src/HOL/BNF/BNF_Util.thy
author traytel
Tue May 07 14:22:54 2013 +0200 (2013-05-07)
changeset 51893 596baae88a88
parent 49510 ba50d204095e
child 53560 4b5f42cfa244
permissions -rw-r--r--
got rid of the set based relator---use (binary) predicate based relator instead
     1 (*  Title:      HOL/BNF/BNF_Util.thy
     2     Author:     Dmitriy Traytel, TU Muenchen
     3     Author:     Jasmin Blanchette, TU Muenchen
     4     Copyright   2012
     5 
     6 Library for bounded natural functors.
     7 *)
     8 
     9 header {* Library for Bounded Natural Functors *}
    10 
    11 theory BNF_Util
    12 imports "../Cardinals/Cardinal_Arithmetic"
    13 begin
    14 
    15 lemma subset_Collect_iff: "B \<subseteq> A \<Longrightarrow> (B \<subseteq> {x \<in> A. P x}) = (\<forall>x \<in> B. P x)"
    16 by blast
    17 
    18 lemma subset_CollectI: "B \<subseteq> A \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> Q x \<Longrightarrow> P x) \<Longrightarrow> ({x \<in> B. Q x} \<subseteq> {x \<in> A. P x})"
    19 by blast
    20 
    21 definition collect where
    22 "collect F x = (\<Union>f \<in> F. f x)"
    23 
    24 (* Weak pullbacks: *)
    25 definition wpull where
    26 "wpull A B1 B2 f1 f2 p1 p2 \<longleftrightarrow>
    27  (\<forall> b1 b2. b1 \<in> B1 \<and> b2 \<in> B2 \<and> f1 b1 = f2 b2 \<longrightarrow> (\<exists> a \<in> A. p1 a = b1 \<and> p2 a = b2))"
    28 
    29 (* Weak pseudo-pullbacks *)
    30 definition wppull where
    31 "wppull A B1 B2 f1 f2 e1 e2 p1 p2 \<longleftrightarrow>
    32  (\<forall> b1 b2. b1 \<in> B1 \<and> b2 \<in> B2 \<and> f1 b1 = f2 b2 \<longrightarrow>
    33            (\<exists> a \<in> A. e1 (p1 a) = e1 b1 \<and> e2 (p2 a) = e2 b2))"
    34 
    35 lemma fst_snd: "\<lbrakk>snd x = (y, z)\<rbrakk> \<Longrightarrow> fst (snd x) = y"
    36 by simp
    37 
    38 lemma snd_snd: "\<lbrakk>snd x = (y, z)\<rbrakk> \<Longrightarrow> snd (snd x) = z"
    39 by simp
    40 
    41 lemma fstI: "x = (y, z) \<Longrightarrow> fst x = y"
    42 by simp
    43 
    44 lemma sndI: "x = (y, z) \<Longrightarrow> snd x = z"
    45 by simp
    46 
    47 lemma bijI: "\<lbrakk>\<And>x y. (f x = f y) = (x = y); \<And>y. \<exists>x. y = f x\<rbrakk> \<Longrightarrow> bij f"
    48 unfolding bij_def inj_on_def by auto blast
    49 
    50 lemma pair_mem_Collect_split:
    51 "(\<lambda>x y. (x, y) \<in> {(x, y). P x y}) = P"
    52 by simp
    53 
    54 lemma Collect_pair_mem_eq: "{(x, y). (x, y) \<in> R} = R"
    55 by simp
    56 
    57 lemma Collect_fst_snd_mem_eq: "{p. (fst p, snd p) \<in> A} = A"
    58 by simp
    59 
    60 (* Operator: *)
    61 definition "Gr A f = {(a, f a) | a. a \<in> A}"
    62 
    63 definition "Grp A f = (\<lambda>a b. b = f a \<and> a \<in> A)"
    64 
    65 ML_file "Tools/bnf_util.ML"
    66 ML_file "Tools/bnf_tactics.ML"
    67 
    68 end