src/HOL/BNF/BNF_Def.thy
 author traytel Sun Jul 28 12:59:59 2013 +0200 (2013-07-28) changeset 52749 ed416f4ac34e parent 52731 dacd47a0633f child 52986 7f7bbeb16538 permissions -rw-r--r--
more converse(p) theorems; tuned proofs;
1 (*  Title:      HOL/BNF/BNF_Def.thy
2     Author:     Dmitriy Traytel, TU Muenchen
5 Definition of bounded natural functors.
6 *)
8 header {* Definition of Bounded Natural Functors *}
10 theory BNF_Def
11 imports BNF_Util
12 keywords
13   "print_bnfs" :: diag and
14   "bnf" :: thy_goal
15 begin
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)
20 definition convol ("<_ , _>") where
21 "<f , g> \<equiv> %a. (f a, g a)"
23 lemma fst_convol:
24 "fst o <f , g> = f"
25 apply(rule ext)
26 unfolding convol_def by simp
28 lemma snd_convol:
29 "snd o <f , g> = g"
30 apply(rule ext)
31 unfolding convol_def by simp
33 lemma convol_mem_GrpI:
34 "\<lbrakk>g x = g' x; x \<in> A\<rbrakk> \<Longrightarrow> <id , g> x \<in> (Collect (split (Grp A g)))"
35 unfolding convol_def Grp_def by auto
37 definition csquare where
38 "csquare A f1 f2 p1 p2 \<longleftrightarrow> (\<forall> a \<in> A. f1 (p1 a) = f2 (p2 a))"
40 (* The pullback of sets *)
41 definition thePull where
42 "thePull B1 B2 f1 f2 = {(b1,b2). b1 \<in> B1 \<and> b2 \<in> B2 \<and> f1 b1 = f2 b2}"
44 lemma wpull_thePull:
45 "wpull (thePull B1 B2 f1 f2) B1 B2 f1 f2 fst snd"
46 unfolding wpull_def thePull_def by auto
48 lemma wppull_thePull:
49 assumes "wppull A B1 B2 f1 f2 e1 e2 p1 p2"
50 shows
51 "\<exists> j. \<forall> a' \<in> thePull B1 B2 f1 f2.
52    j a' \<in> A \<and>
53    e1 (p1 (j a')) = e1 (fst a') \<and> e2 (p2 (j a')) = e2 (snd a')"
54 (is "\<exists> j. \<forall> a' \<in> ?A'. ?phi a' (j a')")
55 proof(rule bchoice[of ?A' ?phi], default)
56   fix a' assume a': "a' \<in> ?A'"
57   hence "fst a' \<in> B1" unfolding thePull_def by auto
58   moreover
59   from a' have "snd a' \<in> B2" unfolding thePull_def by auto
60   moreover have "f1 (fst a') = f2 (snd a')"
61   using a' unfolding csquare_def thePull_def by auto
62   ultimately show "\<exists> ja'. ?phi a' ja'"
63   using assms unfolding wppull_def by blast
64 qed
66 lemma wpull_wppull:
67 assumes wp: "wpull A' B1 B2 f1 f2 p1' p2'" and
68 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')"
69 shows "wppull A B1 B2 f1 f2 e1 e2 p1 p2"
70 unfolding wppull_def proof safe
71   fix b1 b2
72   assume b1: "b1 \<in> B1" and b2: "b2 \<in> B2" and f: "f1 b1 = f2 b2"
73   then obtain a' where a': "a' \<in> A'" and b1: "b1 = p1' a'" and b2: "b2 = p2' a'"
74   using wp unfolding wpull_def by blast
75   show "\<exists>a\<in>A. e1 (p1 a) = e1 b1 \<and> e2 (p2 a) = e2 b2"
76   apply (rule bexI[of _ "j a'"]) unfolding b1 b2 using a' 1 by auto
77 qed
79 lemma wppull_id: "\<lbrakk>wpull UNIV UNIV UNIV f1 f2 p1 p2; e1 = id; e2 = id\<rbrakk> \<Longrightarrow>
80    wppull UNIV UNIV UNIV f1 f2 e1 e2 p1 p2"
81 by (erule wpull_wppull) auto
83 lemma eq_alt: "op = = Grp UNIV id"
84 unfolding Grp_def by auto
86 lemma leq_conversepI: "R = op = \<Longrightarrow> R \<le> R^--1"
87   by auto
89 lemma eq_OOI: "R = op = \<Longrightarrow> R = R OO R"
90   by auto
92 lemma Grp_UNIV_id: "f = id \<Longrightarrow> (Grp UNIV f)^--1 OO Grp UNIV f = Grp UNIV f"
93 unfolding Grp_def by auto
95 lemma Grp_UNIV_idI: "x = y \<Longrightarrow> Grp UNIV id x y"
96 unfolding Grp_def by auto
98 lemma Grp_mono: "A \<le> B \<Longrightarrow> Grp A f \<le> Grp B f"
99 unfolding Grp_def by auto
101 lemma GrpI: "\<lbrakk>f x = y; x \<in> A\<rbrakk> \<Longrightarrow> Grp A f x y"
102 unfolding Grp_def by auto
104 lemma GrpE: "Grp A f x y \<Longrightarrow> (\<lbrakk>f x = y; x \<in> A\<rbrakk> \<Longrightarrow> R) \<Longrightarrow> R"
105 unfolding Grp_def by auto
107 lemma Collect_split_Grp_eqD: "z \<in> Collect (split (Grp A f)) \<Longrightarrow> (f \<circ> fst) z = snd z"
108 unfolding Grp_def o_def by auto
110 lemma Collect_split_Grp_inD: "z \<in> Collect (split (Grp A f)) \<Longrightarrow> fst z \<in> A"
111 unfolding Grp_def o_def by auto
113 lemma wpull_Grp:
114 "wpull (Collect (split (Grp A f))) A (f ` A) f id fst snd"
115 unfolding wpull_def Grp_def by auto
117 definition "pick_middlep P Q a c = (SOME b. P a b \<and> Q b c)"
119 lemma pick_middlep:
120 "(P OO Q) a c \<Longrightarrow> P a (pick_middlep P Q a c) \<and> Q (pick_middlep P Q a c) c"
121 unfolding pick_middlep_def apply(rule someI_ex) by auto
123 definition fstOp where "fstOp P Q ac = (fst ac, pick_middlep P Q (fst ac) (snd ac))"
124 definition sndOp where "sndOp P Q ac = (pick_middlep P Q (fst ac) (snd ac), (snd ac))"
126 lemma fstOp_in: "ac \<in> Collect (split (P OO Q)) \<Longrightarrow> fstOp P Q ac \<in> Collect (split P)"
127 unfolding fstOp_def mem_Collect_eq
128 by (subst (asm) surjective_pairing, unfold prod.cases) (erule pick_middlep[THEN conjunct1])
130 lemma fst_fstOp: "fst bc = (fst \<circ> fstOp P Q) bc"
131 unfolding comp_def fstOp_def by simp
133 lemma snd_sndOp: "snd bc = (snd \<circ> sndOp P Q) bc"
134 unfolding comp_def sndOp_def by simp
136 lemma sndOp_in: "ac \<in> Collect (split (P OO Q)) \<Longrightarrow> sndOp P Q ac \<in> Collect (split Q)"
137 unfolding sndOp_def mem_Collect_eq
138 by (subst (asm) surjective_pairing, unfold prod.cases) (erule pick_middlep[THEN conjunct2])
140 lemma csquare_fstOp_sndOp:
141 "csquare (Collect (split (P OO Q))) snd fst (fstOp P Q) (sndOp P Q)"
142 unfolding csquare_def fstOp_def sndOp_def using pick_middlep by simp
144 lemma wppull_fstOp_sndOp:
145 shows "wppull (Collect (split (P OO Q))) (Collect (split P)) (Collect (split Q))
146   snd fst fst snd (fstOp P Q) (sndOp P Q)"
147 using pick_middlep unfolding wppull_def fstOp_def sndOp_def relcompp.simps by auto
149 lemma snd_fst_flip: "snd xy = (fst o (%(x, y). (y, x))) xy"
150 by (simp split: prod.split)
152 lemma fst_snd_flip: "fst xy = (snd o (%(x, y). (y, x))) xy"
153 by (simp split: prod.split)
155 lemma flip_pred: "A \<subseteq> Collect (split (R ^--1)) \<Longrightarrow> (%(x, y). (y, x)) ` A \<subseteq> Collect (split R)"
156 by auto
158 lemma Collect_split_mono: "A \<le> B \<Longrightarrow> Collect (split A) \<subseteq> Collect (split B)"
159   by auto
161 lemma Collect_split_mono_strong:
162   "\<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>
163   A \<subseteq> Collect (split Q)"
164   by fastforce
166 lemma predicate2_eqD: "A = B \<Longrightarrow> A a b \<longleftrightarrow> B a b"
167 by metis
169 lemma sum_case_o_inj:
170 "sum_case f g \<circ> Inl = f"
171 "sum_case f g \<circ> Inr = g"
172 by auto
174 lemma card_order_csum_cone_cexp_def:
175   "card_order r \<Longrightarrow> ( |A1| +c cone) ^c r = |Func UNIV (Inl ` A1 \<union> {Inr ()})|"
176   unfolding cexp_def cone_def Field_csum Field_card_of by (auto dest: Field_card_order)
178 lemma If_the_inv_into_in_Func:
179   "\<lbrakk>inj_on g C; C \<subseteq> B \<union> {x}\<rbrakk> \<Longrightarrow>
180   (\<lambda>i. if i \<in> g ` C then the_inv_into C g i else x) \<in> Func UNIV (B \<union> {x})"
181 unfolding Func_def by (auto dest: the_inv_into_into)
183 lemma If_the_inv_into_f_f:
184   "\<lbrakk>i \<in> C; inj_on g C\<rbrakk> \<Longrightarrow>
185   ((\<lambda>i. if i \<in> g ` C then the_inv_into C g i else x) o g) i = id i"
186 unfolding Func_def by (auto elim: the_inv_into_f_f)
188 definition vimage2p where
189   "vimage2p f g R = (\<lambda>x y. R (f x) (g y))"
191 lemma vimage2pI: "R (f x) (g y) \<Longrightarrow> vimage2p f g R x y"
192   unfolding vimage2p_def by -
194 lemma vimage2pD: "vimage2p f g R x y \<Longrightarrow> R (f x) (g y)"
195   unfolding vimage2p_def by -
197 lemma fun_rel_iff_leq_vimage2p: "(fun_rel R S) f g = (R \<le> vimage2p f g S)"
198   unfolding fun_rel_def vimage2p_def by auto
200 lemma convol_image_vimage2p: "<f o fst, g o snd> ` Collect (split (vimage2p f g R)) \<subseteq> Collect (split R)"
201   unfolding vimage2p_def convol_def by auto
203 ML_file "Tools/bnf_def_tactics.ML"
204 ML_file "Tools/bnf_def.ML"
207 end