(*  Title:      HOL/Codatatype/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_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 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_memI:

"\<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

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 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:

"(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

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"

unfolding fstO_def

by (subst (asm) surjective_pairing) (rule pick_middle[THEN conjunct1])

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"

unfolding sndO_def

by (subst (asm) surjective_pairing) (rule pick_middle[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 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"

end