author  wenzelm 
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(* Title: HOL/HOLCF/Tutorial/Domain_ex.thy 
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Author: Brian Huffman 
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*) 
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header {* Domain package examples *} 
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theory Domain_ex 
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imports HOLCF 
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begin 
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text {* Domain constructors are strict by default. *} 
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domain d1 = d1a  d1b "d1" "d1" 
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lemma "d1b\<cdot>\<bottom>\<cdot>y = \<bottom>" by simp 
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text {* Constructors can be made lazy using the @{text "lazy"} keyword. *} 
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domain d2 = d2a  d2b (lazy "d2") 
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lemma "d2b\<cdot>x \<noteq> \<bottom>" by simp 
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text {* Strict and lazy arguments may be mixed arbitrarily. *} 
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domain d3 = d3a  d3b (lazy "d2") "d2" 
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lemma "P (d3b\<cdot>x\<cdot>y = \<bottom>) \<longleftrightarrow> P (y = \<bottom>)" by simp 
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text {* Selectors can be used with strict or lazy constructor arguments. *} 
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domain d4 = d4a  d4b (lazy d4b_left :: "d2") (d4b_right :: "d2") 
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lemma "y \<noteq> \<bottom> \<Longrightarrow> d4b_left\<cdot>(d4b\<cdot>x\<cdot>y) = x" by simp 
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text {* Mixfix declarations can be given for data constructors. *} 
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domain d5 = d5a  d5b (lazy "d5") "d5" (infixl ":#:" 70) 
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lemma "d5a \<noteq> x :#: y :#: z" by simp 
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text {* Mixfix declarations can also be given for type constructors. *} 
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domain ('a, 'b) lazypair (infixl ":*:" 25) = 
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lpair (lazy lfst :: 'a) (lazy lsnd :: 'b) (infixl ":*:" 75) 
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lemma "\<forall>p::('a :*: 'b). p \<sqsubseteq> lfst\<cdot>p :*: lsnd\<cdot>p" 
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by (rule allI, case_tac p, simp_all) 
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text {* Nonrecursive constructor arguments can have arbitrary types. *} 
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domain ('a, 'b) d6 = d6 "int lift" "'a \<oplus> 'b u" (lazy "('a :*: 'b) \<times> ('b \<rightarrow> 'a)") 
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text {* 
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Indirect recusion is allowed for sums, products, lifting, and the 
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continuous function space. However, the domain package does not 
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generate an induction rule in terms of the constructors. 
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*} 
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domain 'a d7 = d7a "'a d7 \<oplus> int lift"  d7b "'a \<otimes> 'a d7"  d7c (lazy "'a d7 \<rightarrow> 'a") 
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 "Indirect recursion detected, skipping proofs of (co)induction rules" 
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text {* Note that @{text d7.induct} is absent. *} 
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text {* 
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Indirect recursion is also allowed using previouslydefined datatypes. 
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*} 
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domain 'a slist = SNil  SCons 'a "'a slist" 
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domain 'a stree = STip  SBranch "'a stree slist" 

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text {* Mutuallyrecursive datatypes can be defined using the @{text "and"} keyword. *} 
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domain d8 = d8a  d8b "d9" and d9 = d9a  d9b (lazy "d8") 
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text {* Nonregular recursion is not allowed. *} 
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(* 
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domain ('a, 'b) altlist = ANil  ACons 'a "('b, 'a) altlist" 
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 "illegal direct recursion with different arguments" 
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domain 'a nest = Nest1 'a  Nest2 "'a nest nest" 
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 "illegal direct recursion with different arguments" 
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*) 
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text {* 
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Mutuallyrecursive datatypes must have all the same type arguments, 
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not necessarily in the same order. 
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*} 
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domain ('a, 'b) list1 = Nil1  Cons1 'a "('b, 'a) list2" 
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and ('b, 'a) list2 = Nil2  Cons2 'b "('a, 'b) list1" 
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text {* Induction rules for flat datatypes have no admissibility sidecondition. *} 
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domain 'a flattree = Tip  Branch "'a flattree" "'a flattree" 
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lemma "\<lbrakk>P \<bottom>; P Tip; \<And>x y. \<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>; P x; P y\<rbrakk> \<Longrightarrow> P (Branch\<cdot>x\<cdot>y)\<rbrakk> \<Longrightarrow> P x" 
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by (rule flattree.induct)  "no admissibility requirement" 
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text {* Trivial datatypes will produce a warning message. *} 
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domain triv = Triv triv triv 
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 "domain @{text Domain_ex.triv} is empty!" 
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lemma "(x::triv) = \<bottom>" by (induct x, simp_all) 
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text {* Lazy constructor arguments may have unpointed types. *} 
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domain natlist = nnil  ncons (lazy "nat discr") natlist 
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text {* Class constraints may be given for type parameters on the LHS. *} 

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domain ('a::predomain) box = Box (lazy 'a) 
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domain ('a::countable) stream = snil  scons (lazy "'a discr") "'a stream" 
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subsection {* Generated constants and theorems *} 
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domain 'a tree = Leaf (lazy 'a)  Node (left :: "'a tree") (right :: "'a tree") 
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lemmas tree_abs_bottom_iff = 
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iso.abs_bottom_iff [OF iso.intro [OF tree.abs_iso tree.rep_iso]] 
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text {* Rules about ismorphism *} 
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term tree_rep 
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term tree_abs 
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thm tree.rep_iso 
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thm tree.abs_iso 
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thm tree.iso_rews 
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text {* Rules about constructors *} 
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term Leaf 
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term Node 
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thm Leaf_def Node_def 
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thm tree.nchotomy 
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thm tree.exhaust 
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thm tree.compacts 
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thm tree.con_rews 
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thm tree.dist_les 
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thm tree.dist_eqs 
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thm tree.inverts 
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thm tree.injects 
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143 

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text {* Rules about case combinator *} 
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term tree_case 
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thm tree.tree_case_def 
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thm tree.case_rews 
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148 

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text {* Rules about selectors *} 
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term left 
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term right 
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thm tree.sel_rews 
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153 

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text {* Rules about discriminators *} 
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term is_Leaf 
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term is_Node 
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thm tree.dis_rews 
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158 

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text {* Rules about monadic pattern match combinators *} 
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term match_Leaf 
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term match_Node 
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thm tree.match_rews 
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163 

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text {* Rules about take function *} 
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term tree_take 
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thm tree.take_def 
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thm tree.take_0 
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thm tree.take_Suc 
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thm tree.take_rews 
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thm tree.chain_take 
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thm tree.take_take 
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thm tree.deflation_take 
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thm tree.take_below 
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thm tree.take_lemma 
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thm tree.lub_take 
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thm tree.reach 
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thm tree.finite_induct 
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text {* Rules about finiteness predicate *} 
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term tree_finite 
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thm tree.finite_def 
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thm tree.finite (* only generated for flat datatypes *) 
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183 

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text {* Rules about bisimulation predicate *} 
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term tree_bisim 
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thm tree.bisim_def 
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thm tree.coinduct 
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188 

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text {* Induction rule *} 
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thm tree.induct 
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191 

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192 

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subsection {* Known bugs *} 
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194 

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text {* Declaring a mixfix with spaces causes some strange parse errors. *} 
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(* 
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domain xx = xx ("x y") 
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 "Inner syntax error: unexpected end of input" 
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*) 
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end 