# HG changeset patch
# User nipkow
# Date 1322219919 -3600
# Node ID e5fd20f232412bc3ff8de7ca252d8c81edd41cda
# Parent  ef08425dd2d5c72fae6eddb72f2deb0ed48848c7# Parent  5f85efcb50c19b77925f07d36b5784dd02beb91c
merged

diff -r 5f85efcb50c1 -r e5fd20f23241 src/HOL/Quotient_Examples/Lift_RBT.thy
--- a/src/HOL/Quotient_Examples/Lift_RBT.thy	Fri Nov 25 12:18:33 2011 +0100
+++ b/src/HOL/Quotient_Examples/Lift_RBT.thy	Fri Nov 25 12:18:39 2011 +0100
@@ -53,14 +53,21 @@
 
 declare lookup_def[unfolded map_fun_def comp_def id_def, code]
 
-(* FIXME: some bug in quotient?*)
-(*
+(* FIXME: quotient_definition at the moment requires that types variables match exactly,
+i.e., sort constraints must be annotated to the constant being lifted.
+But, it should be possible to infer the necessary sort constraints, making the explicit
+sort constraints unnecessary.
+
+Hence this unannotated quotient_definition fails:
+
 quotient_definition empty where "empty :: ('a\<Colon>linorder, 'b) rbt"
 is "RBT_Impl.Empty"
+
+Similar issue for quotient_definition for entries, keys, map, and fold.
 *)
 
-definition empty :: "('a\<Colon>linorder, 'b) rbt" where
-  "empty = RBT RBT_Impl.Empty"
+quotient_definition empty where "empty :: ('a\<Colon>linorder, 'b) rbt"
+is "(RBT_Impl.Empty :: ('a\<Colon>linorder, 'b) RBT_Impl.rbt)"
 
 lemma impl_of_empty [code abstract]:
   "impl_of empty = RBT_Impl.Empty"
@@ -80,19 +87,16 @@
   "impl_of (delete k t) = RBT_Impl.delete k (impl_of t)"
   by (simp add: delete_def RBT_inverse)
 
-(* FIXME: bug in quotient? *)
-(*
-quotient_definition entries where "entries :: ('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a \<times> 'b) list"
-is "RBT_Impl.entries"
-*)
-definition entries :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a \<times> 'b) list" where
-  [code]: "entries t = RBT_Impl.entries (impl_of t)"
+(* FIXME: unnecessary type annotations *)
+quotient_definition "entries :: ('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a \<times> 'b) list"
+is "RBT_Impl.entries :: ('a\<Colon>linorder, 'b) RBT_Impl.rbt \<Rightarrow> ('a \<times> 'b) list"
 
-(* FIXME: bug in quotient? *)
-(*
-quotient_definition keys where "keys :: ('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'a list"
-is "RBT_Impl.keys"
-*)
+lemma [code]: "entries t = RBT_Impl.entries (impl_of t)"
+unfolding entries_def map_fun_def comp_def id_def ..
+
+(* FIXME: unnecessary type annotations *)
+quotient_definition "keys :: ('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'a list"
+is "RBT_Impl.keys :: ('a\<Colon>linorder, 'b) RBT_Impl.rbt \<Rightarrow> 'a list"
 
 quotient_definition "bulkload :: ('a\<Colon>linorder \<times> 'b) list \<Rightarrow> ('a, 'b) rbt"
 is "RBT_Impl.bulkload"
@@ -108,28 +112,103 @@
   "impl_of (map_entry k f t) = RBT_Impl.map_entry k f (impl_of t)"
   by (simp add: map_entry_def RBT_inverse)
 
-(* Another bug: map is supposed to be a new definition, not using the old one.
-quotient_definition "map :: ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt"
-is "RBT_Impl.map"
-*)
-(* But this works, and then shows the other bug... *)
-(*
+(* FIXME: unnecesary type annotations *)
 quotient_definition map where "map :: ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt"
-is "RBT_Impl.map"
-*)
-definition map :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
-  "map f t = RBT (RBT_Impl.map f (impl_of t))"
+is "RBT_Impl.map :: ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder, 'b) RBT_Impl.rbt \<Rightarrow> ('a, 'b) RBT_Impl.rbt"
 
 lemma impl_of_map [code abstract]:
   "impl_of (map f t) = RBT_Impl.map f (impl_of t)"
   by (simp add: map_def RBT_inverse)
-(* FIXME: bug?
-quotient_definition fold where "fold :: ('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'c \<Rightarrow> 'c" is "RBT_Impl.fold"
-*)
-(*FIXME: definition of fold should have the code attribute. *)
+
+(* FIXME: unnecessary type annotations *)
+quotient_definition fold where "fold :: ('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'c \<Rightarrow> 'c" is "RBT_Impl.fold :: ('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a\<Colon>linorder, 'b) RBT_Impl.rbt \<Rightarrow> 'c \<Rightarrow> 'c"
+
+lemma [code]: "fold f t = RBT_Impl.fold f (impl_of t)"
+unfolding fold_def map_fun_def comp_def id_def ..
+
+
+subsection {* Derived operations *}
+
+definition is_empty :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> bool" where
+  [code]: "is_empty t = (case impl_of t of RBT_Impl.Empty \<Rightarrow> True | _ \<Rightarrow> False)"
+
+
+subsection {* Abstract lookup properties *}
+
+(* TODO: obtain the following lemmas by lifting existing theorems. *)
+
+lemma lookup_RBT:
+  "is_rbt t \<Longrightarrow> lookup (RBT t) = RBT_Impl.lookup t"
+  by (simp add: lookup_def RBT_inverse)
+
+lemma lookup_impl_of:
+  "RBT_Impl.lookup (impl_of t) = lookup t"
+  by (simp add: lookup_def)
+
+lemma entries_impl_of:
+  "RBT_Impl.entries (impl_of t) = entries t"
+  by (simp add: entries_def)
+
+lemma keys_impl_of:
+  "RBT_Impl.keys (impl_of t) = keys t"
+  by (simp add: keys_def)
+
+lemma lookup_empty [simp]:
+  "lookup empty = Map.empty"
+  by (simp add: empty_def lookup_RBT fun_eq_iff)
+
+lemma lookup_insert [simp]:
+  "lookup (insert k v t) = (lookup t)(k \<mapsto> v)"
+  by (simp add: insert_def lookup_RBT lookup_insert lookup_impl_of)
 
-definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'c \<Rightarrow> 'c" where
-  [code]: "fold f t = RBT_Impl.fold f (impl_of t)"
+lemma lookup_delete [simp]:
+  "lookup (delete k t) = (lookup t)(k := None)"
+  by (simp add: delete_def lookup_RBT RBT_Impl.lookup_delete lookup_impl_of restrict_complement_singleton_eq)
+
+lemma map_of_entries [simp]:
+  "map_of (entries t) = lookup t"
+  by (simp add: entries_def map_of_entries lookup_impl_of)
+
+lemma entries_lookup:
+  "entries t1 = entries t2 \<longleftrightarrow> lookup t1 = lookup t2"
+  by (simp add: entries_def lookup_def entries_lookup)
+
+lemma lookup_bulkload [simp]:
+  "lookup (bulkload xs) = map_of xs"
+  by (simp add: bulkload_def lookup_RBT RBT_Impl.lookup_bulkload)
+
+lemma lookup_map_entry [simp]:
+  "lookup (map_entry k f t) = (lookup t)(k := Option.map f (lookup t k))"
+  by (simp add: map_entry_def lookup_RBT RBT_Impl.lookup_map_entry lookup_impl_of)
+
+lemma lookup_map [simp]:
+  "lookup (map f t) k = Option.map (f k) (lookup t k)"
+  by (simp add: map_def lookup_RBT RBT_Impl.lookup_map lookup_impl_of)
+
+lemma fold_fold:
+  "fold f t = More_List.fold (prod_case f) (entries t)"
+  by (simp add: fold_def fun_eq_iff RBT_Impl.fold_def entries_impl_of)
+
+lemma is_empty_empty [simp]:
+  "is_empty t \<longleftrightarrow> t = empty"
+  by (simp add: rbt_eq_iff is_empty_def impl_of_empty split: rbt.split)
+
+lemma RBT_lookup_empty [simp]: (*FIXME*)
+  "RBT_Impl.lookup t = Map.empty \<longleftrightarrow> t = RBT_Impl.Empty"
+  by (cases t) (auto simp add: fun_eq_iff)
+
+lemma lookup_empty_empty [simp]:
+  "lookup t = Map.empty \<longleftrightarrow> t = empty"
+  by (cases t) (simp add: empty_def lookup_def RBT_inject RBT_inverse)
+
+lemma sorted_keys [iff]:
+  "sorted (keys t)"
+  by (simp add: keys_def RBT_Impl.keys_def sorted_entries)
+
+lemma distinct_keys [iff]:
+  "distinct (keys t)"
+  by (simp add: keys_def RBT_Impl.keys_def distinct_entries)
+
 
 
 end
\ No newline at end of file