(*  Title:      HOL/BNF/Basic_BNFs.thy

     Author:     Dmitriy Traytel, TU Muenchen

     Author:     Andrei Popescu, TU Muenchen

     Author:     Jasmin Blanchette, TU Muenchen

     Copyright   2012

 Registration of basic types as bounded natural functors.

 *)

 header {* Registration of Basic Types as Bounded Natural Functors *}

 theory Basic_BNFs

 imports BNF_Def

    (*FIXME: define relators here, reuse in Lifting_* once this theory is in HOL*)

   Lifting_Sum

   Lifting_Product

   Main

 begin

 lemma wpull_Grp_def: "wpull A B1 B2 f1 f2 p1 p2 \<longleftrightarrow> Grp B1 f1 OO (Grp B2 f2)\<inverse>\<inverse> \<le> (Grp A p1)\<inverse>\<inverse> OO Grp A p2"

   unfolding wpull_def Grp_def by auto

 bnf ID: 'a

   map: "id :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"

   sets: "\<lambda>x. {x}"

   bd: natLeq

   rel: "id :: ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"

 apply (auto simp: Grp_def fun_eq_iff relcompp.simps natLeq_card_order natLeq_cinfinite)

 apply (rule ordLess_imp_ordLeq[OF finite_ordLess_infinite[OF _ natLeq_Well_order]])

 apply (auto simp add: Field_card_of Field_natLeq card_of_well_order_on)[3]

 done

 bnf DEADID: 'a

   map: "id :: 'a \<Rightarrow> 'a"

   bd: "natLeq +c |UNIV :: 'a set|"

   rel: "op = :: 'a \<Rightarrow> 'a \<Rightarrow> bool"

 by (auto simp add: wpull_Grp_def Grp_def

   card_order_csum natLeq_card_order card_of_card_order_on

   cinfinite_csum natLeq_cinfinite)

 definition setl :: "'a + 'b \<Rightarrow> 'a set" where

 "setl x = (case x of Inl z => {z} | _ => {})"

 definition setr :: "'a + 'b \<Rightarrow> 'b set" where

 "setr x = (case x of Inr z => {z} | _ => {})"

 lemmas sum_set_defs = setl_def[abs_def] setr_def[abs_def]

 bnf "'a + 'b"

   map: sum_map

   sets: setl setr

   bd: natLeq

   wits: Inl Inr

   rel: sum_rel

 proof -

   show "sum_map id id = id" by (rule sum_map.id)

 next

   fix f1 :: "'o \<Rightarrow> 's" and f2 :: "'p \<Rightarrow> 't" and g1 :: "'s \<Rightarrow> 'q" and g2 :: "'t \<Rightarrow> 'r"

   show "sum_map (g1 o f1) (g2 o f2) = sum_map g1 g2 o sum_map f1 f2"

     by (rule sum_map.comp[symmetric])

 next

   fix x and f1 :: "'o \<Rightarrow> 'q" and f2 :: "'p \<Rightarrow> 'r" and g1 g2

   assume a1: "\<And>z. z \<in> setl x \<Longrightarrow> f1 z = g1 z" and

          a2: "\<And>z. z \<in> setr x \<Longrightarrow> f2 z = g2 z"

   thus "sum_map f1 f2 x = sum_map g1 g2 x"

   proof (cases x)

     case Inl thus ?thesis using a1 by (clarsimp simp: setl_def)

   next

     case Inr thus ?thesis using a2 by (clarsimp simp: setr_def)

   qed

 next

   fix f1 :: "'o \<Rightarrow> 'q" and f2 :: "'p \<Rightarrow> 'r"

   show "setl o sum_map f1 f2 = image f1 o setl"

     by (rule ext, unfold o_apply) (simp add: setl_def split: sum.split)

 next

   fix f1 :: "'o \<Rightarrow> 'q" and f2 :: "'p \<Rightarrow> 'r"

   show "setr o sum_map f1 f2 = image f2 o setr"

     by (rule ext, unfold o_apply) (simp add: setr_def split: sum.split)

 next

   show "card_order natLeq" by (rule natLeq_card_order)

 next

   show "cinfinite natLeq" by (rule natLeq_cinfinite)

 next

   fix x :: "'o + 'p"

   show "|setl x| \<le>o natLeq"

     apply (rule ordLess_imp_ordLeq)

     apply (rule finite_iff_ordLess_natLeq[THEN iffD1])

     by (simp add: setl_def split: sum.split)

 next

   fix x :: "'o + 'p"

   show "|setr x| \<le>o natLeq"

     apply (rule ordLess_imp_ordLeq)

     apply (rule finite_iff_ordLess_natLeq[THEN iffD1])

     by (simp add: setr_def split: sum.split)

 next

   fix A1 A2 B11 B12 B21 B22 f11 f12 f21 f22 p11 p12 p21 p22

   assume "wpull A1 B11 B21 f11 f21 p11 p21" "wpull A2 B12 B22 f12 f22 p12 p22"

   hence

     pull1: "\<And>b1 b2. \<lbrakk>b1 \<in> B11; b2 \<in> B21; f11 b1 = f21 b2\<rbrakk> \<Longrightarrow> \<exists>a \<in> A1. p11 a = b1 \<and> p21 a = b2"

     and pull2: "\<And>b1 b2. \<lbrakk>b1 \<in> B12; b2 \<in> B22; f12 b1 = f22 b2\<rbrakk> \<Longrightarrow> \<exists>a \<in> A2. p12 a = b1 \<and> p22 a = b2"

     unfolding wpull_def by blast+

   show "wpull {x. setl x \<subseteq> A1 \<and> setr x \<subseteq> A2}

   {x. setl x \<subseteq> B11 \<and> setr x \<subseteq> B12} {x. setl x \<subseteq> B21 \<and> setr x \<subseteq> B22}

   (sum_map f11 f12) (sum_map f21 f22) (sum_map p11 p12) (sum_map p21 p22)"

     (is "wpull ?in ?in1 ?in2 ?mapf1 ?mapf2 ?mapp1 ?mapp2")

   proof (unfold wpull_def)

     { fix B1 B2

       assume *: "B1 \<in> ?in1" "B2 \<in> ?in2" "?mapf1 B1 = ?mapf2 B2"

       have "\<exists>A \<in> ?in. ?mapp1 A = B1 \<and> ?mapp2 A = B2"

       proof (cases B1)

         case (Inl b1)

         { fix b2 assume "B2 = Inr b2"

           with Inl *(3) have False by simp

         } then obtain b2 where Inl': "B2 = Inl b2" by (cases B2) (simp, blast)

         with Inl * have "b1 \<in> B11" "b2 \<in> B21" "f11 b1 = f21 b2"

         by (simp add: setl_def)+

         with pull1 obtain a where "a \<in> A1" "p11 a = b1" "p21 a = b2" by blast+

         with Inl Inl' have "Inl a \<in> ?in" "?mapp1 (Inl a) = B1 \<and> ?mapp2 (Inl a) = B2"

         by (simp add: sum_set_defs)+

         thus ?thesis by blast

       next

         case (Inr b1)

         { fix b2 assume "B2 = Inl b2"

           with Inr *(3) have False by simp

         } then obtain b2 where Inr': "B2 = Inr b2" by (cases B2) (simp, blast)

         with Inr * have "b1 \<in> B12" "b2 \<in> B22" "f12 b1 = f22 b2"

         by (simp add: sum_set_defs)+

         with pull2 obtain a where "a \<in> A2" "p12 a = b1" "p22 a = b2" by blast+

         with Inr Inr' have "Inr a \<in> ?in" "?mapp1 (Inr a) = B1 \<and> ?mapp2 (Inr a) = B2"

         by (simp add: sum_set_defs)+

         thus ?thesis by blast

       qed

     }

     thus "\<forall>B1 B2. B1 \<in> ?in1 \<and> B2 \<in> ?in2 \<and> ?mapf1 B1 = ?mapf2 B2 \<longrightarrow>

       (\<exists>A \<in> ?in. ?mapp1 A = B1 \<and> ?mapp2 A = B2)" by fastforce

   qed

 next

   fix R S

   show "sum_rel R S =

         (Grp {x. setl x \<subseteq> Collect (split R) \<and> setr x \<subseteq> Collect (split S)} (sum_map fst fst))\<inverse>\<inverse> OO

         Grp {x. setl x \<subseteq> Collect (split R) \<and> setr x \<subseteq> Collect (split S)} (sum_map snd snd)"

   unfolding setl_def setr_def sum_rel_def Grp_def relcompp.simps conversep.simps fun_eq_iff

   by (fastforce split: sum.splits)

 qed (auto simp: sum_set_defs)

 definition fsts :: "'a \<times> 'b \<Rightarrow> 'a set" where

 "fsts x = {fst x}"

 definition snds :: "'a \<times> 'b \<Rightarrow> 'b set" where

 "snds x = {snd x}"

 lemmas prod_set_defs = fsts_def[abs_def] snds_def[abs_def]

 bnf "'a \<times> 'b"

   map: map_pair

   sets: fsts snds

   bd: natLeq

   rel: prod_rel

 proof (unfold prod_set_defs)

   show "map_pair id id = id" by (rule map_pair.id)

 next

   fix f1 f2 g1 g2

   show "map_pair (g1 o f1) (g2 o f2) = map_pair g1 g2 o map_pair f1 f2"

     by (rule map_pair.comp[symmetric])

 next

   fix x f1 f2 g1 g2

   assume "\<And>z. z \<in> {fst x} \<Longrightarrow> f1 z = g1 z" "\<And>z. z \<in> {snd x} \<Longrightarrow> f2 z = g2 z"

   thus "map_pair f1 f2 x = map_pair g1 g2 x" by (cases x) simp

 next

   fix f1 f2

   show "(\<lambda>x. {fst x}) o map_pair f1 f2 = image f1 o (\<lambda>x. {fst x})"

     by (rule ext, unfold o_apply) simp

 next

   fix f1 f2

   show "(\<lambda>x. {snd x}) o map_pair f1 f2 = image f2 o (\<lambda>x. {snd x})"

     by (rule ext, unfold o_apply) simp

 next

   show "card_order natLeq" by (rule natLeq_card_order)

 next

   show "cinfinite natLeq" by (rule natLeq_cinfinite)

 next

   fix x

   show "|{fst x}| \<le>o natLeq"

     by (metis ordLess_imp_ordLeq finite_iff_ordLess_natLeq finite.emptyI finite_insert)

 next

   fix x

   show "|{snd x}| \<le>o natLeq"

     by (metis ordLess_imp_ordLeq finite_iff_ordLess_natLeq finite.emptyI finite_insert)

 next

   fix A1 A2 B11 B12 B21 B22 f11 f12 f21 f22 p11 p12 p21 p22

   assume "wpull A1 B11 B21 f11 f21 p11 p21" "wpull A2 B12 B22 f12 f22 p12 p22"

   thus "wpull {x. {fst x} \<subseteq> A1 \<and> {snd x} \<subseteq> A2}

     {x. {fst x} \<subseteq> B11 \<and> {snd x} \<subseteq> B12} {x. {fst x} \<subseteq> B21 \<and> {snd x} \<subseteq> B22}

    (map_pair f11 f12) (map_pair f21 f22) (map_pair p11 p12) (map_pair p21 p22)"

     unfolding wpull_def by simp fast

 next

   fix R S

   show "prod_rel R S =

         (Grp {x. {fst x} \<subseteq> Collect (split R) \<and> {snd x} \<subseteq> Collect (split S)} (map_pair fst fst))\<inverse>\<inverse> OO

         Grp {x. {fst x} \<subseteq> Collect (split R) \<and> {snd x} \<subseteq> Collect (split S)} (map_pair snd snd)"

   unfolding prod_set_defs prod_rel_def Grp_def relcompp.simps conversep.simps fun_eq_iff

   by auto

 qed

 (* Categorical version of pullback: *)

 lemma wpull_cat:

 assumes p: "wpull A B1 B2 f1 f2 p1 p2"

 and c: "f1 o q1 = f2 o q2"

 and r: "range q1 \<subseteq> B1" "range q2 \<subseteq> B2"

 obtains h where "range h \<subseteq> A \<and> q1 = p1 o h \<and> q2 = p2 o h"

 proof-

   have *: "\<forall>d. \<exists>a \<in> A. p1 a = q1 d & p2 a = q2 d"

   proof safe

     fix d

     have "f1 (q1 d) = f2 (q2 d)" using c unfolding comp_def[abs_def] by (rule fun_cong)

     moreover

     have "q1 d : B1" "q2 d : B2" using r unfolding image_def by auto

     ultimately show "\<exists>a \<in> A. p1 a = q1 d \<and> p2 a = q2 d"

       using p unfolding wpull_def by auto

   qed

   then obtain h where "!! d. h d \<in> A & p1 (h d) = q1 d & p2 (h d) = q2 d" by metis

   thus ?thesis using that by fastforce

 qed

 bnf "'a \<Rightarrow> 'b"

   map: "op \<circ>"

   sets: range

   bd: "natLeq +c |UNIV :: 'a set|"

   rel: "fun_rel op ="

 proof

   fix f show "id \<circ> f = id f" by simp

 next

   fix f g show "op \<circ> (g \<circ> f) = op \<circ> g \<circ> op \<circ> f"

   unfolding comp_def[abs_def] ..

 next

   fix x f g

   assume "\<And>z. z \<in> range x \<Longrightarrow> f z = g z"

   thus "f \<circ> x = g \<circ> x" by auto

 next

   fix f show "range \<circ> op \<circ> f = op ` f \<circ> range"

   unfolding image_def comp_def[abs_def] by auto

 next

   show "card_order (natLeq +c |UNIV| )" (is "_ (_ +c ?U)")

   apply (rule card_order_csum)

   apply (rule natLeq_card_order)

   by (rule card_of_card_order_on)

 (*  *)

   show "cinfinite (natLeq +c ?U)"

     apply (rule cinfinite_csum)

     apply (rule disjI1)

     by (rule natLeq_cinfinite)

 next

   fix f :: "'d => 'a"

   have "|range f| \<le>o | (UNIV::'d set) |" (is "_ \<le>o ?U") by (rule card_of_image)

   also have "?U \<le>o natLeq +c ?U" by (rule ordLeq_csum2) (rule card_of_Card_order)

   finally show "|range f| \<le>o natLeq +c ?U" .

 next

   fix A B1 B2 f1 f2 p1 p2 assume p: "wpull A B1 B2 f1 f2 p1 p2"

   show "wpull {h. range h \<subseteq> A} {g1. range g1 \<subseteq> B1} {g2. range g2 \<subseteq> B2}

     (op \<circ> f1) (op \<circ> f2) (op \<circ> p1) (op \<circ> p2)"

   unfolding wpull_def

   proof safe

     fix g1 g2 assume r: "range g1 \<subseteq> B1" "range g2 \<subseteq> B2"

     and c: "f1 \<circ> g1 = f2 \<circ> g2"

     show "\<exists>h \<in> {h. range h \<subseteq> A}. p1 \<circ> h = g1 \<and> p2 \<circ> h = g2"

     using wpull_cat[OF p c r] by simp metis

   qed

 next

   fix R

   show "fun_rel op = R =

         (Grp {x. range x \<subseteq> Collect (split R)} (op \<circ> fst))\<inverse>\<inverse> OO

          Grp {x. range x \<subseteq> Collect (split R)} (op \<circ> snd)"

   unfolding fun_rel_def Grp_def fun_eq_iff relcompp.simps conversep.simps  subset_iff image_iff

   by auto (force, metis (no_types) pair_collapse)

 qed

 end