# HG changeset patch
# User huffman
# Date 1258155080 28800
# Node ID 3317128796660c3573968b9d28325dc211ddc079
# Parent  2a2014cbb2a6aa32599562fe51f0a16e47a2c50a
automate definition of representable domains from algebraic deflations

diff -r 2a2014cbb2a6 -r 331712879666 src/HOLCF/Representable.thy
--- a/src/HOLCF/Representable.thy	Fri Nov 13 15:29:48 2009 -0800
+++ b/src/HOLCF/Representable.thy	Fri Nov 13 15:31:20 2009 -0800
@@ -6,6 +6,7 @@
 
 theory Representable
 imports Algebraic Universal Ssum Sprod One ConvexPD
+uses ("Tools/repdef.ML")
 begin
 
 subsection {* Class of representable types *}
@@ -174,16 +175,21 @@
 setup {* Sign.add_const_constraint
   (@{const_name prj}, SOME @{typ "udom \<rightarrow> 'a::pcpo"}) *}
 
+definition
+  repdef_approx ::
+    "('a::pcpo \<Rightarrow> udom) \<Rightarrow> (udom \<Rightarrow> 'a) \<Rightarrow> udom alg_defl \<Rightarrow> nat \<Rightarrow> 'a \<rightarrow> 'a"
+where
+  "repdef_approx Rep Abs t = (\<lambda>i. \<Lambda> x. Abs (cast\<cdot>(approx i\<cdot>t)\<cdot>(Rep x)))"
+
 lemma typedef_rep_class:
   fixes Rep :: "'a::pcpo \<Rightarrow> udom"
   fixes Abs :: "udom \<Rightarrow> 'a::pcpo"
   fixes t :: TypeRep
   assumes type: "type_definition Rep Abs {x. x ::: t}"
   assumes below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
-  assumes emb: "emb = (\<Lambda> x. Rep x)"
-  assumes prj: "prj = (\<Lambda> x. Abs (cast\<cdot>t\<cdot>x))"
-  assumes approx:
-    "(approx :: nat \<Rightarrow> 'a \<rightarrow> 'a) = (\<lambda>i. prj oo cast\<cdot>(approx i\<cdot>t) oo emb)"
+  assumes emb: "emb \<equiv> (\<Lambda> x. Rep x)"
+  assumes prj: "prj \<equiv> (\<Lambda> x. Abs (cast\<cdot>t\<cdot>x))"
+  assumes approx: "(approx :: nat \<Rightarrow> 'a \<rightarrow> 'a) \<equiv> repdef_approx Rep Abs t"
   shows "OFCLASS('a, rep_class)"
 proof
   have adm: "adm (\<lambda>x. x \<in> {x. x ::: t})"
@@ -199,6 +205,19 @@
     apply (rule typedef_cont_Abs [OF type below adm])
     apply simp_all
     done
+  have cast_cast_approx:
+    "\<And>i x. cast\<cdot>t\<cdot>(cast\<cdot>(approx i\<cdot>t)\<cdot>x) = cast\<cdot>(approx i\<cdot>t)\<cdot>x"
+    apply (rule cast_fixed)
+    apply (rule subdeflationD)
+    apply (rule approx.below)
+    apply (rule cast_in_deflation)
+    done
+  have approx':
+    "approx = (\<lambda>i. \<Lambda>(x::'a). prj\<cdot>(cast\<cdot>(approx i\<cdot>t)\<cdot>(emb\<cdot>x)))"
+    unfolding approx repdef_approx_def
+    apply (subst cast_cast_approx [symmetric])
+    apply (simp add: prj_beta [symmetric] emb_beta [symmetric])
+    done
   have emb_in_deflation: "\<And>x::'a. emb\<cdot>x ::: t"
     using type_definition.Rep [OF type]
     by (simp add: emb_beta)
@@ -216,22 +235,15 @@
     apply (simp add: emb_prj cast.below)
     done
   show "chain (approx :: nat \<Rightarrow> 'a \<rightarrow> 'a)"
-    unfolding approx by simp
+    unfolding approx' by simp
   show "\<And>x::'a. (\<Squnion>i. approx i\<cdot>x) = x"
-    unfolding approx
+    unfolding approx'
     apply (simp add: lub_distribs)
     apply (subst cast_fixed [OF emb_in_deflation])
     apply (rule prj_emb)
     done
-  have cast_cast_approx:
-    "\<And>i x. cast\<cdot>t\<cdot>(cast\<cdot>(approx i\<cdot>t)\<cdot>x) = cast\<cdot>(approx i\<cdot>t)\<cdot>x"
-    apply (rule cast_fixed)
-    apply (rule subdeflationD)
-    apply (rule approx.below)
-    apply (rule cast_in_deflation)
-    done
   show "\<And>(i::nat) (x::'a). approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x"
-    unfolding approx
+    unfolding approx'
     apply simp
     apply (simp add: emb_prj)
     apply (simp add: cast_cast_approx)
@@ -239,7 +251,7 @@
   show "\<And>i::nat. finite {x::'a. approx i\<cdot>x = x}"
     apply (rule_tac B="(\<lambda>x. prj\<cdot>x) ` {x. cast\<cdot>(approx i\<cdot>t)\<cdot>x = x}"
            in finite_subset)
-    apply (clarsimp simp add: approx)
+    apply (clarsimp simp add: approx')
     apply (drule_tac f="\<lambda>x. emb\<cdot>x" in arg_cong)
     apply (rule image_eqI)
     apply (rule prj_emb [symmetric])
@@ -269,8 +281,8 @@
   fixes t :: TypeRep
   assumes type: "type_definition Rep Abs {x. x ::: t}"
   assumes below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
-  assumes emb: "emb = (\<Lambda> x. Rep x)"
-  assumes prj: "prj = (\<Lambda> x. Abs (cast\<cdot>t\<cdot>x))"
+  assumes emb: "emb \<equiv> (\<Lambda> x. Rep x)"
+  assumes prj: "prj \<equiv> (\<Lambda> x. Abs (cast\<cdot>t\<cdot>x))"
   shows "REP('a) = t"
 proof -
   have adm: "adm (\<lambda>x. x \<in> {x. x ::: t})"
@@ -303,6 +315,11 @@
     done
 qed
 
+lemma adm_mem_Collect_in_deflation: "adm (\<lambda>x. x \<in> {x. x ::: A})"
+unfolding mem_Collect_eq by (rule adm_in_deflation)
+
+use "Tools/repdef.ML"
+
 
 subsection {* Instances of class @{text rep} *}
 
diff -r 2a2014cbb2a6 -r 331712879666 src/HOLCF/Tools/repdef.ML
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOLCF/Tools/repdef.ML	Fri Nov 13 15:31:20 2009 -0800
@@ -0,0 +1,181 @@
+(*  Title:      HOLCF/Tools/repdef.ML
+    Author:     Brian Huffman
+
+Defining representable domains using algebraic deflations.
+*)
+
+signature REPDEF =
+sig
+  type rep_info =
+    { emb_def: thm, prj_def: thm, approx_def: thm, REP: thm }
+
+  val add_repdef: bool -> binding option -> binding * string list * mixfix ->
+    term -> (binding * binding) option -> theory ->
+    (Typedef.info * Pcpodef.cpo_info * Pcpodef.pcpo_info * rep_info) * theory
+
+  val repdef_cmd: (bool * binding) * (binding * string list * mixfix) * string
+    * (binding * binding) option -> theory -> theory
+end;
+
+structure Repdef :> REPDEF =
+struct
+
+(** type definitions **)
+
+type rep_info =
+  { emb_def: thm, prj_def: thm, approx_def: thm, REP: thm };
+
+(* building terms *)
+
+fun adm_const T = Const (@{const_name adm}, (T --> HOLogic.boolT) --> HOLogic.boolT);
+fun mk_adm (x, T, P) = adm_const T $ absfree (x, T, P);
+
+fun below_const T = Const (@{const_name below}, T --> T --> HOLogic.boolT);
+
+val natT = @{typ nat};
+val udomT = @{typ udom};
+fun alg_deflT T = Type (@{type_name alg_defl}, [T]);
+fun cfunT (T, U) = Type (@{type_name "->"}, [T, U]);
+fun emb_const T = Const (@{const_name emb}, cfunT (T, udomT));
+fun prj_const T = Const (@{const_name prj}, cfunT (udomT, T));
+fun approx_const T = Const (@{const_name approx}, natT --> cfunT (T, T));
+
+fun LAM_const (T, U) = Const (@{const_name Abs_CFun}, (T --> U) --> cfunT (T, U));
+fun APP_const (T, U) = Const (@{const_name Rep_CFun}, cfunT (T, U) --> (T --> U));
+fun cast_const T = Const (@{const_name cast}, cfunT (alg_deflT T, cfunT (T, T)));
+fun mk_cast (t, x) =
+  APP_const (udomT, udomT)
+  $ (APP_const (alg_deflT udomT, cfunT (udomT, udomT)) $ cast_const udomT $ t)
+  $ x;
+
+(* manipulating theorems *)
+
+(* proving class instances *)
+
+fun declare_type_name a =
+  Variable.declare_constraints (Logic.mk_type (TFree (a, dummyS)));
+
+fun gen_add_repdef
+      (prep_term: Proof.context -> 'a -> term)
+      (def: bool)
+      (name: binding)
+      (typ as (t, vs, mx) : binding * string list * mixfix)
+      (raw_defl: 'a)
+      (opt_morphs: (binding * binding) option)
+      (thy: theory)
+    : (Typedef.info * Pcpodef.cpo_info * Pcpodef.pcpo_info * rep_info) * theory =
+  let
+    val _ = Theory.requires thy "Representable" "repdefs";
+    val ctxt = ProofContext.init thy;
+
+    (*rhs*)
+    val defl = prep_term (ctxt |> fold declare_type_name vs) raw_defl;
+    val deflT = Term.fastype_of defl;
+    val _ = if deflT = @{typ "udom alg_defl"} then ()
+            else error ("Not type udom alg_defl: " ^ quote (Syntax.string_of_typ ctxt deflT));
+    val rhs_tfrees = Term.add_tfrees defl [];
+
+    (*lhs*)
+    val defS = Sign.defaultS thy;
+    val lhs_tfrees = map (fn v => (v, the_default defS (AList.lookup (op =) rhs_tfrees v))) vs;
+    val lhs_sorts = map snd lhs_tfrees;
+    val tname = Binding.map_name (Syntax.type_name mx) t;
+    val full_tname = Sign.full_name thy tname;
+    val newT = Type (full_tname, map TFree lhs_tfrees);
+
+    (*morphisms*)
+    val morphs = opt_morphs
+      |> the_default (Binding.prefix_name "Rep_" name, Binding.prefix_name "Abs_" name);
+
+    (*set*)
+    val in_defl = @{term "in_deflation :: udom => udom alg_defl => bool"};
+    val set = HOLogic.Collect_const udomT $ Abs ("x", udomT, in_defl $ Bound 0 $ defl);
+
+    (*pcpodef*)
+    val tac1 = rtac @{thm CollectI} 1 THEN rtac @{thm bottom_in_deflation} 1;
+    val tac2 = rtac @{thm adm_mem_Collect_in_deflation} 1;
+    val ((info, cpo_info, pcpo_info), thy2) = thy
+      |> Pcpodef.add_pcpodef def (SOME name) typ set (SOME morphs) (tac1, tac2);
+
+    (*definitions*)
+    val Rep_const = Const (#Rep_name info, newT --> udomT);
+    val Abs_const = Const (#Abs_name info, udomT --> newT);
+    val emb_eqn = Logic.mk_equals (emb_const newT, LAM_const (newT, udomT) $ Rep_const);
+    val prj_eqn = Logic.mk_equals (prj_const newT, LAM_const (udomT, newT) $
+      Abs ("x", udomT, Abs_const $ mk_cast (defl, Bound 0)));
+    val repdef_approx_const =
+      Const (@{const_name repdef_approx}, (newT --> udomT) --> (udomT --> newT)
+        --> alg_deflT udomT --> natT --> cfunT (newT, newT));
+    val approx_eqn = Logic.mk_equals (approx_const newT,
+      repdef_approx_const $ Rep_const $ Abs_const $ defl);
+
+    (*instantiate class rep*)
+    val name_def = Binding.suffix_name "_def" name;
+    val ([emb_ldef, prj_ldef, approx_ldef], lthy3) = thy2
+      |> Theory_Target.instantiation ([full_tname], lhs_tfrees, @{sort rep})
+      |> fold_map Specification.definition
+          [ (NONE, ((Binding.prefix_name "emb_" name_def, []), emb_eqn))
+          , (NONE, ((Binding.prefix_name "prj_" name_def, []), prj_eqn))
+          , (NONE, ((Binding.prefix_name "approx_" name_def, []), approx_eqn)) ]
+      |>> map (snd o snd);
+    val ctxt_thy = ProofContext.init (ProofContext.theory_of lthy3);
+    val [emb_def, prj_def, approx_def] =
+      ProofContext.export lthy3 ctxt_thy [emb_ldef, prj_ldef, approx_ldef];
+    val typedef_thms =
+      [#type_definition info, #below_def cpo_info, emb_def, prj_def, approx_def];
+    val thy4 = lthy3
+      |> Class.prove_instantiation_instance
+          (K (Tactic.rtac (@{thm typedef_rep_class} OF typedef_thms) 1))
+      |> LocalTheory.exit_global;
+
+    (*other theorems*)
+    val typedef_thms' = map (Thm.transfer thy4)
+      [#type_definition info, #below_def cpo_info, emb_def, prj_def];
+    val ([REP_thm], thy5) = thy4
+      |> Sign.add_path (Binding.name_of name)
+      |> PureThy.add_thms
+        [((Binding.prefix_name "REP_" name,
+          Drule.standard (@{thm typedef_REP} OF typedef_thms')), [])]
+      ||> Sign.parent_path;
+
+    val rep_info =
+      { emb_def = emb_def, prj_def = prj_def, approx_def = approx_def, REP = REP_thm };
+  in
+    ((info, cpo_info, pcpo_info, rep_info), thy5)
+  end
+  handle ERROR msg =>
+    cat_error msg ("The error(s) above occurred in repdef " ^ quote (Binding.str_of name));
+
+fun add_repdef def opt_name typ defl opt_morphs thy =
+  let
+    val name = the_default (#1 typ) opt_name;
+  in
+    gen_add_repdef Syntax.check_term def name typ defl opt_morphs thy
+  end;
+
+fun repdef_cmd ((def, name), typ, A, morphs) =
+  snd o gen_add_repdef Syntax.read_term def name typ A morphs;
+
+(** outer syntax **)
+
+local structure P = OuterParse and K = OuterKeyword in
+
+val repdef_decl =
+  Scan.optional (P.$$$ "(" |--
+      ((P.$$$ "open" >> K false) -- Scan.option P.binding || P.binding >> (fn s => (true, SOME s)))
+        --| P.$$$ ")") (true, NONE) --
+    (P.type_args -- P.binding) -- P.opt_infix -- (P.$$$ "=" |-- P.term) --
+    Scan.option (P.$$$ "morphisms" |-- P.!!! (P.binding -- P.binding));
+
+fun mk_repdef ((((((def, opt_name), (vs, t)), mx), A), morphs)) =
+  repdef_cmd
+    ((def, the_default (Binding.map_name (Syntax.type_name mx) t) opt_name), (t, vs, mx), A, morphs);
+
+val _ =
+  OuterSyntax.command "repdef" "HOLCF definition of representable domains" K.thy_goal
+    (repdef_decl >>
+      (Toplevel.print oo (Toplevel.theory o mk_repdef)));
+
+end;
+
+end;
diff -r 2a2014cbb2a6 -r 331712879666 src/HOLCF/ex/Domain_Proofs.thy
--- a/src/HOLCF/ex/Domain_Proofs.thy	Fri Nov 13 15:29:48 2009 -0800
+++ b/src/HOLCF/ex/Domain_Proofs.thy	Fri Nov 13 15:31:20 2009 -0800
@@ -94,13 +94,13 @@
 begin
 
 definition emb_foo :: "'a foo \<rightarrow> udom"
-where "emb_foo = (\<Lambda> x. Rep_foo x)"
+where "emb_foo \<equiv> (\<Lambda> x. Rep_foo x)"
 
 definition prj_foo :: "udom \<rightarrow> 'a foo"
-where "prj_foo = (\<Lambda> y. Abs_foo (cast\<cdot>(foo_typ\<cdot>REP('a))\<cdot>y))"
+where "prj_foo \<equiv> (\<Lambda> y. Abs_foo (cast\<cdot>(foo_typ\<cdot>REP('a))\<cdot>y))"
 
 definition approx_foo :: "nat \<Rightarrow> 'a foo \<rightarrow> 'a foo"
-where "approx_foo = (\<lambda>i. prj oo cast\<cdot>(approx i\<cdot>(foo_typ\<cdot>REP('a))) oo emb)"
+where "approx_foo \<equiv> repdef_approx Rep_foo Abs_foo (foo_typ\<cdot>REP('a))"
 
 instance
 apply (rule typedef_rep_class)
@@ -117,13 +117,13 @@
 begin
 
 definition emb_bar :: "'a bar \<rightarrow> udom"
-where "emb_bar = (\<Lambda> x. Rep_bar x)"
+where "emb_bar \<equiv> (\<Lambda> x. Rep_bar x)"
 
 definition prj_bar :: "udom \<rightarrow> 'a bar"
-where "prj_bar = (\<Lambda> y. Abs_bar (cast\<cdot>(bar_typ\<cdot>REP('a))\<cdot>y))"
+where "prj_bar \<equiv> (\<Lambda> y. Abs_bar (cast\<cdot>(bar_typ\<cdot>REP('a))\<cdot>y))"
 
 definition approx_bar :: "nat \<Rightarrow> 'a bar \<rightarrow> 'a bar"
-where "approx_bar = (\<lambda>i. prj oo cast\<cdot>(approx i\<cdot>(bar_typ\<cdot>REP('a))) oo emb)"
+where "approx_bar \<equiv> repdef_approx Rep_bar Abs_bar (bar_typ\<cdot>REP('a))"
 
 instance
 apply (rule typedef_rep_class)
@@ -140,13 +140,13 @@
 begin
 
 definition emb_baz :: "'a baz \<rightarrow> udom"
-where "emb_baz = (\<Lambda> x. Rep_baz x)"
+where "emb_baz \<equiv> (\<Lambda> x. Rep_baz x)"
 
 definition prj_baz :: "udom \<rightarrow> 'a baz"
-where "prj_baz = (\<Lambda> y. Abs_baz (cast\<cdot>(baz_typ\<cdot>REP('a))\<cdot>y))"
+where "prj_baz \<equiv> (\<Lambda> y. Abs_baz (cast\<cdot>(baz_typ\<cdot>REP('a))\<cdot>y))"
 
 definition approx_baz :: "nat \<Rightarrow> 'a baz \<rightarrow> 'a baz"
-where "approx_baz = (\<lambda>i. prj oo cast\<cdot>(approx i\<cdot>(baz_typ\<cdot>REP('a))) oo emb)"
+where "approx_baz \<equiv> repdef_approx Rep_baz Abs_baz (baz_typ\<cdot>REP('a))"
 
 instance
 apply (rule typedef_rep_class)