blanchet@55075: (*  Title:      HOL/Basic_BNFs.thy
blanchet@48975:     Author:     Dmitriy Traytel, TU Muenchen
blanchet@48975:     Author:     Andrei Popescu, TU Muenchen
blanchet@48975:     Author:     Jasmin Blanchette, TU Muenchen
blanchet@48975:     Copyright   2012
blanchet@48975: 
blanchet@49309: Registration of basic types as bounded natural functors.
blanchet@48975: *)
blanchet@48975: 
blanchet@49309: header {* Registration of Basic Types as Bounded Natural Functors *}
blanchet@48975: 
blanchet@48975: theory Basic_BNFs
blanchet@49310: imports BNF_Def
blanchet@48975: begin
blanchet@48975: 
blanchet@49451: definition setl :: "'a + 'b \<Rightarrow> 'a set" where
blanchet@49451: "setl x = (case x of Inl z => {z} | _ => {})"
blanchet@48975: 
blanchet@49451: definition setr :: "'a + 'b \<Rightarrow> 'b set" where
blanchet@49451: "setr x = (case x of Inr z => {z} | _ => {})"
blanchet@48975: 
blanchet@49451: lemmas sum_set_defs = setl_def[abs_def] setr_def[abs_def]
blanchet@48975: 
blanchet@55083: definition
blanchet@55943:    rel_sum :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'a + 'b \<Rightarrow> 'c + 'd \<Rightarrow> bool"
blanchet@55083: where
blanchet@55943:    "rel_sum R1 R2 x y =
blanchet@55083:      (case (x, y) of (Inl x, Inl y) \<Rightarrow> R1 x y
blanchet@55083:      | (Inr x, Inr y) \<Rightarrow> R2 x y
blanchet@55083:      | _ \<Rightarrow> False)"
blanchet@55083: 
blanchet@55943: lemma rel_sum_simps[simp]:
blanchet@55943:   "rel_sum R1 R2 (Inl a1) (Inl b1) = R1 a1 b1"
blanchet@55943:   "rel_sum R1 R2 (Inl a1) (Inr b2) = False"
blanchet@55943:   "rel_sum R1 R2 (Inr a2) (Inl b1) = False"
blanchet@55943:   "rel_sum R1 R2 (Inr a2) (Inr b2) = R2 a2 b2"
blanchet@55943:   unfolding rel_sum_def by simp_all
blanchet@55083: 
traytel@54421: bnf "'a + 'b"
blanchet@55931:   map: map_sum
traytel@54421:   sets: setl setr
traytel@54421:   bd: natLeq
traytel@54421:   wits: Inl Inr
blanchet@55943:   rel: rel_sum
blanchet@48975: proof -
blanchet@55931:   show "map_sum id id = id" by (rule map_sum.id)
blanchet@48975: next
blanchet@54486:   fix f1 :: "'o \<Rightarrow> 's" and f2 :: "'p \<Rightarrow> 't" and g1 :: "'s \<Rightarrow> 'q" and g2 :: "'t \<Rightarrow> 'r"
blanchet@55931:   show "map_sum (g1 o f1) (g2 o f2) = map_sum g1 g2 o map_sum f1 f2"
blanchet@55931:     by (rule map_sum.comp[symmetric])
blanchet@48975: next
blanchet@54486:   fix x and f1 :: "'o \<Rightarrow> 'q" and f2 :: "'p \<Rightarrow> 'r" and g1 g2
blanchet@49451:   assume a1: "\<And>z. z \<in> setl x \<Longrightarrow> f1 z = g1 z" and
blanchet@49451:          a2: "\<And>z. z \<in> setr x \<Longrightarrow> f2 z = g2 z"
blanchet@55931:   thus "map_sum f1 f2 x = map_sum g1 g2 x"
blanchet@48975:   proof (cases x)
blanchet@49451:     case Inl thus ?thesis using a1 by (clarsimp simp: setl_def)
blanchet@48975:   next
blanchet@49451:     case Inr thus ?thesis using a2 by (clarsimp simp: setr_def)
blanchet@48975:   qed
blanchet@48975: next
blanchet@54486:   fix f1 :: "'o \<Rightarrow> 'q" and f2 :: "'p \<Rightarrow> 'r"
blanchet@55931:   show "setl o map_sum f1 f2 = image f1 o setl"
blanchet@49451:     by (rule ext, unfold o_apply) (simp add: setl_def split: sum.split)
blanchet@48975: next
blanchet@54486:   fix f1 :: "'o \<Rightarrow> 'q" and f2 :: "'p \<Rightarrow> 'r"
blanchet@55931:   show "setr o map_sum f1 f2 = image f2 o setr"
blanchet@49451:     by (rule ext, unfold o_apply) (simp add: setr_def split: sum.split)
blanchet@48975: next
blanchet@48975:   show "card_order natLeq" by (rule natLeq_card_order)
blanchet@48975: next
blanchet@48975:   show "cinfinite natLeq" by (rule natLeq_cinfinite)
blanchet@48975: next
blanchet@54486:   fix x :: "'o + 'p"
blanchet@49451:   show "|setl x| \<le>o natLeq"
blanchet@48975:     apply (rule ordLess_imp_ordLeq)
blanchet@48975:     apply (rule finite_iff_ordLess_natLeq[THEN iffD1])
blanchet@49451:     by (simp add: setl_def split: sum.split)
blanchet@48975: next
blanchet@54486:   fix x :: "'o + 'p"
blanchet@49451:   show "|setr x| \<le>o natLeq"
blanchet@48975:     apply (rule ordLess_imp_ordLeq)
blanchet@48975:     apply (rule finite_iff_ordLess_natLeq[THEN iffD1])
blanchet@49451:     by (simp add: setr_def split: sum.split)
blanchet@48975: next
traytel@54841:   fix R1 R2 S1 S2
blanchet@55943:   show "rel_sum R1 R2 OO rel_sum S1 S2 \<le> rel_sum (R1 OO S1) (R2 OO S2)"
blanchet@55943:     by (auto simp: rel_sum_def split: sum.splits)
blanchet@49453: next
blanchet@49453:   fix R S
blanchet@55943:   show "rel_sum R S =
blanchet@55931:         (Grp {x. setl x \<subseteq> Collect (split R) \<and> setr x \<subseteq> Collect (split S)} (map_sum fst fst))\<inverse>\<inverse> OO
blanchet@55931:         Grp {x. setl x \<subseteq> Collect (split R) \<and> setr x \<subseteq> Collect (split S)} (map_sum snd snd)"
blanchet@55943:   unfolding setl_def setr_def rel_sum_def Grp_def relcompp.simps conversep.simps fun_eq_iff
blanchet@49453:   by (fastforce split: sum.splits)
blanchet@48975: qed (auto simp: sum_set_defs)
blanchet@48975: 
blanchet@48975: definition fsts :: "'a \<times> 'b \<Rightarrow> 'a set" where
blanchet@48975: "fsts x = {fst x}"
blanchet@48975: 
blanchet@48975: definition snds :: "'a \<times> 'b \<Rightarrow> 'b set" where
blanchet@48975: "snds x = {snd x}"
blanchet@48975: 
blanchet@48975: lemmas prod_set_defs = fsts_def[abs_def] snds_def[abs_def]
blanchet@48975: 
blanchet@55083: definition
blanchet@55944:   rel_prod :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'b \<times> 'd \<Rightarrow> bool"
blanchet@55083: where
blanchet@55944:   "rel_prod R1 R2 = (\<lambda>(a, b) (c, d). R1 a c \<and> R2 b d)"
blanchet@55083: 
blanchet@55944: lemma rel_prod_apply [simp]:
blanchet@55944:   "rel_prod R1 R2 (a, b) (c, d) \<longleftrightarrow> R1 a c \<and> R2 b d"
blanchet@55944:   by (simp add: rel_prod_def)
blanchet@55083: 
traytel@54421: bnf "'a \<times> 'b"
blanchet@55932:   map: map_prod
traytel@54421:   sets: fsts snds
traytel@54421:   bd: natLeq
blanchet@55944:   rel: rel_prod
blanchet@48975: proof (unfold prod_set_defs)
blanchet@55932:   show "map_prod id id = id" by (rule map_prod.id)
blanchet@48975: next
blanchet@48975:   fix f1 f2 g1 g2
blanchet@55932:   show "map_prod (g1 o f1) (g2 o f2) = map_prod g1 g2 o map_prod f1 f2"
blanchet@55932:     by (rule map_prod.comp[symmetric])
blanchet@48975: next
blanchet@48975:   fix x f1 f2 g1 g2
blanchet@48975:   assume "\<And>z. z \<in> {fst x} \<Longrightarrow> f1 z = g1 z" "\<And>z. z \<in> {snd x} \<Longrightarrow> f2 z = g2 z"
blanchet@55932:   thus "map_prod f1 f2 x = map_prod g1 g2 x" by (cases x) simp
blanchet@48975: next
blanchet@48975:   fix f1 f2
blanchet@55932:   show "(\<lambda>x. {fst x}) o map_prod f1 f2 = image f1 o (\<lambda>x. {fst x})"
blanchet@48975:     by (rule ext, unfold o_apply) simp
blanchet@48975: next
blanchet@48975:   fix f1 f2
blanchet@55932:   show "(\<lambda>x. {snd x}) o map_prod f1 f2 = image f2 o (\<lambda>x. {snd x})"
blanchet@48975:     by (rule ext, unfold o_apply) simp
blanchet@48975: next
traytel@52635:   show "card_order natLeq" by (rule natLeq_card_order)
blanchet@48975: next
traytel@52635:   show "cinfinite natLeq" by (rule natLeq_cinfinite)
blanchet@48975: next
blanchet@48975:   fix x
traytel@52635:   show "|{fst x}| \<le>o natLeq"
traytel@55811:     by (rule ordLess_imp_ordLeq) (simp add: finite_iff_ordLess_natLeq[symmetric])
blanchet@48975: next
traytel@52635:   fix x
traytel@52635:   show "|{snd x}| \<le>o natLeq"
traytel@55811:     by (rule ordLess_imp_ordLeq) (simp add: finite_iff_ordLess_natLeq[symmetric])
blanchet@48975: next
traytel@54841:   fix R1 R2 S1 S2
blanchet@55944:   show "rel_prod R1 R2 OO rel_prod S1 S2 \<le> rel_prod (R1 OO S1) (R2 OO S2)" by auto
blanchet@49453: next
blanchet@49453:   fix R S
blanchet@55944:   show "rel_prod R S =
blanchet@55932:         (Grp {x. {fst x} \<subseteq> Collect (split R) \<and> {snd x} \<subseteq> Collect (split S)} (map_prod fst fst))\<inverse>\<inverse> OO
blanchet@55932:         Grp {x. {fst x} \<subseteq> Collect (split R) \<and> {snd x} \<subseteq> Collect (split S)} (map_prod snd snd)"
blanchet@55944:   unfolding prod_set_defs rel_prod_def Grp_def relcompp.simps conversep.simps fun_eq_iff
blanchet@49453:   by auto
traytel@54189: qed
blanchet@48975: 
traytel@54421: bnf "'a \<Rightarrow> 'b"
traytel@54421:   map: "op \<circ>"
traytel@54421:   sets: range
traytel@54421:   bd: "natLeq +c |UNIV :: 'a set|"
blanchet@55945:   rel: "rel_fun op ="
blanchet@48975: proof
blanchet@48975:   fix f show "id \<circ> f = id f" by simp
blanchet@48975: next
blanchet@48975:   fix f g show "op \<circ> (g \<circ> f) = op \<circ> g \<circ> op \<circ> f"
blanchet@48975:   unfolding comp_def[abs_def] ..
blanchet@48975: next
blanchet@48975:   fix x f g
blanchet@48975:   assume "\<And>z. z \<in> range x \<Longrightarrow> f z = g z"
blanchet@48975:   thus "f \<circ> x = g \<circ> x" by auto
blanchet@48975: next
blanchet@48975:   fix f show "range \<circ> op \<circ> f = op ` f \<circ> range"
haftmann@56077:     by (auto simp add: fun_eq_iff)
blanchet@48975: next
blanchet@48975:   show "card_order (natLeq +c |UNIV| )" (is "_ (_ +c ?U)")
blanchet@48975:   apply (rule card_order_csum)
blanchet@48975:   apply (rule natLeq_card_order)
blanchet@48975:   by (rule card_of_card_order_on)
blanchet@48975: (*  *)
blanchet@48975:   show "cinfinite (natLeq +c ?U)"
blanchet@48975:     apply (rule cinfinite_csum)
blanchet@48975:     apply (rule disjI1)
blanchet@48975:     by (rule natLeq_cinfinite)
blanchet@48975: next
blanchet@48975:   fix f :: "'d => 'a"
blanchet@48975:   have "|range f| \<le>o | (UNIV::'d set) |" (is "_ \<le>o ?U") by (rule card_of_image)
blanchet@54486:   also have "?U \<le>o natLeq +c ?U" by (rule ordLeq_csum2) (rule card_of_Card_order)
blanchet@48975:   finally show "|range f| \<le>o natLeq +c ?U" .
blanchet@48975: next
traytel@54841:   fix R S
blanchet@55945:   show "rel_fun op = R OO rel_fun op = S \<le> rel_fun op = (R OO S)" by (auto simp: rel_fun_def)
blanchet@49453: next
blanchet@49463:   fix R
blanchet@55945:   show "rel_fun op = R =
traytel@51893:         (Grp {x. range x \<subseteq> Collect (split R)} (op \<circ> fst))\<inverse>\<inverse> OO
traytel@51893:          Grp {x. range x \<subseteq> Collect (split R)} (op \<circ> snd)"
blanchet@55945:   unfolding rel_fun_def Grp_def fun_eq_iff relcompp.simps conversep.simps subset_iff image_iff
traytel@55811:     comp_apply mem_Collect_eq split_beta bex_UNIV
traytel@55811:   proof (safe, unfold fun_eq_iff[symmetric])
traytel@55811:     fix x xa a b c xb y aa ba
traytel@55811:     assume *: "x = a" "xa = c" "a = ba" "b = aa" "c = (\<lambda>x. snd (b x))" "ba = (\<lambda>x. fst (aa x))" and
traytel@55811:        **: "\<forall>t. (\<exists>x. t = aa x) \<longrightarrow> R (fst t) (snd t)"
traytel@55811:     show "R (x y) (xa y)" unfolding * by (rule mp[OF spec[OF **]]) blast
traytel@55811:   qed force
traytel@54189: qed
traytel@54191: 
blanchet@48975: end