src/HOL/BNF/BNF_Comp.thy
author hoelzl
Tue Nov 05 09:44:57 2013 +0100 (2013-11-05)
changeset 54257 5c7a3b6b05a9
parent 52660 7f7311d04727
child 54485 b61b8c9e4cf7
permissions -rw-r--r--
generalize SUP and INF to the syntactic type classes Sup and Inf
     1 (*  Title:      HOL/BNF/BNF_Comp.thy
     2     Author:     Dmitriy Traytel, TU Muenchen
     3     Copyright   2012
     4 
     5 Composition of bounded natural functors.
     6 *)
     7 
     8 header {* Composition of Bounded Natural Functors *}
     9 
    10 theory BNF_Comp
    11 imports Basic_BNFs
    12 begin
    13 
    14 lemma empty_natural: "(\<lambda>_. {}) o f = image g o (\<lambda>_. {})"
    15 by (rule ext) simp
    16 
    17 lemma Union_natural: "Union o image (image f) = image f o Union"
    18 by (rule ext) (auto simp only: o_apply)
    19 
    20 lemma in_Union_o_assoc: "x \<in> (Union o gset o gmap) A \<Longrightarrow> x \<in> (Union o (gset o gmap)) A"
    21 by (unfold o_assoc)
    22 
    23 lemma comp_single_set_bd:
    24   assumes fbd_Card_order: "Card_order fbd" and
    25     fset_bd: "\<And>x. |fset x| \<le>o fbd" and
    26     gset_bd: "\<And>x. |gset x| \<le>o gbd"
    27   shows "|\<Union>(fset ` gset x)| \<le>o gbd *c fbd"
    28 apply (subst sym[OF SUP_def])
    29 apply (rule ordLeq_transitive)
    30 apply (rule card_of_UNION_Sigma)
    31 apply (subst SIGMA_CSUM)
    32 apply (rule ordLeq_transitive)
    33 apply (rule card_of_Csum_Times')
    34 apply (rule fbd_Card_order)
    35 apply (rule ballI)
    36 apply (rule fset_bd)
    37 apply (rule ordLeq_transitive)
    38 apply (rule cprod_mono1)
    39 apply (rule gset_bd)
    40 apply (rule ordIso_imp_ordLeq)
    41 apply (rule ordIso_refl)
    42 apply (rule Card_order_cprod)
    43 done
    44 
    45 lemma Union_image_insert: "\<Union>(f ` insert a B) = f a \<union> \<Union>(f ` B)"
    46 by simp
    47 
    48 lemma Union_image_empty: "A \<union> \<Union>(f ` {}) = A"
    49 by simp
    50 
    51 lemma image_o_collect: "collect ((\<lambda>f. image g o f) ` F) = image g o collect F"
    52 by (rule ext) (auto simp add: collect_def)
    53 
    54 lemma conj_subset_def: "A \<subseteq> {x. P x \<and> Q x} = (A \<subseteq> {x. P x} \<and> A \<subseteq> {x. Q x})"
    55 by blast
    56 
    57 lemma UN_image_subset: "\<Union>(f ` g x) \<subseteq> X = (g x \<subseteq> {x. f x \<subseteq> X})"
    58 by blast
    59 
    60 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"
    61 by (unfold o_apply collect_def SUP_def)
    62 
    63 lemma wpull_cong:
    64 "\<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"
    65 by simp
    66 
    67 lemma Grp_fst_snd: "(Grp (Collect (split R)) fst)^--1 OO Grp (Collect (split R)) snd = R"
    68 unfolding Grp_def fun_eq_iff relcompp.simps by auto
    69 
    70 lemma OO_Grp_cong: "A = B \<Longrightarrow> (Grp A f)^--1 OO Grp A g = (Grp B f)^--1 OO Grp B g"
    71 by simp
    72 
    73 ML_file "Tools/bnf_comp_tactics.ML"
    74 ML_file "Tools/bnf_comp.ML"
    75 
    76 end