(*  Title:      HOL/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.

 *)

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

 theory Basic_BNFs

 imports BNF_Def

 begin

 inductive_set setl :: "'a + 'b \<Rightarrow> 'a set" for s :: "'a + 'b" where

   "s = Inl x \<Longrightarrow> x \<in> setl s"

 inductive_set setr :: "'a + 'b \<Rightarrow> 'b set" for s :: "'a + 'b" where

   "s = Inr x \<Longrightarrow> x \<in> setr s"

 lemma sum_set_defs[code]:

   "setl = (\<lambda>x. case x of Inl z => {z} | _ => {})"

   "setr = (\<lambda>x. case x of Inr z => {z} | _ => {})"

   by (auto simp: fun_eq_iff intro: setl.intros setr.intros elim: setl.cases setr.cases split: sum.splits)

 lemma rel_sum_simps[code, simp]:

   "rel_sum R1 R2 (Inl a1) (Inl b1) = R1 a1 b1"

   "rel_sum R1 R2 (Inl a1) (Inr b2) = False"

   "rel_sum R1 R2 (Inr a2) (Inl b1) = False"

   "rel_sum R1 R2 (Inr a2) (Inr b2) = R2 a2 b2"

   by (auto intro: rel_sum.intros elim: rel_sum.cases)

 bnf "'a + 'b"

   map: map_sum

   sets: setl setr

   bd: natLeq

   wits: Inl Inr

   rel: rel_sum

 proof -

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

 next

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

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

     by (rule map_sum.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 "map_sum f1 f2 x = map_sum g1 g2 x"

   proof (cases x)

     case Inl thus ?thesis using a1 by (clarsimp simp: sum_set_defs(1))

   next

     case Inr thus ?thesis using a2 by (clarsimp simp: sum_set_defs(2))

   qed

 next

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

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

     by (rule ext, unfold o_apply) (simp add: sum_set_defs(1) split: sum.split)

 next

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

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

     by (rule ext, unfold o_apply) (simp add: sum_set_defs(2) 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: sum_set_defs(1) 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: sum_set_defs(2) split: sum.split)

 next

   fix R1 R2 S1 S2

   show "rel_sum R1 R2 OO rel_sum S1 S2 \<le> rel_sum (R1 OO S1) (R2 OO S2)"

     by (force elim: rel_sum.cases)

 next

   fix R S

   show "rel_sum R S =

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

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

   unfolding sum_set_defs Grp_def relcompp.simps conversep.simps fun_eq_iff

   by (fastforce elim: rel_sum.cases split: sum.splits)

 qed (auto simp: sum_set_defs)

 inductive_set fsts :: "'a \<times> 'b \<Rightarrow> 'a set" for p :: "'a \<times> 'b" where

   "fst p \<in> fsts p"

 inductive_set snds :: "'a \<times> 'b \<Rightarrow> 'b set" for p :: "'a \<times> 'b" where

   "snd p \<in> snds p"

 lemma prod_set_defs[code]: "fsts = (\<lambda>p. {fst p})" "snds = (\<lambda>p. {snd p})"

   by (auto intro: fsts.intros snds.intros elim: fsts.cases snds.cases)

 inductive

   rel_prod :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'b \<times> 'd \<Rightarrow> bool" for R1 R2

 where

   "\<lbrakk>R1 a b; R2 c d\<rbrakk> \<Longrightarrow> rel_prod R1 R2 (a, c) (b, d)"

 hide_fact rel_prod_def

 lemma rel_prod_apply [code, simp]:

   "rel_prod R1 R2 (a, b) (c, d) \<longleftrightarrow> R1 a c \<and> R2 b d"

   by (auto intro: rel_prod.intros elim: rel_prod.cases)

 lemma rel_prod_conv:

   "rel_prod R1 R2 = (\<lambda>(a, b) (c, d). R1 a c \<and> R2 b d)"

   by (rule ext, rule ext) auto

 bnf "'a \<times> 'b"

   map: map_prod

   sets: fsts snds

   bd: natLeq

   rel: rel_prod

 proof (unfold prod_set_defs)

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

 next

   fix f1 f2 g1 g2

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

     by (rule map_prod.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_prod f1 f2 x = map_prod g1 g2 x" by (cases x) simp

 next

   fix f1 f2

   show "(\<lambda>x. {fst x}) o map_prod 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_prod 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 (rule ordLess_imp_ordLeq) (simp add: finite_iff_ordLess_natLeq[symmetric])

 next

   fix x

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

     by (rule ordLess_imp_ordLeq) (simp add: finite_iff_ordLess_natLeq[symmetric])

 next

   fix R1 R2 S1 S2

   show "rel_prod R1 R2 OO rel_prod S1 S2 \<le> rel_prod (R1 OO S1) (R2 OO S2)" by auto

 next

   fix R S

   show "rel_prod R S =

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

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

   unfolding prod_set_defs rel_prod_apply Grp_def relcompp.simps conversep.simps fun_eq_iff

   by auto

 qed

 bnf "'a \<Rightarrow> 'b"

   map: "op \<circ>"

   sets: range

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

   rel: "rel_fun 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"

     by (auto simp add: fun_eq_iff)

 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 R S

   show "rel_fun op = R OO rel_fun op = S \<le> rel_fun op = (R OO S)" by (auto simp: rel_fun_def)

 next

   fix R

   show "rel_fun 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 rel_fun_def Grp_def fun_eq_iff relcompp.simps conversep.simps subset_iff image_iff

     comp_apply mem_Collect_eq split_beta bex_UNIV

   proof (safe, unfold fun_eq_iff[symmetric])

     fix x xa a b c xb y aa ba

     assume *: "x = a" "xa = c" "a = ba" "b = aa" "c = (\<lambda>x. snd (b x))" "ba = (\<lambda>x. fst (aa x))" and

        **: "\<forall>t. (\<exists>x. t = aa x) \<longrightarrow> R (fst t) (snd t)"

     show "R (x y) (xa y)" unfolding * by (rule mp[OF spec[OF **]]) blast

   qed force

 qed

 end