src/HOL/HOLCF/ex/Powerdomain_ex.thy
author kuncar
Fri Dec 09 18:07:04 2011 +0100 (2011-12-09)
changeset 45802 b16f976db515
parent 42151 4da4fc77664b
child 58880 0baae4311a9f
permissions -rw-r--r--
Quotient_Info stores only relation maps
     1 (*  Title:      HOL/HOLCF/ex/Powerdomain_ex.thy
     2     Author:     Brian Huffman
     3 *)
     4 
     5 header {* Powerdomain examples *}
     6 
     7 theory Powerdomain_ex
     8 imports HOLCF
     9 begin
    10 
    11 subsection {* Monadic sorting example *}
    12 
    13 domain ordering = LT | EQ | GT
    14 
    15 definition
    16   compare :: "int lift \<rightarrow> int lift \<rightarrow> ordering" where
    17   "compare = (FLIFT x y. if x < y then LT else if x = y then EQ else GT)"
    18 
    19 definition
    20   is_le :: "int lift \<rightarrow> int lift \<rightarrow> tr" where
    21   "is_le = (\<Lambda> x y. case compare\<cdot>x\<cdot>y of LT \<Rightarrow> TT | EQ \<Rightarrow> TT | GT \<Rightarrow> FF)"
    22 
    23 definition
    24   is_less :: "int lift \<rightarrow> int lift \<rightarrow> tr" where
    25   "is_less = (\<Lambda> x y. case compare\<cdot>x\<cdot>y of LT \<Rightarrow> TT | EQ \<Rightarrow> FF | GT \<Rightarrow> FF)"
    26 
    27 definition
    28   r1 :: "(int lift \<times> 'a) \<rightarrow> (int lift \<times> 'a) \<rightarrow> tr convex_pd" where
    29   "r1 = (\<Lambda> (x,_) (y,_). case compare\<cdot>x\<cdot>y of
    30           LT \<Rightarrow> {TT}\<natural> |
    31           EQ \<Rightarrow> {TT, FF}\<natural> |
    32           GT \<Rightarrow> {FF}\<natural>)"
    33 
    34 definition
    35   r2 :: "(int lift \<times> 'a) \<rightarrow> (int lift \<times> 'a) \<rightarrow> tr convex_pd" where
    36   "r2 = (\<Lambda> (x,_) (y,_). {is_le\<cdot>x\<cdot>y, is_less\<cdot>x\<cdot>y}\<natural>)"
    37 
    38 lemma r1_r2: "r1\<cdot>(x,a)\<cdot>(y,b) = (r2\<cdot>(x,a)\<cdot>(y,b) :: tr convex_pd)"
    39 apply (simp add: r1_def r2_def)
    40 apply (simp add: is_le_def is_less_def)
    41 apply (cases "compare\<cdot>x\<cdot>y")
    42 apply simp_all
    43 done
    44 
    45 
    46 subsection {* Picking a leaf from a tree *}
    47 
    48 domain 'a tree =
    49   Node (lazy "'a tree") (lazy "'a tree") |
    50   Leaf (lazy "'a")
    51 
    52 fixrec
    53   mirror :: "'a tree \<rightarrow> 'a tree"
    54 where
    55   mirror_Leaf: "mirror\<cdot>(Leaf\<cdot>a) = Leaf\<cdot>a"
    56 | mirror_Node: "mirror\<cdot>(Node\<cdot>l\<cdot>r) = Node\<cdot>(mirror\<cdot>r)\<cdot>(mirror\<cdot>l)"
    57 
    58 lemma mirror_strict [simp]: "mirror\<cdot>\<bottom> = \<bottom>"
    59 by fixrec_simp
    60 
    61 fixrec
    62   pick :: "'a tree \<rightarrow> 'a convex_pd"
    63 where
    64   pick_Leaf: "pick\<cdot>(Leaf\<cdot>a) = {a}\<natural>"
    65 | pick_Node: "pick\<cdot>(Node\<cdot>l\<cdot>r) = pick\<cdot>l \<union>\<natural> pick\<cdot>r"
    66 
    67 lemma pick_strict [simp]: "pick\<cdot>\<bottom> = \<bottom>"
    68 by fixrec_simp
    69 
    70 lemma pick_mirror: "pick\<cdot>(mirror\<cdot>t) = pick\<cdot>t"
    71 by (induct t) (simp_all add: convex_plus_ac)
    72 
    73 fixrec tree1 :: "int lift tree"
    74 where "tree1 = Node\<cdot>(Node\<cdot>(Leaf\<cdot>(Def 1))\<cdot>(Leaf\<cdot>(Def 2)))
    75                    \<cdot>(Node\<cdot>(Leaf\<cdot>(Def 3))\<cdot>(Leaf\<cdot>(Def 4)))"
    76 
    77 fixrec tree2 :: "int lift tree"
    78 where "tree2 = Node\<cdot>(Node\<cdot>(Leaf\<cdot>(Def 1))\<cdot>(Leaf\<cdot>(Def 2)))
    79                    \<cdot>(Node\<cdot>\<bottom>\<cdot>(Leaf\<cdot>(Def 4)))"
    80 
    81 fixrec tree3 :: "int lift tree"
    82 where "tree3 = Node\<cdot>(Node\<cdot>(Leaf\<cdot>(Def 1))\<cdot>tree3)
    83                    \<cdot>(Node\<cdot>(Leaf\<cdot>(Def 3))\<cdot>(Leaf\<cdot>(Def 4)))"
    84 
    85 declare tree1.simps tree2.simps tree3.simps [simp del]
    86 
    87 lemma pick_tree1:
    88   "pick\<cdot>tree1 = {Def 1, Def 2, Def 3, Def 4}\<natural>"
    89 apply (subst tree1.simps)
    90 apply simp
    91 apply (simp add: convex_plus_ac)
    92 done
    93 
    94 lemma pick_tree2:
    95   "pick\<cdot>tree2 = {Def 1, Def 2, \<bottom>, Def 4}\<natural>"
    96 apply (subst tree2.simps)
    97 apply simp
    98 apply (simp add: convex_plus_ac)
    99 done
   100 
   101 lemma pick_tree3:
   102   "pick\<cdot>tree3 = {Def 1, \<bottom>, Def 3, Def 4}\<natural>"
   103 apply (subst tree3.simps)
   104 apply simp
   105 apply (induct rule: tree3.induct)
   106 apply simp
   107 apply simp
   108 apply (simp add: convex_plus_ac)
   109 apply simp
   110 apply (simp add: convex_plus_ac)
   111 done
   112 
   113 end