src/HOL/BNF/BNF_GFP.thy
 author traytel Mon Jul 15 15:50:39 2013 +0200 (2013-07-15) changeset 52660 7f7311d04727 parent 52659 58b87aa4dc3b child 52731 dacd47a0633f permissions -rw-r--r--
killed unused theorems
```     1 (*  Title:      HOL/BNF/BNF_GFP.thy
```
```     2     Author:     Dmitriy Traytel, TU Muenchen
```
```     3     Copyright   2012
```
```     4
```
```     5 Greatest fixed point operation on bounded natural functors.
```
```     6 *)
```
```     7
```
```     8 header {* Greatest Fixed Point Operation on Bounded Natural Functors *}
```
```     9
```
```    10 theory BNF_GFP
```
```    11 imports BNF_FP_Basic Equiv_Relations_More "~~/src/HOL/Library/Sublist"
```
```    12 keywords
```
```    13   "codatatype" :: thy_decl
```
```    14 begin
```
```    15
```
```    16 lemma o_sum_case: "h o sum_case f g = sum_case (h o f) (h o g)"
```
```    17 unfolding o_def by (auto split: sum.splits)
```
```    18
```
```    19 lemma sum_case_expand_Inr: "f o Inl = g \<Longrightarrow> f x = sum_case g (f o Inr) x"
```
```    20 by (auto split: sum.splits)
```
```    21
```
```    22 lemma sum_case_expand_Inr': "f o Inl = g \<Longrightarrow> h = f o Inr \<longleftrightarrow> sum_case g h = f"
```
```    23 by (metis sum_case_o_inj(1,2) surjective_sum)
```
```    24
```
```    25 lemma converse_Times: "(A \<times> B) ^-1 = B \<times> A"
```
```    26 by auto
```
```    27
```
```    28 lemma equiv_triv1:
```
```    29 assumes "equiv A R" and "(a, b) \<in> R" and "(a, c) \<in> R"
```
```    30 shows "(b, c) \<in> R"
```
```    31 using assms unfolding equiv_def sym_def trans_def by blast
```
```    32
```
```    33 lemma equiv_triv2:
```
```    34 assumes "equiv A R" and "(a, b) \<in> R" and "(b, c) \<in> R"
```
```    35 shows "(a, c) \<in> R"
```
```    36 using assms unfolding equiv_def trans_def by blast
```
```    37
```
```    38 lemma equiv_proj:
```
```    39   assumes e: "equiv A R" and "z \<in> R"
```
```    40   shows "(proj R o fst) z = (proj R o snd) z"
```
```    41 proof -
```
```    42   from assms(2) have z: "(fst z, snd z) \<in> R" by auto
```
```    43   have P: "\<And>x. (fst z, x) \<in> R \<Longrightarrow> (snd z, x) \<in> R" by (erule equiv_triv1[OF e z])
```
```    44   have "\<And>x. (snd z, x) \<in> R \<Longrightarrow> (fst z, x) \<in> R" by (erule equiv_triv2[OF e z])
```
```    45   with P show ?thesis unfolding proj_def[abs_def] by auto
```
```    46 qed
```
```    47
```
```    48 (* Operators: *)
```
```    49 definition image2 where "image2 A f g = {(f a, g a) | a. a \<in> A}"
```
```    50
```
```    51
```
```    52 lemma Id_onD: "(a, b) \<in> Id_on A \<Longrightarrow> a = b"
```
```    53 unfolding Id_on_def by simp
```
```    54
```
```    55 lemma Id_onD': "x \<in> Id_on A \<Longrightarrow> fst x = snd x"
```
```    56 unfolding Id_on_def by auto
```
```    57
```
```    58 lemma Id_on_fst: "x \<in> Id_on A \<Longrightarrow> fst x \<in> A"
```
```    59 unfolding Id_on_def by auto
```
```    60
```
```    61 lemma Id_on_UNIV: "Id_on UNIV = Id"
```
```    62 unfolding Id_on_def by auto
```
```    63
```
```    64 lemma Id_on_Comp: "Id_on A = Id_on A O Id_on A"
```
```    65 unfolding Id_on_def by auto
```
```    66
```
```    67 lemma Id_on_Gr: "Id_on A = Gr A id"
```
```    68 unfolding Id_on_def Gr_def by auto
```
```    69
```
```    70 lemma Id_on_UNIV_I: "x = y \<Longrightarrow> (x, y) \<in> Id_on UNIV"
```
```    71 unfolding Id_on_def by auto
```
```    72
```
```    73 lemma image2_eqI: "\<lbrakk>b = f x; c = g x; x \<in> A\<rbrakk> \<Longrightarrow> (b, c) \<in> image2 A f g"
```
```    74 unfolding image2_def by auto
```
```    75
```
```    76 lemma eq_subset: "op = \<le> (\<lambda>a b. P a b \<or> a = b)"
```
```    77 by auto
```
```    78
```
```    79 lemma IdD: "(a, b) \<in> Id \<Longrightarrow> a = b"
```
```    80 by auto
```
```    81
```
```    82 lemma image2_Gr: "image2 A f g = (Gr A f)^-1 O (Gr A g)"
```
```    83 unfolding image2_def Gr_def by auto
```
```    84
```
```    85 lemma GrD1: "(x, fx) \<in> Gr A f \<Longrightarrow> x \<in> A"
```
```    86 unfolding Gr_def by simp
```
```    87
```
```    88 lemma GrD2: "(x, fx) \<in> Gr A f \<Longrightarrow> f x = fx"
```
```    89 unfolding Gr_def by simp
```
```    90
```
```    91 lemma Gr_incl: "Gr A f \<subseteq> A <*> B \<longleftrightarrow> f ` A \<subseteq> B"
```
```    92 unfolding Gr_def by auto
```
```    93
```
```    94 lemma in_rel_Collect_split_eq: "in_rel (Collect (split X)) = X"
```
```    95 unfolding fun_eq_iff by auto
```
```    96
```
```    97 lemma Collect_split_in_rel_leI: "X \<subseteq> Y \<Longrightarrow> X \<subseteq> Collect (split (in_rel Y))"
```
```    98 by auto
```
```    99
```
```   100 lemma Collect_split_in_rel_leE: "X \<subseteq> Collect (split (in_rel Y)) \<Longrightarrow> (X \<subseteq> Y \<Longrightarrow> R) \<Longrightarrow> R"
```
```   101 by force
```
```   102
```
```   103 lemma Collect_split_in_relI: "x \<in> X \<Longrightarrow> x \<in> Collect (split (in_rel X))"
```
```   104 by auto
```
```   105
```
```   106 lemma conversep_in_rel: "(in_rel R)\<inverse>\<inverse> = in_rel (R\<inverse>)"
```
```   107 unfolding fun_eq_iff by auto
```
```   108
```
```   109 lemmas conversep_in_rel_Id_on =
```
```   110   trans[OF conversep_in_rel arg_cong[of _ _ in_rel, OF converse_Id_on]]
```
```   111
```
```   112 lemma relcompp_in_rel: "in_rel R OO in_rel S = in_rel (R O S)"
```
```   113 unfolding fun_eq_iff by auto
```
```   114
```
```   115 lemmas relcompp_in_rel_Id_on =
```
```   116   trans[OF relcompp_in_rel arg_cong[of _ _ in_rel, OF Id_on_Comp[symmetric]]]
```
```   117
```
```   118 lemma in_rel_Gr: "in_rel (Gr A f) = Grp A f"
```
```   119 unfolding Gr_def Grp_def fun_eq_iff by auto
```
```   120
```
```   121 lemma in_rel_Id_on_UNIV: "in_rel (Id_on UNIV) = op ="
```
```   122 unfolding fun_eq_iff by auto
```
```   123
```
```   124 definition relImage where
```
```   125 "relImage R f \<equiv> {(f a1, f a2) | a1 a2. (a1,a2) \<in> R}"
```
```   126
```
```   127 definition relInvImage where
```
```   128 "relInvImage A R f \<equiv> {(a1, a2) | a1 a2. a1 \<in> A \<and> a2 \<in> A \<and> (f a1, f a2) \<in> R}"
```
```   129
```
```   130 lemma relImage_Gr:
```
```   131 "\<lbrakk>R \<subseteq> A \<times> A\<rbrakk> \<Longrightarrow> relImage R f = (Gr A f)^-1 O R O Gr A f"
```
```   132 unfolding relImage_def Gr_def relcomp_def by auto
```
```   133
```
```   134 lemma relInvImage_Gr: "\<lbrakk>R \<subseteq> B \<times> B\<rbrakk> \<Longrightarrow> relInvImage A R f = Gr A f O R O (Gr A f)^-1"
```
```   135 unfolding Gr_def relcomp_def image_def relInvImage_def by auto
```
```   136
```
```   137 lemma relImage_mono:
```
```   138 "R1 \<subseteq> R2 \<Longrightarrow> relImage R1 f \<subseteq> relImage R2 f"
```
```   139 unfolding relImage_def by auto
```
```   140
```
```   141 lemma relInvImage_mono:
```
```   142 "R1 \<subseteq> R2 \<Longrightarrow> relInvImage A R1 f \<subseteq> relInvImage A R2 f"
```
```   143 unfolding relInvImage_def by auto
```
```   144
```
```   145 lemma relInvImage_Id_on:
```
```   146 "(\<And>a1 a2. f a1 = f a2 \<longleftrightarrow> a1 = a2) \<Longrightarrow> relInvImage A (Id_on B) f \<subseteq> Id"
```
```   147 unfolding relInvImage_def Id_on_def by auto
```
```   148
```
```   149 lemma relInvImage_UNIV_relImage:
```
```   150 "R \<subseteq> relInvImage UNIV (relImage R f) f"
```
```   151 unfolding relInvImage_def relImage_def by auto
```
```   152
```
```   153 lemma equiv_Image: "equiv A R \<Longrightarrow> (\<And>a b. (a, b) \<in> R \<Longrightarrow> a \<in> A \<and> b \<in> A \<and> R `` {a} = R `` {b})"
```
```   154 unfolding equiv_def refl_on_def Image_def by (auto intro: transD symD)
```
```   155
```
```   156 lemma relImage_proj:
```
```   157 assumes "equiv A R"
```
```   158 shows "relImage R (proj R) \<subseteq> Id_on (A//R)"
```
```   159 unfolding relImage_def Id_on_def
```
```   160 using proj_iff[OF assms] equiv_class_eq_iff[OF assms]
```
```   161 by (auto simp: proj_preserves)
```
```   162
```
```   163 lemma relImage_relInvImage:
```
```   164 assumes "R \<subseteq> f ` A <*> f ` A"
```
```   165 shows "relImage (relInvImage A R f) f = R"
```
```   166 using assms unfolding relImage_def relInvImage_def by fastforce
```
```   167
```
```   168 lemma subst_Pair: "P x y \<Longrightarrow> a = (x, y) \<Longrightarrow> P (fst a) (snd a)"
```
```   169 by simp
```
```   170
```
```   171 lemma fst_diag_id: "(fst \<circ> (%x. (x, x))) z = id z"
```
```   172 by simp
```
```   173
```
```   174 lemma snd_diag_id: "(snd \<circ> (%x. (x, x))) z = id z"
```
```   175 by simp
```
```   176
```
```   177 lemma image_convolD: "\<lbrakk>(a, b) \<in> <f, g> ` X\<rbrakk> \<Longrightarrow> \<exists>x. x \<in> X \<and> a = f x \<and> b = g x"
```
```   178 unfolding convol_def by auto
```
```   179
```
```   180 (*Extended Sublist*)
```
```   181
```
```   182 definition prefCl where
```
```   183   "prefCl Kl = (\<forall> kl1 kl2. prefixeq kl1 kl2 \<and> kl2 \<in> Kl \<longrightarrow> kl1 \<in> Kl)"
```
```   184 definition PrefCl where
```
```   185   "PrefCl A n = (\<forall>kl kl'. kl \<in> A n \<and> prefixeq kl' kl \<longrightarrow> (\<exists>m\<le>n. kl' \<in> A m))"
```
```   186
```
```   187 lemma prefCl_UN:
```
```   188   "\<lbrakk>\<And>n. PrefCl A n\<rbrakk> \<Longrightarrow> prefCl (\<Union>n. A n)"
```
```   189 unfolding prefCl_def PrefCl_def by fastforce
```
```   190
```
```   191 definition Succ where "Succ Kl kl = {k . kl @ [k] \<in> Kl}"
```
```   192 definition Shift where "Shift Kl k = {kl. k # kl \<in> Kl}"
```
```   193 definition shift where "shift lab k = (\<lambda>kl. lab (k # kl))"
```
```   194
```
```   195 lemma empty_Shift: "\<lbrakk>[] \<in> Kl; k \<in> Succ Kl []\<rbrakk> \<Longrightarrow> [] \<in> Shift Kl k"
```
```   196 unfolding Shift_def Succ_def by simp
```
```   197
```
```   198 lemma Shift_clists: "Kl \<subseteq> Field (clists r) \<Longrightarrow> Shift Kl k \<subseteq> Field (clists r)"
```
```   199 unfolding Shift_def clists_def Field_card_of by auto
```
```   200
```
```   201 lemma Shift_prefCl: "prefCl Kl \<Longrightarrow> prefCl (Shift Kl k)"
```
```   202 unfolding prefCl_def Shift_def
```
```   203 proof safe
```
```   204   fix kl1 kl2
```
```   205   assume "\<forall>kl1 kl2. prefixeq kl1 kl2 \<and> kl2 \<in> Kl \<longrightarrow> kl1 \<in> Kl"
```
```   206     "prefixeq kl1 kl2" "k # kl2 \<in> Kl"
```
```   207   thus "k # kl1 \<in> Kl" using Cons_prefixeq_Cons[of k kl1 k kl2] by blast
```
```   208 qed
```
```   209
```
```   210 lemma not_in_Shift: "kl \<notin> Shift Kl x \<Longrightarrow> x # kl \<notin> Kl"
```
```   211 unfolding Shift_def by simp
```
```   212
```
```   213 lemma SuccD: "k \<in> Succ Kl kl \<Longrightarrow> kl @ [k] \<in> Kl"
```
```   214 unfolding Succ_def by simp
```
```   215
```
```   216 lemmas SuccE = SuccD[elim_format]
```
```   217
```
```   218 lemma SuccI: "kl @ [k] \<in> Kl \<Longrightarrow> k \<in> Succ Kl kl"
```
```   219 unfolding Succ_def by simp
```
```   220
```
```   221 lemma ShiftD: "kl \<in> Shift Kl k \<Longrightarrow> k # kl \<in> Kl"
```
```   222 unfolding Shift_def by simp
```
```   223
```
```   224 lemma Succ_Shift: "Succ (Shift Kl k) kl = Succ Kl (k # kl)"
```
```   225 unfolding Succ_def Shift_def by auto
```
```   226
```
```   227 lemma Nil_clists: "{[]} \<subseteq> Field (clists r)"
```
```   228 unfolding clists_def Field_card_of by auto
```
```   229
```
```   230 lemma Cons_clists:
```
```   231   "\<lbrakk>x \<in> Field r; xs \<in> Field (clists r)\<rbrakk> \<Longrightarrow> x # xs \<in> Field (clists r)"
```
```   232 unfolding clists_def Field_card_of by auto
```
```   233
```
```   234 lemma length_Cons: "length (x # xs) = Suc (length xs)"
```
```   235 by simp
```
```   236
```
```   237 lemma length_append_singleton: "length (xs @ [x]) = Suc (length xs)"
```
```   238 by simp
```
```   239
```
```   240 (*injection into the field of a cardinal*)
```
```   241 definition "toCard_pred A r f \<equiv> inj_on f A \<and> f ` A \<subseteq> Field r \<and> Card_order r"
```
```   242 definition "toCard A r \<equiv> SOME f. toCard_pred A r f"
```
```   243
```
```   244 lemma ex_toCard_pred:
```
```   245 "\<lbrakk>|A| \<le>o r; Card_order r\<rbrakk> \<Longrightarrow> \<exists> f. toCard_pred A r f"
```
```   246 unfolding toCard_pred_def
```
```   247 using card_of_ordLeq[of A "Field r"]
```
```   248       ordLeq_ordIso_trans[OF _ card_of_unique[of "Field r" r], of "|A|"]
```
```   249 by blast
```
```   250
```
```   251 lemma toCard_pred_toCard:
```
```   252   "\<lbrakk>|A| \<le>o r; Card_order r\<rbrakk> \<Longrightarrow> toCard_pred A r (toCard A r)"
```
```   253 unfolding toCard_def using someI_ex[OF ex_toCard_pred] .
```
```   254
```
```   255 lemma toCard_inj: "\<lbrakk>|A| \<le>o r; Card_order r; x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow>
```
```   256   toCard A r x = toCard A r y \<longleftrightarrow> x = y"
```
```   257 using toCard_pred_toCard unfolding inj_on_def toCard_pred_def by blast
```
```   258
```
```   259 lemma toCard: "\<lbrakk>|A| \<le>o r; Card_order r; b \<in> A\<rbrakk> \<Longrightarrow> toCard A r b \<in> Field r"
```
```   260 using toCard_pred_toCard unfolding toCard_pred_def by blast
```
```   261
```
```   262 definition "fromCard A r k \<equiv> SOME b. b \<in> A \<and> toCard A r b = k"
```
```   263
```
```   264 lemma fromCard_toCard:
```
```   265 "\<lbrakk>|A| \<le>o r; Card_order r; b \<in> A\<rbrakk> \<Longrightarrow> fromCard A r (toCard A r b) = b"
```
```   266 unfolding fromCard_def by (rule some_equality) (auto simp add: toCard_inj)
```
```   267
```
```   268 (* pick according to the weak pullback *)
```
```   269 definition pickWP where
```
```   270 "pickWP A p1 p2 b1 b2 \<equiv> SOME a. a \<in> A \<and> p1 a = b1 \<and> p2 a = b2"
```
```   271
```
```   272 lemma pickWP_pred:
```
```   273 assumes "wpull A B1 B2 f1 f2 p1 p2" and
```
```   274 "b1 \<in> B1" and "b2 \<in> B2" and "f1 b1 = f2 b2"
```
```   275 shows "\<exists> a. a \<in> A \<and> p1 a = b1 \<and> p2 a = b2"
```
```   276 using assms unfolding wpull_def by blast
```
```   277
```
```   278 lemma pickWP:
```
```   279 assumes "wpull A B1 B2 f1 f2 p1 p2" and
```
```   280 "b1 \<in> B1" and "b2 \<in> B2" and "f1 b1 = f2 b2"
```
```   281 shows "pickWP A p1 p2 b1 b2 \<in> A"
```
```   282       "p1 (pickWP A p1 p2 b1 b2) = b1"
```
```   283       "p2 (pickWP A p1 p2 b1 b2) = b2"
```
```   284 unfolding pickWP_def using assms someI_ex[OF pickWP_pred] by fastforce+
```
```   285
```
```   286 lemma Inl_Field_csum: "a \<in> Field r \<Longrightarrow> Inl a \<in> Field (r +c s)"
```
```   287 unfolding Field_card_of csum_def by auto
```
```   288
```
```   289 lemma Inr_Field_csum: "a \<in> Field s \<Longrightarrow> Inr a \<in> Field (r +c s)"
```
```   290 unfolding Field_card_of csum_def by auto
```
```   291
```
```   292 lemma nat_rec_0: "f = nat_rec f1 (%n rec. f2 n rec) \<Longrightarrow> f 0 = f1"
```
```   293 by auto
```
```   294
```
```   295 lemma nat_rec_Suc: "f = nat_rec f1 (%n rec. f2 n rec) \<Longrightarrow> f (Suc n) = f2 n (f n)"
```
```   296 by auto
```
```   297
```
```   298 lemma list_rec_Nil: "f = list_rec f1 (%x xs rec. f2 x xs rec) \<Longrightarrow> f [] = f1"
```
```   299 by auto
```
```   300
```
```   301 lemma list_rec_Cons: "f = list_rec f1 (%x xs rec. f2 x xs rec) \<Longrightarrow> f (x # xs) = f2 x xs (f xs)"
```
```   302 by auto
```
```   303
```
```   304 lemma not_arg_cong_Inr: "x \<noteq> y \<Longrightarrow> Inr x \<noteq> Inr y"
```
```   305 by simp
```
```   306
```
```   307 lemma Collect_splitD: "x \<in> Collect (split A) \<Longrightarrow> A (fst x) (snd x)"
```
```   308 by auto
```
```   309
```
```   310 ML_file "Tools/bnf_gfp_util.ML"
```
```   311 ML_file "Tools/bnf_gfp_tactics.ML"
```
```   312 ML_file "Tools/bnf_gfp.ML"
```
```   313
```
```   314 end
```