(* 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>_. {}) \<circ> f = image g \<circ> (\<lambda>_. {})"
by (rule ext) simp
lemma Union_natural: "Union \<circ> image (image f) = image f \<circ> Union"
by (rule ext) (auto simp only: comp_apply)
lemma in_Union_o_assoc: "x \<in> (Union \<circ> gset \<circ> gmap) A \<Longrightarrow> x \<in> (Union \<circ> (gset \<circ> 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 \<circ> f) ` F) = image g \<circ> 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 Collect_inj: "Collect P = Collect Q \<Longrightarrow> P = Q"
by blast
lemma Grp_fst_snd: "(Grp (Collect (case_prod R)) fst)^--1 OO Grp (Collect (case_prod 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 \<circ> M \<circ> g) x = (f \<circ> N \<circ> h) x"
by auto
lemma type_copy_set_bd: "(\<And>y. |S y| \<le>o bd) \<Longrightarrow> |(S \<circ> Rep) x| \<le>o bd"
by auto
lemma vimage2p_cong: "R = S \<Longrightarrow> vimage2p f g R = vimage2p f g S"
by simp
lemma Ball_comp_iff: "(\<lambda>x. Ball (A x) f) \<circ> g = (\<lambda>x. Ball ((A \<circ> g) x) f)"
unfolding o_def by auto
lemma conj_comp_iff: "(\<lambda>x. P x \<and> Q x) \<circ> g = (\<lambda>x. (P \<circ> g) x \<and> (Q \<circ> g) x)"
unfolding o_def by auto
context
fixes Rep Abs
assumes type_copy: "type_definition Rep Abs UNIV"
begin
lemma type_copy_map_id0: "M = id \<Longrightarrow> Abs \<circ> M \<circ> Rep = id"
using type_definition.Rep_inverse[OF type_copy] by auto
lemma type_copy_map_comp0: "M = M1 \<circ> M2 \<Longrightarrow> f \<circ> M \<circ> g = (f \<circ> M1 \<circ> Rep) \<circ> (Abs \<circ> M2 \<circ> g)"
using type_definition.Abs_inverse[OF type_copy UNIV_I] by auto
lemma type_copy_set_map0: "S \<circ> M = image f \<circ> S' \<Longrightarrow> (S \<circ> Rep) \<circ> (Abs \<circ> M \<circ> g) = image f \<circ> (S' \<circ> 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 \<circ> 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 \<circ> h \<circ> 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 \<circ> h \<circ> 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: "(=) :: 'a \<Rightarrow> 'a \<Rightarrow> bool"
by (auto simp add: 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"
pred: "id_bnf :: ('a \<Rightarrow> bool) \<Rightarrow> 'a \<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