src/HOL/BNF/BNF_Def.thy

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(*  Title:      HOL/BNF/BNF_Def.thy
    Author:     Dmitriy Traytel, TU Muenchen
    Copyright   2012

Definition of bounded natural functors.
*)

header {* Definition of Bounded Natural Functors *}

theory BNF_Def
imports BNF_Util
keywords
  "print_bnfs" :: diag and
  "bnf" :: 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)

definition convol ("<_ , _>") where
"<f , g> \<equiv> %a. (f a, g a)"

lemma fst_convol:
"fst o <f , g> = f"
apply(rule ext)
unfolding convol_def by simp

lemma snd_convol:
"snd o <f , g> = g"
apply(rule ext)
unfolding convol_def by simp

lemma convol_mem_GrpI:
"x \<in> A \<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))"

(* 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 blast
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 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 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 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

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.cases) (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.cases) (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 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)

lemma fst_snd_flip: "fst xy = (snd o (%(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>\<forall>a\<in>fst ` A. \<forall>b \<in> snd ` A. 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 metis

lemma sum_case_o_inj:
"sum_case f g \<circ> Inl = f"
"sum_case 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) o g) i = id i"
unfolding Func_def by (auto elim: the_inv_into_f_f)

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

lemma vimage2pI: "R (f x) (g y) \<Longrightarrow> vimage2p f g R x y"
  unfolding vimage2p_def by -

lemma vimage2pD: "vimage2p f g R x y \<Longrightarrow> R (f x) (g y)"
  unfolding vimage2p_def by -

lemma fun_rel_iff_leq_vimage2p: "(fun_rel R S) f g = (R \<le> vimage2p f g S)"
  unfolding fun_rel_def vimage2p_def by auto

lemma convol_image_vimage2p: "<f o fst, g o snd> ` Collect (split (vimage2p f g R)) \<subseteq> Collect (split R)"
  unfolding vimage2p_def convol_def by auto

ML_file "Tools/bnf_def_tactics.ML"
ML_file "Tools/bnf_def.ML"


end