(*  Title:      HOL/BNF_Def.thy

     Author:     Dmitriy Traytel, TU Muenchen

     Author:     Jasmin Blanchette, TU Muenchen

     Copyright   2012

 Definition of bounded natural functors.

 *)

 header {* Definition of Bounded Natural Functors *}

 theory BNF_Def

 imports BNF_Cardinal_Arithmetic Fun_Def_Base

 keywords

   "print_bnfs" :: diag and

   "bnf" :: thy_goal

 begin

 definition

   rel_fun :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"

 where

   "rel_fun A B = (\<lambda>f g. \<forall>x y. A x y \<longrightarrow> B (f x) (g y))"

 lemma rel_funI [intro]:

   assumes "\<And>x y. A x y \<Longrightarrow> B (f x) (g y)"

   shows "rel_fun A B f g"

   using assms by (simp add: rel_fun_def)

 lemma rel_funD:

   assumes "rel_fun A B f g" and "A x y"

   shows "B (f x) (g y)"

   using assms by (simp add: rel_fun_def)

 definition collect where

 "collect F x = (\<Union>f \<in> F. f x)"

 lemma fstI: "x = (y, z) \<Longrightarrow> fst x = y"

 by simp

 lemma sndI: "x = (y, z) \<Longrightarrow> snd x = z"

 by simp

 lemma bijI': "\<lbrakk>\<And>x y. (f x = f y) = (x = y); \<And>y. \<exists>x. y = f x\<rbrakk> \<Longrightarrow> bij f"

 unfolding bij_def inj_on_def by auto blast

 (* Operator: *)

 definition "Gr A f = {(a, f a) | a. a \<in> A}"

 definition "Grp A f = (\<lambda>a b. b = f a \<and> a \<in> A)"

 definition vimage2p where

   "vimage2p f g R = (\<lambda>x y. R (f x) (g y))"

 lemma collect_comp: "collect F \<circ> g = collect ((\<lambda>f. f \<circ> g) ` F)"

   by (rule ext) (auto simp only: comp_apply collect_def)

 definition convol ("\<langle>(_,/ _)\<rangle>") where

 "\<langle>f, g\<rangle> \<equiv> \<lambda>a. (f a, g a)"

 lemma fst_convol:

 "fst \<circ> \<langle>f, g\<rangle> = f"

 apply(rule ext)

 unfolding convol_def by simp

 lemma snd_convol:

 "snd \<circ> \<langle>f, g\<rangle> = g"

 apply(rule ext)

 unfolding convol_def by simp

 lemma convol_mem_GrpI:

 "x \<in> A \<Longrightarrow> \<langle>id, g\<rangle> 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))"

 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 leq_OOI: "R = op = \<Longrightarrow> R \<le> R OO R"

   by auto

 lemma OO_Grp_alt: "(Grp A f)^--1 OO Grp A g = (\<lambda>x y. \<exists>z. z \<in> A \<and> f z = x \<and> g z = y)"

   unfolding Grp_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 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 comp_def by auto

 lemma Collect_split_Grp_inD: "z \<in> Collect (split (Grp A f)) \<Longrightarrow> fst z \<in> A"

 unfolding Grp_def comp_def by auto

 definition "pick_middlep P Q a c = (SOME b. P a b \<and> Q b c)"

 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 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 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.case) (erule pick_middlep[THEN conjunct1])

 lemma fst_fstOp: "fst bc = (fst \<circ> fstOp P Q) bc"

 unfolding comp_def fstOp_def by simp

 lemma snd_sndOp: "snd bc = (snd \<circ> sndOp P Q) bc"

 unfolding comp_def sndOp_def by simp

 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.case) (erule pick_middlep[THEN conjunct2])

 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 snd_fst_flip: "snd xy = (fst \<circ> (%(x, y). (y, x))) xy"

 by (simp split: prod.split)

 lemma fst_snd_flip: "fst xy = (snd \<circ> (%(x, y). (y, x))) xy"

 by (simp split: prod.split)

 lemma flip_pred: "A \<subseteq> Collect (split (R ^--1)) \<Longrightarrow> (%(x, y). (y, x)) ` A \<subseteq> Collect (split R)"

 by auto

 lemma Collect_split_mono: "A \<le> B \<Longrightarrow> Collect (split A) \<subseteq> Collect (split B)"

   by auto

 lemma Collect_split_mono_strong: 

   "\<lbrakk>X = fst ` A; Y = snd ` A; \<forall>a\<in>X. \<forall>b \<in> Y. P a b \<longrightarrow> Q a b; A \<subseteq> Collect (split P)\<rbrakk> \<Longrightarrow>

   A \<subseteq> Collect (split Q)"

   by fastforce

 lemma predicate2_eqD: "A = B \<Longrightarrow> A a b \<longleftrightarrow> B a b"

 by simp

 lemma case_sum_o_inj:

 "case_sum f g \<circ> Inl = f"

 "case_sum f g \<circ> Inr = g"

 by auto

 lemma card_order_csum_cone_cexp_def:

   "card_order r \<Longrightarrow> ( |A1| +c cone) ^c r = |Func UNIV (Inl ` A1 \<union> {Inr ()})|"

   unfolding cexp_def cone_def Field_csum Field_card_of by (auto dest: Field_card_order)

 lemma If_the_inv_into_in_Func:

   "\<lbrakk>inj_on g C; C \<subseteq> B \<union> {x}\<rbrakk> \<Longrightarrow>

   (\<lambda>i. if i \<in> g ` C then the_inv_into C g i else x) \<in> Func UNIV (B \<union> {x})"

 unfolding Func_def by (auto dest: the_inv_into_into)

 lemma If_the_inv_into_f_f:

   "\<lbrakk>i \<in> C; inj_on g C\<rbrakk> \<Longrightarrow>

   ((\<lambda>i. if i \<in> g ` C then the_inv_into C g i else x) \<circ> g) i = id i"

 unfolding Func_def by (auto elim: the_inv_into_f_f)

 lemma the_inv_f_o_f_id: "inj f \<Longrightarrow> (the_inv f \<circ> f) z = id z"

   by (simp add: the_inv_f_f)

 lemma vimage2pI: "R (f x) (g y) \<Longrightarrow> vimage2p f g R x y"

   unfolding vimage2p_def by -

 lemma rel_fun_iff_leq_vimage2p: "(rel_fun R S) f g = (R \<le> vimage2p f g S)"

   unfolding rel_fun_def vimage2p_def by auto

 lemma convol_image_vimage2p: "\<langle>f \<circ> fst, g \<circ> snd\<rangle> ` Collect (split (vimage2p f g R)) \<subseteq> Collect (split R)"

   unfolding vimage2p_def convol_def by auto

 lemma vimage2p_Grp: "vimage2p f g P = Grp UNIV f OO P OO (Grp UNIV g)\<inverse>\<inverse>"

   unfolding vimage2p_def Grp_def by auto

 ML_file "Tools/BNF/bnf_util.ML"

 ML_file "Tools/BNF/bnf_tactics.ML"

 ML_file "Tools/BNF/bnf_def_tactics.ML"

 ML_file "Tools/BNF/bnf_def.ML"

 end