--- a/src/HOLCF/Cont.thy Sat Nov 27 00:00:54 2010 +0100
+++ b/src/HOLCF/Cont.thy Fri Nov 26 15:49:59 2010 -0800
@@ -97,22 +97,26 @@
done
lemma contI2:
+ fixes f :: "'a::cpo \<Rightarrow> 'b::cpo"
assumes mono: "monofun f"
assumes below: "\<And>Y. \<lbrakk>chain Y; chain (\<lambda>i. f (Y i))\<rbrakk>
\<Longrightarrow> f (\<Squnion>i. Y i) \<sqsubseteq> (\<Squnion>i. f (Y i))"
shows "cont f"
-apply (rule contI)
-apply (rule thelubE)
-apply (erule ch2ch_monofun [OF mono])
-apply (rule below_antisym)
-apply (rule is_lub_thelub)
-apply (erule ch2ch_monofun [OF mono])
-apply (rule ub2ub_monofun [OF mono])
-apply (rule is_lubD1)
-apply (erule cpo_lubI)
-apply (rule below, assumption)
-apply (erule ch2ch_monofun [OF mono])
-done
+proof (rule contI)
+ fix Y :: "nat \<Rightarrow> 'a"
+ assume Y: "chain Y"
+ with mono have fY: "chain (\<lambda>i. f (Y i))"
+ by (rule ch2ch_monofun)
+ have "(\<Squnion>i. f (Y i)) = f (\<Squnion>i. Y i)"
+ apply (rule below_antisym)
+ apply (rule lub_below [OF fY])
+ apply (rule monofunE [OF mono])
+ apply (rule is_ub_thelub [OF Y])
+ apply (rule below [OF Y fY])
+ done
+ with fY show "range (\<lambda>i. f (Y i)) <<| f (\<Squnion>i. Y i)"
+ by (rule thelubE)
+qed
subsection {* Collection of continuity rules *}
--- a/src/HOLCF/Domain.thy Sat Nov 27 00:00:54 2010 +0100
+++ b/src/HOLCF/Domain.thy Fri Nov 26 15:49:59 2010 -0800
@@ -340,13 +340,13 @@
deflation_sprod_map deflation_cprod_map deflation_u_map
setup {*
- fold Domain_Take_Proofs.add_map_function
- [(@{type_name cfun}, @{const_name cfun_map}, [true, true]),
- (@{type_name "sfun"}, @{const_name sfun_map}, [true, true]),
- (@{type_name ssum}, @{const_name ssum_map}, [true, true]),
- (@{type_name sprod}, @{const_name sprod_map}, [true, true]),
- (@{type_name prod}, @{const_name cprod_map}, [true, true]),
- (@{type_name "u"}, @{const_name u_map}, [true])]
+ fold Domain_Take_Proofs.add_rec_type
+ [(@{type_name cfun}, [true, true]),
+ (@{type_name "sfun"}, [true, true]),
+ (@{type_name ssum}, [true, true]),
+ (@{type_name sprod}, [true, true]),
+ (@{type_name prod}, [true, true]),
+ (@{type_name "u"}, [true])]
*}
end
--- a/src/HOLCF/Fixrec.thy Sat Nov 27 00:00:54 2010 +0100
+++ b/src/HOLCF/Fixrec.thy Fri Nov 26 15:49:59 2010 -0800
@@ -26,10 +26,6 @@
succeed :: "'a \<rightarrow> 'a match" where
"succeed = (\<Lambda> x. Abs_match (sinr\<cdot>(up\<cdot>x)))"
-definition
- match_case :: "'b \<rightarrow> ('a \<rightarrow> 'b) \<rightarrow> 'a match \<rightarrow> 'b::pcpo" where
- "match_case = (\<Lambda> f r m. sscase\<cdot>(\<Lambda> x. f)\<cdot>(fup\<cdot>r)\<cdot>(Rep_match m))"
-
lemma matchE [case_names bottom fail succeed, cases type: match]:
"\<lbrakk>p = \<bottom> \<Longrightarrow> Q; p = fail \<Longrightarrow> Q; \<And>x. p = succeed\<cdot>x \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
unfolding fail_def succeed_def
@@ -52,40 +48,31 @@
"succeed\<cdot>x \<noteq> fail" "fail \<noteq> succeed\<cdot>x"
by (simp_all add: succeed_def fail_def cont_Abs_match Abs_match_inject)
-lemma match_case_simps [simp]:
- "match_case\<cdot>f\<cdot>r\<cdot>\<bottom> = \<bottom>"
- "match_case\<cdot>f\<cdot>r\<cdot>fail = f"
- "match_case\<cdot>f\<cdot>r\<cdot>(succeed\<cdot>x) = r\<cdot>x"
-by (simp_all add: succeed_def fail_def match_case_def cont_Rep_match
- cont2cont_LAM
- cont_Abs_match Abs_match_inverse Rep_match_strict)
-
-translations
- "case m of XCONST fail \<Rightarrow> t1 | XCONST succeed\<cdot>x \<Rightarrow> t2"
- == "CONST match_case\<cdot>t1\<cdot>(\<Lambda> x. t2)\<cdot>m"
-
subsubsection {* Run operator *}
definition
run :: "'a match \<rightarrow> 'a::pcpo" where
- "run = match_case\<cdot>\<bottom>\<cdot>ID"
+ "run = (\<Lambda> m. sscase\<cdot>\<bottom>\<cdot>(fup\<cdot>ID)\<cdot>(Rep_match m))"
text {* rewrite rules for run *}
lemma run_strict [simp]: "run\<cdot>\<bottom> = \<bottom>"
-by (simp add: run_def)
+unfolding run_def
+by (simp add: cont_Rep_match Rep_match_strict)
lemma run_fail [simp]: "run\<cdot>fail = \<bottom>"
-by (simp add: run_def)
+unfolding run_def fail_def
+by (simp add: cont_Rep_match Abs_match_inverse)
lemma run_succeed [simp]: "run\<cdot>(succeed\<cdot>x) = x"
-by (simp add: run_def)
+unfolding run_def succeed_def
+by (simp add: cont_Rep_match cont_Abs_match Abs_match_inverse)
subsubsection {* Monad plus operator *}
definition
mplus :: "'a match \<rightarrow> 'a match \<rightarrow> 'a match" where
- "mplus = (\<Lambda> m1 m2. case m1 of fail \<Rightarrow> m2 | succeed\<cdot>x \<Rightarrow> m1)"
+ "mplus = (\<Lambda> m1 m2. sscase\<cdot>(\<Lambda> _. m2)\<cdot>(\<Lambda> _. m1)\<cdot>(Rep_match m1))"
abbreviation
mplus_syn :: "['a match, 'a match] \<Rightarrow> 'a match" (infixr "+++" 65) where
@@ -93,14 +80,19 @@
text {* rewrite rules for mplus *}
+lemmas cont2cont_Rep_match = cont_Rep_match [THEN cont_compose]
+
lemma mplus_strict [simp]: "\<bottom> +++ m = \<bottom>"
-by (simp add: mplus_def)
+unfolding mplus_def
+by (simp add: cont2cont_Rep_match Rep_match_strict)
lemma mplus_fail [simp]: "fail +++ m = m"
-by (simp add: mplus_def)
+unfolding mplus_def fail_def
+by (simp add: cont2cont_Rep_match Abs_match_inverse)
lemma mplus_succeed [simp]: "succeed\<cdot>x +++ m = succeed\<cdot>x"
-by (simp add: mplus_def)
+unfolding mplus_def succeed_def
+by (simp add: cont2cont_Rep_match cont_Abs_match Abs_match_inverse)
lemma mplus_fail2 [simp]: "m +++ fail = m"
by (cases m, simp_all)
--- a/src/HOLCF/LowerPD.thy Sat Nov 27 00:00:54 2010 +0100
+++ b/src/HOLCF/LowerPD.thy Fri Nov 26 15:49:59 2010 -0800
@@ -195,7 +195,7 @@
apply (erule (1) monofun_cfun [OF monofun_cfun_arg])
done
-lemma lower_plus_below_iff:
+lemma lower_plus_below_iff [simp]:
"xs +\<flat> ys \<sqsubseteq> zs \<longleftrightarrow> xs \<sqsubseteq> zs \<and> ys \<sqsubseteq> zs"
apply safe
apply (erule below_trans [OF lower_plus_below1])
@@ -203,7 +203,7 @@
apply (erule (1) lower_plus_least)
done
-lemma lower_unit_below_plus_iff:
+lemma lower_unit_below_plus_iff [simp]:
"{x}\<flat> \<sqsubseteq> ys +\<flat> zs \<longleftrightarrow> {x}\<flat> \<sqsubseteq> ys \<or> {x}\<flat> \<sqsubseteq> zs"
apply (induct x rule: compact_basis.principal_induct, simp)
apply (induct ys rule: lower_pd.principal_induct, simp)
@@ -328,7 +328,7 @@
apply (erule lower_le_induct, safe)
apply (simp add: monofun_cfun)
apply (simp add: rev_below_trans [OF lower_plus_below1])
-apply (simp add: lower_plus_below_iff)
+apply simp
done
definition
@@ -389,14 +389,14 @@
apply default
apply (induct_tac x rule: lower_pd_induct, simp_all add: ep_pair.e_inverse)
apply (induct_tac y rule: lower_pd_induct)
-apply (simp_all add: ep_pair.e_p_below monofun_cfun)
+apply (simp_all add: ep_pair.e_p_below monofun_cfun del: lower_plus_below_iff)
done
lemma deflation_lower_map: "deflation d \<Longrightarrow> deflation (lower_map\<cdot>d)"
apply default
apply (induct_tac x rule: lower_pd_induct, simp_all add: deflation.idem)
apply (induct_tac x rule: lower_pd_induct)
-apply (simp_all add: deflation.below monofun_cfun)
+apply (simp_all add: deflation.below monofun_cfun del: lower_plus_below_iff)
done
(* FIXME: long proof! *)
--- a/src/HOLCF/Powerdomains.thy Sat Nov 27 00:00:54 2010 +0100
+++ b/src/HOLCF/Powerdomains.thy Fri Nov 26 15:49:59 2010 -0800
@@ -42,10 +42,10 @@
deflation_upper_map deflation_lower_map deflation_convex_map
setup {*
- fold Domain_Take_Proofs.add_map_function
- [(@{type_name "upper_pd"}, @{const_name upper_map}, [true]),
- (@{type_name "lower_pd"}, @{const_name lower_map}, [true]),
- (@{type_name "convex_pd"}, @{const_name convex_map}, [true])]
+ fold Domain_Take_Proofs.add_rec_type
+ [(@{type_name "upper_pd"}, [true]),
+ (@{type_name "lower_pd"}, [true]),
+ (@{type_name "convex_pd"}, [true])]
*}
end
--- a/src/HOLCF/Tools/Domain/domain.ML Sat Nov 27 00:00:54 2010 +0100
+++ b/src/HOLCF/Tools/Domain/domain.ML Fri Nov 26 15:49:59 2010 -0800
@@ -118,14 +118,14 @@
(* test for free type variables, illegal sort constraints on rhs,
non-pcpo-types and invalid use of recursive type;
replace sorts in type variables on rhs *)
- val map_tab = Domain_Take_Proofs.get_map_tab thy;
+ val rec_tab = Domain_Take_Proofs.get_rec_tab thy;
fun check_rec rec_ok (T as TFree (v,_)) =
if AList.defined (op =) sorts v then T
else error ("Free type variable " ^ quote v ^ " on rhs.")
| check_rec rec_ok (T as Type (s, Ts)) =
(case AList.lookup (op =) dtnvs' s of
NONE =>
- let val rec_ok' = rec_ok andalso Symtab.defined map_tab s;
+ let val rec_ok' = rec_ok andalso Symtab.defined rec_tab s;
in Type (s, map (check_rec rec_ok') Ts) end
| SOME typevars =>
if typevars <> Ts
--- a/src/HOLCF/Tools/Domain/domain_isomorphism.ML Sat Nov 27 00:00:54 2010 +0100
+++ b/src/HOLCF/Tools/Domain/domain_isomorphism.ML Fri Nov 26 15:49:59 2010 -0800
@@ -350,17 +350,17 @@
(Global_Theory.add_thms o map (Thm.no_attributes o apsnd Drule.zero_var_indexes))
(conjuncts deflation_map_binds deflation_map_thm);
- (* register map functions in theory data *)
+ (* register indirect recursion in theory data *)
local
- fun register_map ((dname, map_name), args) =
- Domain_Take_Proofs.add_map_function (dname, map_name, args);
+ fun register_map (dname, args) =
+ Domain_Take_Proofs.add_rec_type (dname, args);
val dnames = map (fst o dest_Type o fst) dom_eqns;
val map_names = map (fst o dest_Const) map_consts;
fun args (T, _) = case T of Type (_, Ts) => map (is_cpo thy) Ts | _ => [];
val argss = map args dom_eqns;
in
val thy =
- fold register_map (dnames ~~ map_names ~~ argss) thy;
+ fold register_map (dnames ~~ argss) thy;
end;
(* register deflation theorems *)
--- a/src/HOLCF/Tools/Domain/domain_take_proofs.ML Sat Nov 27 00:00:54 2010 +0100
+++ b/src/HOLCF/Tools/Domain/domain_take_proofs.ML Fri Nov 26 15:49:59 2010 -0800
@@ -55,8 +55,8 @@
val map_of_typ :
theory -> (typ * term) list -> typ -> term
- val add_map_function : (string * string * bool list) -> theory -> theory
- val get_map_tab : theory -> (string * bool list) Symtab.table
+ val add_rec_type : (string * bool list) -> theory -> theory
+ val get_rec_tab : theory -> (bool list) Symtab.table
val add_deflation_thm : thm -> theory -> theory
val get_deflation_thms : theory -> thm list
val map_ID_add : attribute
@@ -119,12 +119,10 @@
(******************************** theory data *********************************)
(******************************************************************************)
-structure MapData = Theory_Data
+structure Rec_Data = Theory_Data
(
- (* constant names like "foo_map" *)
- (* list indicates which type arguments correspond to map arguments *)
- (* alternatively, which type arguments allow indirect recursion *)
- type T = (string * bool list) Symtab.table;
+ (* list indicates which type arguments allow indirect recursion *)
+ type T = (bool list) Symtab.table;
val empty = Symtab.empty;
val extend = I;
fun merge data = Symtab.merge (K true) data;
@@ -142,13 +140,13 @@
val description = "theorems like foo_map$ID = ID"
);
-fun add_map_function (tname, map_name, bs) =
- MapData.map (Symtab.insert (K true) (tname, (map_name, bs)));
+fun add_rec_type (tname, bs) =
+ Rec_Data.map (Symtab.insert (K true) (tname, bs));
fun add_deflation_thm thm =
Context.theory_map (DeflMapData.add_thm thm);
-val get_map_tab = MapData.get;
+val get_rec_tab = Rec_Data.get;
fun get_deflation_thms thy = DeflMapData.get (ProofContext.init_global thy);
val map_ID_add = Map_Id_Data.add;
--- a/src/HOLCF/UpperPD.thy Sat Nov 27 00:00:54 2010 +0100
+++ b/src/HOLCF/UpperPD.thy Fri Nov 26 15:49:59 2010 -0800
@@ -193,7 +193,7 @@
apply (erule (1) monofun_cfun [OF monofun_cfun_arg])
done
-lemma upper_below_plus_iff:
+lemma upper_below_plus_iff [simp]:
"xs \<sqsubseteq> ys +\<sharp> zs \<longleftrightarrow> xs \<sqsubseteq> ys \<and> xs \<sqsubseteq> zs"
apply safe
apply (erule below_trans [OF _ upper_plus_below1])
@@ -201,7 +201,7 @@
apply (erule (1) upper_plus_greatest)
done
-lemma upper_plus_below_unit_iff:
+lemma upper_plus_below_unit_iff [simp]:
"xs +\<sharp> ys \<sqsubseteq> {z}\<sharp> \<longleftrightarrow> xs \<sqsubseteq> {z}\<sharp> \<or> ys \<sqsubseteq> {z}\<sharp>"
apply (induct xs rule: upper_pd.principal_induct, simp)
apply (induct ys rule: upper_pd.principal_induct, simp)
@@ -323,7 +323,7 @@
apply (erule upper_le_induct, safe)
apply (simp add: monofun_cfun)
apply (simp add: below_trans [OF upper_plus_below1])
-apply (simp add: upper_below_plus_iff)
+apply simp
done
definition
@@ -384,14 +384,14 @@
apply default
apply (induct_tac x rule: upper_pd_induct, simp_all add: ep_pair.e_inverse)
apply (induct_tac y rule: upper_pd_induct)
-apply (simp_all add: ep_pair.e_p_below monofun_cfun)
+apply (simp_all add: ep_pair.e_p_below monofun_cfun del: upper_below_plus_iff)
done
lemma deflation_upper_map: "deflation d \<Longrightarrow> deflation (upper_map\<cdot>d)"
apply default
apply (induct_tac x rule: upper_pd_induct, simp_all add: deflation.idem)
apply (induct_tac x rule: upper_pd_induct)
-apply (simp_all add: deflation.below monofun_cfun)
+apply (simp_all add: deflation.below monofun_cfun del: upper_below_plus_iff)
done
(* FIXME: long proof! *)
--- a/src/HOLCF/ex/Pattern_Match.thy Sat Nov 27 00:00:54 2010 +0100
+++ b/src/HOLCF/ex/Pattern_Match.thy Fri Nov 26 15:49:59 2010 -0800
@@ -53,11 +53,24 @@
lemmas run_fatbar_simps [simp] = run_fatbar1 run_fatbar2 run_fatbar3
+subsection {* Bind operator for match monad *}
+
+definition match_bind :: "'a match \<rightarrow> ('a \<rightarrow> 'b match) \<rightarrow> 'b match" where
+ "match_bind = (\<Lambda> m k. sscase\<cdot>(\<Lambda> _. fail)\<cdot>(fup\<cdot>k)\<cdot>(Rep_match m))"
+
+lemma match_bind_simps [simp]:
+ "match_bind\<cdot>\<bottom>\<cdot>k = \<bottom>"
+ "match_bind\<cdot>fail\<cdot>k = fail"
+ "match_bind\<cdot>(succeed\<cdot>x)\<cdot>k = k\<cdot>x"
+unfolding match_bind_def fail_def succeed_def
+by (simp_all add: cont2cont_Rep_match cont_Abs_match
+ Rep_match_strict Abs_match_inverse)
+
subsection {* Case branch combinator *}
definition
branch :: "('a \<rightarrow> 'b match) \<Rightarrow> ('b \<rightarrow> 'c) \<rightarrow> ('a \<rightarrow> 'c match)" where
- "branch p \<equiv> \<Lambda> r x. match_case\<cdot>fail\<cdot>(\<Lambda> y. succeed\<cdot>(r\<cdot>y))\<cdot>(p\<cdot>x)"
+ "branch p \<equiv> \<Lambda> r x. match_bind\<cdot>(p\<cdot>x)\<cdot>(\<Lambda> y. succeed\<cdot>(r\<cdot>y))"
lemma branch_simps:
"p\<cdot>x = \<bottom> \<Longrightarrow> branch p\<cdot>r\<cdot>x = \<bottom>"
@@ -72,7 +85,7 @@
definition
cases :: "'a match \<rightarrow> 'a::pcpo" where
- "cases = match_case\<cdot>\<bottom>\<cdot>ID"
+ "cases = Fixrec.run"
text {* rewrite rules for cases *}
@@ -165,7 +178,7 @@
definition
cpair_pat :: "('a, 'c) pat \<Rightarrow> ('b, 'd) pat \<Rightarrow> ('a \<times> 'b, 'c \<times> 'd) pat" where
"cpair_pat p1 p2 = (\<Lambda>(x, y).
- match_case\<cdot>fail\<cdot>(\<Lambda> a. match_case\<cdot>fail\<cdot>(\<Lambda> b. succeed\<cdot>(a, b))\<cdot>(p2\<cdot>y))\<cdot>(p1\<cdot>x))"
+ match_bind\<cdot>(p1\<cdot>x)\<cdot>(\<Lambda> a. match_bind\<cdot>(p2\<cdot>y)\<cdot>(\<Lambda> b. succeed\<cdot>(a, b))))"
definition
spair_pat ::
@@ -313,7 +326,7 @@
definition
as_pat :: "('a \<rightarrow> 'b match) \<Rightarrow> 'a \<rightarrow> ('a \<times> 'b) match" where
- "as_pat p = (\<Lambda> x. match_case\<cdot>fail\<cdot>(\<Lambda> a. succeed\<cdot>(x, a))\<cdot>(p\<cdot>x))"
+ "as_pat p = (\<Lambda> x. match_bind\<cdot>(p\<cdot>x)\<cdot>(\<Lambda> a. succeed\<cdot>(x, a)))"
definition
lazy_pat :: "('a \<rightarrow> 'b::pcpo match) \<Rightarrow> ('a \<rightarrow> 'b match)" where
@@ -544,7 +557,7 @@
val (fun1, fun2, taken) = pat_lhs (pat, args);
val defs = @{thm branch_def} :: pat_defs;
val goal = mk_trp (mk_strict fun1);
- val rules = @{thms match_case_simps} @ case_rews;
+ val rules = @{thms match_bind_simps} @ case_rews;
val tacs = [simp_tac (beta_ss addsimps rules) 1];
in prove thy defs goal (K tacs) end;
fun pat_apps (i, (pat, (con, args))) =
@@ -559,7 +572,7 @@
val concl = mk_trp (mk_eq (fun1 ` con_app, rhs));
val goal = Logic.list_implies (assms, concl);
val defs = @{thm branch_def} :: pat_defs;
- val rules = @{thms match_case_simps} @ case_rews;
+ val rules = @{thms match_bind_simps} @ case_rews;
val tacs = [asm_simp_tac (beta_ss addsimps rules) 1];
in prove thy defs goal (K tacs) end;
in map_index pat_app spec end;