unfolding wpull_def by blast
```
```    83   show "\<exists>a\<in>A. e1 (p1 a) = e1 b1 \<and> e2 (p2 a) = e2 b2"
```
```    84   apply (rule bexI[of _ "j a'"]) unfolding b1 b2 using a' 1 by auto
```
```    85 qed
```
```    86
```
```    87 lemma wppull_id: "\<lbrakk>wpull UNIV UNIV UNIV f1 f2 p1 p2; e1 = id; e2 = id\<rbrakk> \<Longrightarrow>
```
```    88    wppull UNIV UNIV UNIV f1 f2 e1 e2 p1 p2"
```
```    89 by (erule wpull_wppull) auto
```
```    90
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```    91 lemma Id_alt: "Id = Gr UNIV id"
```
```    92 unfolding Gr_def by auto
```
```    93
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```    94 lemma Gr_UNIV_id: "f = id \<Longrightarrow> (Gr UNIV f)^-1 O Gr UNIV f = Gr UNIV f"
```
```    95 unfolding Gr_def by auto
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```    96
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```    97 lemma Gr_mono: "A \<subseteq> B \<Longrightarrow> Gr A f \<subseteq> Gr B f"
```
```    98 unfolding Gr_def by auto
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```    99
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```   100 lemma wpull_Gr:
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```   101 "wpull (Gr A f) A (f ` A) f id fst snd"
```
```   102 unfolding wpull_def Gr_def by auto
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```   103
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```   104 definition "pick_middle P Q a c = (SOME b. (a,b) \<in> P \<and> (b,c) \<in> Q)"
```
```   105
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```   106 lemma pick_middle:
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```   107 "(a,c) \<in> P O Q \<Longrightarrow> (a, pick_middle P Q a c) \<in> P \<and> (pick_middle P Q a c, c) \<in> Q"
```
```   108 unfolding pick_middle_def apply(rule someI_ex)
```
```   109 using assms unfolding relcomp_def by auto
```
```   110
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```   111 definition fstO where "fstO P Q ac = (fst ac, pick_middle P Q (fst ac) (snd ac))"
```
```   112 definition sndO where "sndO P Q ac = (pick_middle P Q (fst ac) (snd ac), snd ac)"
```
```   113
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```   114 lemma fstO_in: "ac \<in> P O Q \<Longrightarrow> fstO P Q ac \<in> P"
```
```   115 unfolding fstO_def
```
```   116 by (subst (asm) surjective_pairing) (rule pick_middle[THEN conjunct1])
```
```   117
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```   118 lemma fst_fstO: "fst bc = (fst \<circ> fstO P Q) bc"
```
```   119 unfolding comp_def fstO_def by simp
```
```   120
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```   121 lemma snd_sndO: "snd bc = (snd \<circ> sndO P Q) bc"
```
```   122 unfolding comp_def sndO_def by simp
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```   123
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```   124 lemma sndO_in: "ac \<in> P O Q \<Longrightarrow> sndO P Q ac \<in> Q"
```
```   125 unfolding sndO_def
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```   126 by (subst (asm) surjective_pairing) (rule pick_middle[THEN conjunct2])
```
```   127
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```   128 lemma csquare_fstO_sndO:
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```   129 "csquare (P O Q) snd fst (fstO P Q) (sndO P Q)"
```
```   130 unfolding csquare_def fstO_def sndO_def using pick_middle by simp
```
```   131
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```   132 lemma wppull_fstO_sndO:
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```   133 shows "wppull (P O Q) P Q snd fst fst snd (fstO P Q) (sndO P Q)"
```
```   134 using pick_middle unfolding wppull_def fstO_def sndO_def relcomp_def by auto
```
```   135
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```   136 lemma snd_fst_flip: "snd xy = (fst o (%(x, y). (y, x))) xy"
```
```   137 by (simp split: prod.split)
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```   138
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```   139 lemma fst_snd_flip: "fst xy = (snd o (%(x, y). (y, x))) xy"
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```   140 by (simp split: prod.split)
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```   141
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```   142 lemma flip_rel: "A \<subseteq> (R ^-1) \<Longrightarrow> (%(x, y). (y, x)) ` A \<subseteq> R"
```
```   143 by auto
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```   144
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```   145 lemma pointfreeE: "f o g = f' o g' \<Longrightarrow> f (g x) = f' (g' x)"
```
```   146 unfolding o_def fun_eq_iff by simp
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```   147
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```   148 ML_file "Tools/bnf_def_tactics.ML"
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```   149 ML_file"Tools/bnf_def.ML"
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```   150
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```   151 end
```