author | wenzelm |
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permissions | -rw-r--r-- |
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(* Title: HOL/HOLCF/ex/Powerdomain_ex.thy |
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Author: Brian Huffman |
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*) |
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section {* Powerdomain examples *} |
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theory Powerdomain_ex |
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imports HOLCF |
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begin |
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subsection {* Monadic sorting example *} |
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domain ordering = LT | EQ | GT |
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definition |
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compare :: "int lift \<rightarrow> int lift \<rightarrow> ordering" where |
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"compare = (FLIFT x y. if x < y then LT else if x = y then EQ else GT)" |
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definition |
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is_le :: "int lift \<rightarrow> int lift \<rightarrow> tr" where |
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"is_le = (\<Lambda> x y. case compare\<cdot>x\<cdot>y of LT \<Rightarrow> TT | EQ \<Rightarrow> TT | GT \<Rightarrow> FF)" |
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definition |
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is_less :: "int lift \<rightarrow> int lift \<rightarrow> tr" where |
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"is_less = (\<Lambda> x y. case compare\<cdot>x\<cdot>y of LT \<Rightarrow> TT | EQ \<Rightarrow> FF | GT \<Rightarrow> FF)" |
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definition |
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r1 :: "(int lift \<times> 'a) \<rightarrow> (int lift \<times> 'a) \<rightarrow> tr convex_pd" where |
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"r1 = (\<Lambda> (x,_) (y,_). case compare\<cdot>x\<cdot>y of |
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LT \<Rightarrow> {TT}\<natural> | |
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EQ \<Rightarrow> {TT, FF}\<natural> | |
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GT \<Rightarrow> {FF}\<natural>)" |
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definition |
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r2 :: "(int lift \<times> 'a) \<rightarrow> (int lift \<times> 'a) \<rightarrow> tr convex_pd" where |
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"r2 = (\<Lambda> (x,_) (y,_). {is_le\<cdot>x\<cdot>y, is_less\<cdot>x\<cdot>y}\<natural>)" |
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lemma r1_r2: "r1\<cdot>(x,a)\<cdot>(y,b) = (r2\<cdot>(x,a)\<cdot>(y,b) :: tr convex_pd)" |
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apply (simp add: r1_def r2_def) |
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apply (simp add: is_le_def is_less_def) |
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apply (cases "compare\<cdot>x\<cdot>y") |
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apply simp_all |
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done |
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subsection {* Picking a leaf from a tree *} |
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domain 'a tree = |
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Node (lazy "'a tree") (lazy "'a tree") | |
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Leaf (lazy "'a") |
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fixrec |
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mirror :: "'a tree \<rightarrow> 'a tree" |
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where |
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mirror_Leaf: "mirror\<cdot>(Leaf\<cdot>a) = Leaf\<cdot>a" |
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| mirror_Node: "mirror\<cdot>(Node\<cdot>l\<cdot>r) = Node\<cdot>(mirror\<cdot>r)\<cdot>(mirror\<cdot>l)" |
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lemma mirror_strict [simp]: "mirror\<cdot>\<bottom> = \<bottom>" |
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by fixrec_simp |
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fixrec |
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pick :: "'a tree \<rightarrow> 'a convex_pd" |
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where |
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pick_Leaf: "pick\<cdot>(Leaf\<cdot>a) = {a}\<natural>" |
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| pick_Node: "pick\<cdot>(Node\<cdot>l\<cdot>r) = pick\<cdot>l \<union>\<natural> pick\<cdot>r" |
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lemma pick_strict [simp]: "pick\<cdot>\<bottom> = \<bottom>" |
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by fixrec_simp |
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lemma pick_mirror: "pick\<cdot>(mirror\<cdot>t) = pick\<cdot>t" |
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by (induct t) (simp_all add: convex_plus_ac) |
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fixrec tree1 :: "int lift tree" |
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where "tree1 = Node\<cdot>(Node\<cdot>(Leaf\<cdot>(Def 1))\<cdot>(Leaf\<cdot>(Def 2))) |
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\<cdot>(Node\<cdot>(Leaf\<cdot>(Def 3))\<cdot>(Leaf\<cdot>(Def 4)))" |
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fixrec tree2 :: "int lift tree" |
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where "tree2 = Node\<cdot>(Node\<cdot>(Leaf\<cdot>(Def 1))\<cdot>(Leaf\<cdot>(Def 2))) |
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\<cdot>(Node\<cdot>\<bottom>\<cdot>(Leaf\<cdot>(Def 4)))" |
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fixrec tree3 :: "int lift tree" |
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where "tree3 = Node\<cdot>(Node\<cdot>(Leaf\<cdot>(Def 1))\<cdot>tree3) |
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\<cdot>(Node\<cdot>(Leaf\<cdot>(Def 3))\<cdot>(Leaf\<cdot>(Def 4)))" |
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declare tree1.simps tree2.simps tree3.simps [simp del] |
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lemma pick_tree1: |
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"pick\<cdot>tree1 = {Def 1, Def 2, Def 3, Def 4}\<natural>" |
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apply (subst tree1.simps) |
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apply simp |
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apply (simp add: convex_plus_ac) |
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done |
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lemma pick_tree2: |
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"pick\<cdot>tree2 = {Def 1, Def 2, \<bottom>, Def 4}\<natural>" |
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apply (subst tree2.simps) |
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apply simp |
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apply (simp add: convex_plus_ac) |
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done |
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lemma pick_tree3: |
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"pick\<cdot>tree3 = {Def 1, \<bottom>, Def 3, Def 4}\<natural>" |
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apply (subst tree3.simps) |
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apply simp |
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apply (induct rule: tree3.induct) |
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apply simp |
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apply simp |
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apply (simp add: convex_plus_ac) |
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apply simp |
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apply (simp add: convex_plus_ac) |
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done |
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end |