--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/HOLCF/Tutorial/Domain_ex.thy Sat Nov 27 16:08:10 2010 -0800
@@ -0,0 +1,201 @@
+(* Title: HOLCF/ex/Domain_ex.thy
+ Author: Brian Huffman
+*)
+
+header {* Domain package examples *}
+
+theory Domain_ex
+imports HOLCF
+begin
+
+text {* Domain constructors are strict by default. *}
+
+domain d1 = d1a | d1b "d1" "d1"
+
+lemma "d1b\<cdot>\<bottom>\<cdot>y = \<bottom>" by simp
+
+text {* Constructors can be made lazy using the @{text "lazy"} keyword. *}
+
+domain d2 = d2a | d2b (lazy "d2")
+
+lemma "d2b\<cdot>x \<noteq> \<bottom>" by simp
+
+text {* Strict and lazy arguments may be mixed arbitrarily. *}
+
+domain d3 = d3a | d3b (lazy "d2") "d2"
+
+lemma "P (d3b\<cdot>x\<cdot>y = \<bottom>) \<longleftrightarrow> P (y = \<bottom>)" by simp
+
+text {* Selectors can be used with strict or lazy constructor arguments. *}
+
+domain d4 = d4a | d4b (lazy d4b_left :: "d2") (d4b_right :: "d2")
+
+lemma "y \<noteq> \<bottom> \<Longrightarrow> d4b_left\<cdot>(d4b\<cdot>x\<cdot>y) = x" by simp
+
+text {* Mixfix declarations can be given for data constructors. *}
+
+domain d5 = d5a | d5b (lazy "d5") "d5" (infixl ":#:" 70)
+
+lemma "d5a \<noteq> x :#: y :#: z" by simp
+
+text {* Mixfix declarations can also be given for type constructors. *}
+
+domain ('a, 'b) lazypair (infixl ":*:" 25) =
+ lpair (lazy lfst :: 'a) (lazy lsnd :: 'b) (infixl ":*:" 75)
+
+lemma "\<forall>p::('a :*: 'b). p \<sqsubseteq> lfst\<cdot>p :*: lsnd\<cdot>p"
+by (rule allI, case_tac p, simp_all)
+
+text {* Non-recursive constructor arguments can have arbitrary types. *}
+
+domain ('a, 'b) d6 = d6 "int lift" "'a \<oplus> 'b u" (lazy "('a :*: 'b) \<times> ('b \<rightarrow> 'a)")
+
+text {*
+ Indirect recusion is allowed for sums, products, lifting, and the
+ continuous function space. However, the domain package does not
+ generate an induction rule in terms of the constructors.
+*}
+
+domain 'a d7 = d7a "'a d7 \<oplus> int lift" | d7b "'a \<otimes> 'a d7" | d7c (lazy "'a d7 \<rightarrow> 'a")
+ -- "Indirect recursion detected, skipping proofs of (co)induction rules"
+
+text {* Note that @{text d7.induct} is absent. *}
+
+text {*
+ Indirect recursion is also allowed using previously-defined datatypes.
+*}
+
+domain 'a slist = SNil | SCons 'a "'a slist"
+
+domain 'a stree = STip | SBranch "'a stree slist"
+
+text {* Mutually-recursive datatypes can be defined using the @{text "and"} keyword. *}
+
+domain d8 = d8a | d8b "d9" and d9 = d9a | d9b (lazy "d8")
+
+text {* Non-regular recursion is not allowed. *}
+(*
+domain ('a, 'b) altlist = ANil | ACons 'a "('b, 'a) altlist"
+ -- "illegal direct recursion with different arguments"
+domain 'a nest = Nest1 'a | Nest2 "'a nest nest"
+ -- "illegal direct recursion with different arguments"
+*)
+
+text {*
+ Mutually-recursive datatypes must have all the same type arguments,
+ not necessarily in the same order.
+*}
+
+domain ('a, 'b) list1 = Nil1 | Cons1 'a "('b, 'a) list2"
+ and ('b, 'a) list2 = Nil2 | Cons2 'b "('a, 'b) list1"
+
+text {* Induction rules for flat datatypes have no admissibility side-condition. *}
+
+domain 'a flattree = Tip | Branch "'a flattree" "'a flattree"
+
+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"
+by (rule flattree.induct) -- "no admissibility requirement"
+
+text {* Trivial datatypes will produce a warning message. *}
+
+domain triv = Triv triv triv
+ -- "domain @{text Domain_ex.triv} is empty!"
+
+lemma "(x::triv) = \<bottom>" by (induct x, simp_all)
+
+text {* Lazy constructor arguments may have unpointed types. *}
+
+domain natlist = nnil | ncons (lazy "nat discr") natlist
+
+text {* Class constraints may be given for type parameters on the LHS. *}
+
+domain ('a::predomain) box = Box (lazy 'a)
+
+domain ('a::countable) stream = snil | scons (lazy "'a discr") "'a stream"
+
+
+subsection {* Generated constants and theorems *}
+
+domain 'a tree = Leaf (lazy 'a) | Node (left :: "'a tree") (right :: "'a tree")
+
+lemmas tree_abs_bottom_iff =
+ iso.abs_bottom_iff [OF iso.intro [OF tree.abs_iso tree.rep_iso]]
+
+text {* Rules about ismorphism *}
+term tree_rep
+term tree_abs
+thm tree.rep_iso
+thm tree.abs_iso
+thm tree.iso_rews
+
+text {* Rules about constructors *}
+term Leaf
+term Node
+thm Leaf_def Node_def
+thm tree.nchotomy
+thm tree.exhaust
+thm tree.compacts
+thm tree.con_rews
+thm tree.dist_les
+thm tree.dist_eqs
+thm tree.inverts
+thm tree.injects
+
+text {* Rules about case combinator *}
+term tree_case
+thm tree.tree_case_def
+thm tree.case_rews
+
+text {* Rules about selectors *}
+term left
+term right
+thm tree.sel_rews
+
+text {* Rules about discriminators *}
+term is_Leaf
+term is_Node
+thm tree.dis_rews
+
+text {* Rules about monadic pattern match combinators *}
+term match_Leaf
+term match_Node
+thm tree.match_rews
+
+text {* Rules about take function *}
+term tree_take
+thm tree.take_def
+thm tree.take_0
+thm tree.take_Suc
+thm tree.take_rews
+thm tree.chain_take
+thm tree.take_take
+thm tree.deflation_take
+thm tree.take_below
+thm tree.take_lemma
+thm tree.lub_take
+thm tree.reach
+thm tree.finite_induct
+
+text {* Rules about finiteness predicate *}
+term tree_finite
+thm tree.finite_def
+thm tree.finite (* only generated for flat datatypes *)
+
+text {* Rules about bisimulation predicate *}
+term tree_bisim
+thm tree.bisim_def
+thm tree.coinduct
+
+text {* Induction rule *}
+thm tree.induct
+
+
+subsection {* Known bugs *}
+
+text {* Declaring a mixfix with spaces causes some strange parse errors. *}
+(*
+domain xx = xx ("x y")
+ -- "Inner syntax error: unexpected end of input"
+*)
+
+end