diff -r e5432ec161ff -r 596baae88a88 src/HOL/BNF/BNF_Def.thy
--- a/src/HOL/BNF/BNF_Def.thy	Tue May 07 11:27:29 2013 +0200
+++ b/src/HOL/BNF/BNF_Def.thy	Tue May 07 14:22:54 2013 +0200
@@ -21,10 +21,18 @@
 "R1 ^-1 \<subseteq> R2 ^-1 \<longleftrightarrow> R1 \<subseteq> R2"
 unfolding converse_def by auto
 
+lemma conversep_mono:
+"R1 ^--1 \<le> R2 ^--1 \<longleftrightarrow> R1 \<le> R2"
+unfolding conversep.simps by auto
+
 lemma converse_shift:
 "R1 \<subseteq> R2 ^-1 \<Longrightarrow> R1 ^-1 \<subseteq> R2"
 unfolding converse_def by auto
 
+lemma conversep_shift:
+"R1 \<le> R2 ^--1 \<Longrightarrow> R1 ^--1 \<le> R2"
+unfolding conversep.simps by auto
+
 definition convol ("<_ , _>") where
 "<f , g> \<equiv> %a. (f a, g a)"
 
@@ -42,6 +50,10 @@
 "\<lbrakk>f x = f' x; g x = g' x; P x\<rbrakk> \<Longrightarrow> <f , g> x \<in> {(f' a, g' a) |a. P a}"
 unfolding convol_def by auto
 
+lemma convol_mem_GrpI:
+"\<lbrakk>g x = g' x; x \<in> A\<rbrakk> \<Longrightarrow> <id , g> x \<in> (Collect (split (Grp A g)))"
+unfolding convol_def Grp_def by auto
+
 definition csquare where
 "csquare A f1 f2 p1 p2 \<longleftrightarrow> (\<forall> a \<in> A. f1 (p1 a) = f2 (p2 a))"
 
@@ -91,48 +103,111 @@
 lemma Id_alt: "Id = Gr UNIV id"
 unfolding Gr_def by auto
 
+lemma eq_alt: "op = = Grp UNIV id"
+unfolding Grp_def by auto
+
+lemma leq_conversepI: "R = op = \<Longrightarrow> R \<le> R^--1"
+  by auto
+
+lemma eq_OOI: "R = op = \<Longrightarrow> R = R OO R"
+  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 Grp_UNIV_id: "f = id \<Longrightarrow> (Grp UNIV f)^--1 OO Grp UNIV f = Grp UNIV f"
+unfolding Grp_def by auto
+
+lemma Grp_UNIV_idI: "x = y \<Longrightarrow> Grp UNIV id x y"
+unfolding Grp_def by auto
+
 lemma Gr_mono: "A \<subseteq> B \<Longrightarrow> Gr A f \<subseteq> Gr B f"
 unfolding Gr_def by auto
 
+lemma Grp_mono: "A \<le> B \<Longrightarrow> Grp A f \<le> Grp B f"
+unfolding Grp_def by auto
+
+lemma GrpI: "\<lbrakk>f x = y; x \<in> A\<rbrakk> \<Longrightarrow> Grp A f x y"
+unfolding Grp_def by auto
+
+lemma GrpE: "Grp A f x y \<Longrightarrow> (\<lbrakk>f x = y; x \<in> A\<rbrakk> \<Longrightarrow> R) \<Longrightarrow> R"
+unfolding Grp_def by auto
+
+lemma Collect_split_Grp_eqD: "z \<in> Collect (split (Grp A f)) \<Longrightarrow> (f \<circ> fst) z = snd z"
+unfolding Grp_def o_def by auto
+
+lemma Collect_split_Grp_inD: "z \<in> Collect (split (Grp A f)) \<Longrightarrow> fst z \<in> A"
+unfolding Grp_def o_def by auto
+
 lemma wpull_Gr:
 "wpull (Gr A f) A (f ` A) f id fst snd"
 unfolding wpull_def Gr_def by auto
 
+lemma wpull_Grp:
+"wpull (Collect (split (Grp A f))) A (f ` A) f id fst snd"
+unfolding wpull_def Grp_def by auto
+
 definition "pick_middle P Q a c = (SOME b. (a,b) \<in> P \<and> (b,c) \<in> Q)"
 
+definition "pick_middlep P Q a c = (SOME b. P a b \<and> Q b c)"
+
 lemma pick_middle:
 "(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"
-unfolding pick_middle_def apply(rule someI_ex)
-using assms unfolding relcomp_def by auto
+unfolding pick_middle_def apply(rule someI_ex) by auto
+
+lemma pick_middlep:
+"(P OO Q) a c \<Longrightarrow> P a (pick_middlep P Q a c) \<and> Q (pick_middlep P Q a c) c"
+unfolding pick_middlep_def apply(rule someI_ex) 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)"
 
+definition fstOp where "fstOp P Q ac = (fst ac, pick_middlep P Q (fst ac) (snd ac))"
+definition sndOp where "sndOp P Q ac = (pick_middlep P Q (fst ac) (snd ac), (snd ac))"
+
 lemma fstO_in: "ac \<in> P O Q \<Longrightarrow> fstO P Q ac \<in> P"
-unfolding fstO_def
-by (subst (asm) surjective_pairing) (rule pick_middle[THEN conjunct1])
+unfolding fstO_def by (subst (asm) surjective_pairing) (rule pick_middle[THEN conjunct1])
+
+lemma fstOp_in: "ac \<in> Collect (split (P OO Q)) \<Longrightarrow> fstOp P Q ac \<in> Collect (split P)"
+unfolding fstOp_def mem_Collect_eq
+by (subst (asm) surjective_pairing, unfold prod.cases) (erule pick_middlep[THEN conjunct1])
 
 lemma fst_fstO: "fst bc = (fst \<circ> fstO P Q) bc"
 unfolding comp_def fstO_def by simp
 
+lemma fst_fstOp: "fst bc = (fst \<circ> fstOp P Q) bc"
+unfolding comp_def fstOp_def by simp
+
 lemma snd_sndO: "snd bc = (snd \<circ> sndO P Q) bc"
 unfolding comp_def sndO_def by simp
 
+lemma snd_sndOp: "snd bc = (snd \<circ> sndOp P Q) bc"
+unfolding comp_def sndOp_def by simp
+
 lemma sndO_in: "ac \<in> P O Q \<Longrightarrow> sndO P Q ac \<in> Q"
 unfolding sndO_def
 by (subst (asm) surjective_pairing) (rule pick_middle[THEN conjunct2])
 
+lemma sndOp_in: "ac \<in> Collect (split (P OO Q)) \<Longrightarrow> sndOp P Q ac \<in> Collect (split Q)"
+unfolding sndOp_def mem_Collect_eq
+by (subst (asm) surjective_pairing, unfold prod.cases) (erule pick_middlep[THEN conjunct2])
+
 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 simp
 
+lemma csquare_fstOp_sndOp:
+"csquare (Collect (split (P OO Q))) snd fst (fstOp P Q) (sndOp P Q)"
+unfolding csquare_def fstOp_def sndOp_def using pick_middlep by simp
+
 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 wppull_fstOp_sndOp:
+shows "wppull (Collect (split (P OO Q))) (Collect (split P)) (Collect (split Q)) snd fst fst snd (fstOp P Q) (sndOp P Q)"
+using pick_middlep unfolding wppull_def fstOp_def sndOp_def relcompp.simps by auto
+
 lemma snd_fst_flip: "snd xy = (fst o (%(x, y). (y, x))) xy"
 by (simp split: prod.split)
 
@@ -142,11 +217,17 @@
 lemma flip_rel: "A \<subseteq> (R ^-1) \<Longrightarrow> (%(x, y). (y, x)) ` A \<subseteq> R"
 by auto
 
+lemma flip_pred: "A \<subseteq> Collect (split (R ^--1)) \<Longrightarrow> (%(x, y). (y, x)) ` A \<subseteq> Collect (split 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
 
-lemma eqset_imp_iff_pair: "A = B \<Longrightarrow> (a, b) \<in> A \<longleftrightarrow> (a, b) \<in> B"
-by (rule eqset_imp_iff)
+lemma Collect_split_mono: "A \<le> B \<Longrightarrow> Collect (split A) \<subseteq> Collect (split B)"
+  by auto
+
+lemma predicate2_cong: "A = B \<Longrightarrow> A a b \<longleftrightarrow> B a b"
+by metis
 
 lemma fun_cong_pair: "f = g \<Longrightarrow> f {(a, b). R a b} = g {(a, b). R a b}"
 by (rule fun_cong)
@@ -161,4 +242,5 @@
 ML_file "Tools/bnf_def_tactics.ML"
 ML_file "Tools/bnf_def.ML"
 
+
 end