
diff -r 56fcd826f90c -r c874ff5658dc src/HOL/Codatatype/BNF_Def.thy
--- a/src/HOL/Codatatype/BNF_Def.thy	Wed Sep 12 06:27:36 2012 +0200
+++ b/src/HOL/Codatatype/BNF_Def.thy	Wed Sep 12 06:27:48 2012 +0200
@@ -14,6 +14,119 @@
   "bnf_def" :: thy_goal
 begin
 
+lemma collect_o: "collect F o g = collect ((\<lambda>f. f o g) ` F)"
+by (rule ext) (auto simp only: o_apply collect_def)
+
+lemma converse_mono:
+"R1 ^-1 \<subseteq> R2 ^-1 \<longleftrightarrow> R1 \<subseteq> R2"
+unfolding converse_def by auto
+
+lemma converse_shift:
+"R1 \<subseteq> R2 ^-1 \<Longrightarrow> R1 ^-1 \<subseteq> R2"
+unfolding converse_def by auto
+
+definition csquare where
+"csquare A f1 f2 p1 p2 \<longleftrightarrow> (\<forall> a \<in> A. f1 (p1 a) = f2 (p2 a))"
+
+(* The pullback of sets *)
+definition thePull where
+"thePull B1 B2 f1 f2 = {(b1,b2). b1 \<in> B1 \<and> b2 \<in> B2 \<and> f1 b1 = f2 b2}"
+
+lemma wpull_thePull:
+"wpull (thePull B1 B2 f1 f2) B1 B2 f1 f2 fst snd"
+unfolding wpull_def thePull_def by auto
+
+lemma wppull_thePull:
+assumes "wppull A B1 B2 f1 f2 e1 e2 p1 p2"
+shows
+"\<exists> j. \<forall> a' \<in> thePull B1 B2 f1 f2.
+   j a' \<in> A \<and>
+   e1 (p1 (j a')) = e1 (fst a') \<and> e2 (p2 (j a')) = e2 (snd a')"
+(is "\<exists> j. \<forall> a' \<in> ?A'. ?phi a' (j a')")
+proof(rule bchoice[of ?A' ?phi], default)
+  fix a' assume a': "a' \<in> ?A'"
+  hence "fst a' \<in> B1" unfolding thePull_def by auto
+  moreover
+  from a' have "snd a' \<in> B2" unfolding thePull_def by auto
+  moreover have "f1 (fst a') = f2 (snd a')"
+  using a' unfolding csquare_def thePull_def by auto
+  ultimately show "\<exists> ja'. ?phi a' ja'"
+  using assms unfolding wppull_def by auto
+qed
+
+lemma wpull_wppull:
+assumes wp: "wpull A' B1 B2 f1 f2 p1' p2'" and
+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')"
+shows "wppull A B1 B2 f1 f2 e1 e2 p1 p2"
+unfolding wppull_def proof safe
+  fix b1 b2
+  assume b1: "b1 \<in> B1" and b2: "b2 \<in> B2" and f: "f1 b1 = f2 b2"
+  then obtain a' where a': "a' \<in> A'" and b1: "b1 = p1' a'" and b2: "b2 = p2' a'"
+  using wp unfolding wpull_def by blast
+  show "\<exists>a\<in>A. e1 (p1 a) = e1 b1 \<and> e2 (p2 a) = e2 b2"
+  apply(rule bexI[of _ "j a'"]) unfolding b1 b2 using a' 1 by auto
+qed
+
+lemma wppull_id: "\<lbrakk>wpull UNIV UNIV UNIV f1 f2 p1 p2; e1 = id; e2 = id\<rbrakk> \<Longrightarrow>
+   wppull UNIV UNIV UNIV f1 f2 e1 e2 p1 p2"
+by (erule wpull_wppull) auto
+
+lemma Id_alt: "Id = Gr UNIV id"
+unfolding Gr_def by auto
+
+lemma Gr_UNIV_id: "f = id \<Longrightarrow> (Gr UNIV f)^-1 O Gr UNIV f = Gr UNIV f"
+unfolding Gr_def by auto
+
+lemma Gr_mono: "A \<subseteq> B \<Longrightarrow> Gr A f \<subseteq> Gr B f"
+unfolding Gr_def by auto
+
+lemma wpull_Gr:
+"wpull (Gr A f) A (f ` A) f id fst snd"
+unfolding wpull_def Gr_def by auto
+
+definition "pick_middle P Q a c = (SOME b. (a,b) \<in> P \<and> (b,c) \<in> Q)"
+
+lemma pick_middle:
+assumes "(a,c) \<in> P O Q"
+shows "(a, pick_middle P Q a c) \<in> P \<and> (pick_middle P Q a c, c) \<in> Q"
+unfolding pick_middle_def apply(rule someI_ex)
+using assms unfolding relcomp_def by auto
+
+definition fstO where "fstO P Q ac = (fst ac, pick_middle P Q (fst ac) (snd ac))"
+definition sndO where "sndO P Q ac = (pick_middle P Q (fst ac) (snd ac), snd ac)"
+
+lemma fstO_in: "ac \<in> P O Q \<Longrightarrow> fstO P Q ac \<in> P"
+by (metis assms fstO_def pick_middle surjective_pairing)
+
+lemma fst_fstO: "fst bc = (fst \<circ> fstO P Q) bc"
+unfolding comp_def fstO_def by simp
+
+lemma snd_sndO: "snd bc = (snd \<circ> sndO P Q) bc"
+unfolding comp_def sndO_def by simp
+
+lemma sndO_in: "ac \<in> P O Q \<Longrightarrow> sndO P Q ac \<in> Q"
+by (metis assms sndO_def pick_middle surjective_pairing)
+
+lemma csquare_fstO_sndO:
+"csquare (P O Q) snd fst (fstO P Q) (sndO P Q)"
+unfolding csquare_def fstO_def sndO_def using pick_middle by auto
+
+lemma wppull_fstO_sndO:
+shows "wppull (P O Q) P Q snd fst fst snd (fstO P Q) (sndO P Q)"
+using pick_middle unfolding wppull_def fstO_def sndO_def relcomp_def by auto
+
+lemma snd_fst_flip: "snd xy = (fst o (%(x, y). (y, x))) xy"
+by (simp split: prod.split)
+
+lemma fst_snd_flip: "fst xy = (snd o (%(x, y). (y, x))) xy"
+by (simp split: prod.split)
+
+lemma flip_rel: "A \<subseteq> (R ^-1) \<Longrightarrow> (%(x, y). (y, x)) ` A \<subseteq> R"
+by auto
+
+lemma pointfreeE: "f o g = f' o g' \<Longrightarrow> f (g x) = f' (g' x)"
+unfolding o_def fun_eq_iff by simp
+
 ML_file "Tools/bnf_def_tactics.ML"
 ML_file"Tools/bnf_def.ML"
 




