(*  Title:      HOL/BNF_Composition.thy

     Author:     Dmitriy Traytel, TU Muenchen

     Author:     Jasmin Blanchette, TU Muenchen

     Copyright   2012, 2013, 2014

 Composition of bounded natural functors.

 *)

 section \<open>Composition of Bounded Natural Functors\<close>

 theory BNF_Composition

 imports BNF_Def

 keywords

   "copy_bnf" :: thy_decl and

   "lift_bnf" :: thy_goal

 begin

 lemma ssubst_mem: "\<lbrakk>t = s; s \<in> X\<rbrakk> \<Longrightarrow> t \<in> X"

   by simp

 lemma empty_natural: "(\<lambda>_. {}) o f = image g o (\<lambda>_. {})"

   by (rule ext) simp

 lemma Union_natural: "Union o image (image f) = image f o Union"

   by (rule ext) (auto simp only: comp_apply)

 lemma in_Union_o_assoc: "x \<in> (Union o gset o gmap) A \<Longrightarrow> x \<in> (Union o (gset o gmap)) A"

   by (unfold comp_assoc)

 lemma comp_single_set_bd:

   assumes fbd_Card_order: "Card_order fbd" and

     fset_bd: "\<And>x. |fset x| \<le>o fbd" and

     gset_bd: "\<And>x. |gset x| \<le>o gbd"

   shows "|\<Union>(fset ` gset x)| \<le>o gbd *c fbd"

   apply simp

   apply (rule ordLeq_transitive)

   apply (rule card_of_UNION_Sigma)

   apply (subst SIGMA_CSUM)

   apply (rule ordLeq_transitive)

   apply (rule card_of_Csum_Times')

   apply (rule fbd_Card_order)

   apply (rule ballI)

   apply (rule fset_bd)

   apply (rule ordLeq_transitive)

   apply (rule cprod_mono1)

   apply (rule gset_bd)

   apply (rule ordIso_imp_ordLeq)

   apply (rule ordIso_refl)

   apply (rule Card_order_cprod)

   done

 lemma csum_dup: "cinfinite r \<Longrightarrow> Card_order r \<Longrightarrow> p +c p' =o r +c r \<Longrightarrow> p +c p' =o r"

   apply (erule ordIso_transitive)

   apply (frule csum_absorb2')

   apply (erule ordLeq_refl)

   by simp

 lemma cprod_dup: "cinfinite r \<Longrightarrow> Card_order r \<Longrightarrow> p *c p' =o r *c r \<Longrightarrow> p *c p' =o r"

   apply (erule ordIso_transitive)

   apply (rule cprod_infinite)

   by simp

 lemma Union_image_insert: "\<Union>(f ` insert a B) = f a \<union> \<Union>(f ` B)"

   by simp

 lemma Union_image_empty: "A \<union> \<Union>(f ` {}) = A"

   by simp

 lemma image_o_collect: "collect ((\<lambda>f. image g o f) ` F) = image g o collect F"

   by (rule ext) (auto simp add: collect_def)

 lemma conj_subset_def: "A \<subseteq> {x. P x \<and> Q x} = (A \<subseteq> {x. P x} \<and> A \<subseteq> {x. Q x})"

   by blast

 lemma UN_image_subset: "\<Union>(f ` g x) \<subseteq> X = (g x \<subseteq> {x. f x \<subseteq> X})"

   by blast

 lemma comp_set_bd_Union_o_collect: "|\<Union>\<Union>((\<lambda>f. f x) ` X)| \<le>o hbd \<Longrightarrow> |(Union \<circ> collect X) x| \<le>o hbd"

   by (unfold comp_apply collect_def) simp

 lemma wpull_cong:

   "\<lbrakk>A' = A; B1' = B1; B2' = B2; wpull A B1 B2 f1 f2 p1 p2\<rbrakk> \<Longrightarrow> wpull A' B1' B2' f1 f2 p1 p2"

   by simp

 lemma Grp_fst_snd: "(Grp (Collect (split R)) fst)^--1 OO Grp (Collect (split R)) snd = R"

   unfolding Grp_def fun_eq_iff relcompp.simps by auto

 lemma OO_Grp_cong: "A = B \<Longrightarrow> (Grp A f)^--1 OO Grp A g = (Grp B f)^--1 OO Grp B g"

   by (rule arg_cong)

 lemma vimage2p_relcompp_mono: "R OO S \<le> T \<Longrightarrow>

   vimage2p f g R OO vimage2p g h S \<le> vimage2p f h T"

   unfolding vimage2p_def by auto

 lemma type_copy_map_cong0: "M (g x) = N (h x) \<Longrightarrow> (f o M o g) x = (f o N o h) x"

   by auto

 lemma type_copy_set_bd: "(\<And>y. |S y| \<le>o bd) \<Longrightarrow> |(S o Rep) x| \<le>o bd"

   by auto

 lemma vimage2p_cong: "R = S \<Longrightarrow> vimage2p f g R = vimage2p f g S"

   by simp

 context

   fixes Rep Abs

   assumes type_copy: "type_definition Rep Abs UNIV"

 begin

 lemma type_copy_map_id0: "M = id \<Longrightarrow> Abs o M o Rep = id"

   using type_definition.Rep_inverse[OF type_copy] by auto

 lemma type_copy_map_comp0: "M = M1 o M2 \<Longrightarrow> f o M o g = (f o M1 o Rep) o (Abs o M2 o g)"

   using type_definition.Abs_inverse[OF type_copy UNIV_I] by auto

 lemma type_copy_set_map0: "S o M = image f o S' \<Longrightarrow> (S o Rep) o (Abs o M o g) = image f o (S' o g)"

   using type_definition.Abs_inverse[OF type_copy UNIV_I] by (auto simp: o_def fun_eq_iff)

 lemma type_copy_wit: "x \<in> (S o Rep) (Abs y) \<Longrightarrow> x \<in> S y"

   using type_definition.Abs_inverse[OF type_copy UNIV_I] by auto

 lemma type_copy_vimage2p_Grp_Rep: "vimage2p f Rep (Grp (Collect P) h) =

     Grp (Collect (\<lambda>x. P (f x))) (Abs o h o f)"

   unfolding vimage2p_def Grp_def fun_eq_iff

   by (auto simp: type_definition.Abs_inverse[OF type_copy UNIV_I]

    type_definition.Rep_inverse[OF type_copy] dest: sym)

 lemma type_copy_vimage2p_Grp_Abs:

   "\<And>h. vimage2p g Abs (Grp (Collect P) h) = Grp (Collect (\<lambda>x. P (g x))) (Rep o h o g)"

   unfolding vimage2p_def Grp_def fun_eq_iff

   by (auto simp: type_definition.Abs_inverse[OF type_copy UNIV_I]

    type_definition.Rep_inverse[OF type_copy] dest: sym)

 lemma type_copy_ex_RepI: "(\<exists>b. F b) = (\<exists>b. F (Rep b))"

 proof safe

   fix b assume "F b"

   show "\<exists>b'. F (Rep b')"

   proof (rule exI)

     from \<open>F b\<close> show "F (Rep (Abs b))" using type_definition.Abs_inverse[OF type_copy] by auto

   qed

 qed blast

 lemma vimage2p_relcompp_converse:

   "vimage2p f g (R^--1 OO S) = (vimage2p Rep f R)^--1 OO vimage2p Rep g S"

   unfolding vimage2p_def relcompp.simps conversep.simps fun_eq_iff image_def

   by (auto simp: type_copy_ex_RepI)

 end

 bnf DEADID: 'a

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

   bd: natLeq

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

   by (auto simp add: Grp_def natLeq_card_order natLeq_cinfinite)

 definition id_bnf :: "'a \<Rightarrow> 'a" where

   "id_bnf \<equiv> (\<lambda>x. x)"

 lemma id_bnf_apply: "id_bnf x = x"

   unfolding id_bnf_def by simp

 bnf ID: 'a

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

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

   bd: natLeq

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

   unfolding id_bnf_def

   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

 lemma type_definition_id_bnf_UNIV: "type_definition id_bnf id_bnf UNIV"

   unfolding id_bnf_def by unfold_locales auto

 ML_file "Tools/BNF/bnf_comp_tactics.ML"

 ML_file "Tools/BNF/bnf_comp.ML"

 ML_file "Tools/BNF/bnf_lift.ML"

 hide_fact

   DEADID.inj_map DEADID.inj_map_strong DEADID.map_comp DEADID.map_cong DEADID.map_cong0

   DEADID.map_cong_simp DEADID.map_id DEADID.map_id0 DEADID.map_ident DEADID.map_transfer

   DEADID.rel_Grp DEADID.rel_compp DEADID.rel_compp_Grp DEADID.rel_conversep DEADID.rel_eq

   DEADID.rel_flip DEADID.rel_map DEADID.rel_mono DEADID.rel_transfer

   ID.inj_map ID.inj_map_strong ID.map_comp ID.map_cong ID.map_cong0 ID.map_cong_simp ID.map_id

   ID.map_id0 ID.map_ident ID.map_transfer ID.rel_Grp ID.rel_compp ID.rel_compp_Grp ID.rel_conversep

   ID.rel_eq ID.rel_flip ID.rel_map ID.rel_mono ID.rel_transfer ID.set_map ID.set_transfer

 end