(* Title: HOL/Library/AssocList.thy
Author: Norbert Schirmer, Tobias Nipkow, Martin Wildmoser, TU Muenchen
*)
header {* Map operations implemented on association lists*}
theory AssocList
imports Main Mapping
begin
text {*
The operations preserve distinctness of keys and
function @{term "clearjunk"} distributes over them. Since
@{term clearjunk} enforces distinctness of keys it can be used
to establish the invariant, e.g. for inductive proofs.
*}
subsection {* @{text update} and @{text updates} *}
primrec update :: "'key \ 'val \ ('key \ 'val) list \ ('key \ 'val) list" where
"update k v [] = [(k, v)]"
| "update k v (p#ps) = (if fst p = k then (k, v) # ps else p # update k v ps)"
lemma update_conv': "map_of (update k v al) = (map_of al)(k\v)"
by (induct al) (auto simp add: expand_fun_eq)
corollary update_conv: "map_of (update k v al) k' = ((map_of al)(k\v)) k'"
by (simp add: update_conv')
lemma dom_update: "fst ` set (update k v al) = {k} \ fst ` set al"
by (induct al) auto
lemma update_keys:
"map fst (update k v al) =
(if k \ set (map fst al) then map fst al else map fst al @ [k])"
by (induct al) simp_all
lemma distinct_update:
assumes "distinct (map fst al)"
shows "distinct (map fst (update k v al))"
using assms by (simp add: update_keys)
lemma update_filter:
"a\k \ update k v [q\ps . fst q \ a] = [q\update k v ps . fst q \ a]"
by (induct ps) auto
lemma update_triv: "map_of al k = Some v \ update k v al = al"
by (induct al) auto
lemma update_nonempty [simp]: "update k v al \ []"
by (induct al) auto
lemma update_eqD: "update k v al = update k v' al' \ v = v'"
proof (induct al arbitrary: al')
case Nil thus ?case
by (cases al') (auto split: split_if_asm)
next
case Cons thus ?case
by (cases al') (auto split: split_if_asm)
qed
lemma update_last [simp]: "update k v (update k v' al) = update k v al"
by (induct al) auto
text {* Note that the lists are not necessarily the same:
@{term "update k v (update k' v' []) = [(k', v'), (k, v)]"} and
@{term "update k' v' (update k v []) = [(k, v), (k', v')]"}.*}
lemma update_swap: "k\k'
\ map_of (update k v (update k' v' al)) = map_of (update k' v' (update k v al))"
by (simp add: update_conv' expand_fun_eq)
lemma update_Some_unfold:
"map_of (update k v al) x = Some y \
x = k \ v = y \ x \ k \ map_of al x = Some y"
by (simp add: update_conv' map_upd_Some_unfold)
lemma image_update [simp]:
"x \ A \ map_of (update x y al) ` A = map_of al ` A"
by (simp add: update_conv' image_map_upd)
definition updates :: "'key list \ 'val list \ ('key \ 'val) list \ ('key \ 'val) list" where
"updates ks vs al = foldl (\al (k, v). update k v al) al (zip ks vs)"
lemma updates_simps [simp]:
"updates [] vs ps = ps"
"updates ks [] ps = ps"
"updates (k#ks) (v#vs) ps = updates ks vs (update k v ps)"
by (simp_all add: updates_def)
lemma updates_key_simp [simp]:
"updates (k # ks) vs ps =
(case vs of [] \ ps | v # vs \ updates ks vs (update k v ps))"
by (cases vs) simp_all
lemma updates_conv': "map_of (updates ks vs al) = (map_of al)(ks[\]vs)"
proof -
have "foldl (\f (k, v). f(k \ v)) (map_of al) (zip ks vs) =
map_of (foldl (\al (k, v). update k v al) al (zip ks vs))"
by (rule foldl_apply) (auto simp add: expand_fun_eq update_conv')
then show ?thesis
by (simp add: updates_def map_upds_fold_map_upd)
qed
lemma updates_conv: "map_of (updates ks vs al) k = ((map_of al)(ks[\]vs)) k"
by (simp add: updates_conv')
lemma distinct_updates:
assumes "distinct (map fst al)"
shows "distinct (map fst (updates ks vs al))"
proof -
from assms have "distinct (foldl
(\al (k, v). if k \ set al then al else al @ [k])
(map fst al) (zip ks vs))"
by (rule foldl_invariant) auto
moreover have "foldl (\al (k, v). if k \ set al then al else al @ [k])
(map fst al) (zip ks vs)
= map fst (foldl (\al (k, v). update k v al) al (zip ks vs))"
by (rule foldl_apply) (simp add: update_keys split_def comp_def)
ultimately show ?thesis by (simp add: updates_def)
qed
lemma updates_append1[simp]: "size ks < size vs \
updates (ks@[k]) vs al = update k (vs!size ks) (updates ks vs al)"
by (induct ks arbitrary: vs al) (auto split: list.splits)
lemma updates_list_update_drop[simp]:
"\size ks \ i; i < size vs\
\ updates ks (vs[i:=v]) al = updates ks vs al"
by (induct ks arbitrary: al vs i) (auto split:list.splits nat.splits)
lemma update_updates_conv_if: "
map_of (updates xs ys (update x y al)) =
map_of (if x \ set(take (length ys) xs) then updates xs ys al
else (update x y (updates xs ys al)))"
by (simp add: updates_conv' update_conv' map_upd_upds_conv_if)
lemma updates_twist [simp]:
"k \ set ks \
map_of (updates ks vs (update k v al)) = map_of (update k v (updates ks vs al))"
by (simp add: updates_conv' update_conv' map_upds_twist)
lemma updates_apply_notin[simp]:
"k \ set ks ==> map_of (updates ks vs al) k = map_of al k"
by (simp add: updates_conv)
lemma updates_append_drop[simp]:
"size xs = size ys \ updates (xs@zs) ys al = updates xs ys al"
by (induct xs arbitrary: ys al) (auto split: list.splits)
lemma updates_append2_drop[simp]:
"size xs = size ys \ updates xs (ys@zs) al = updates xs ys al"
by (induct xs arbitrary: ys al) (auto split: list.splits)
subsection {* @{text delete} *}
definition delete :: "'key \ ('key \ 'val) list \ ('key \ 'val) list" where
delete_eq: "delete k = filter (\(k', _). k \ k')"
lemma delete_simps [simp]:
"delete k [] = []"
"delete k (p#ps) = (if fst p = k then delete k ps else p # delete k ps)"
by (auto simp add: delete_eq)
lemma delete_conv': "map_of (delete k al) = (map_of al)(k := None)"
by (induct al) (auto simp add: expand_fun_eq)
corollary delete_conv: "map_of (delete k al) k' = ((map_of al)(k := None)) k'"
by (simp add: delete_conv')
lemma delete_keys:
"map fst (delete k al) = removeAll k (map fst al)"
by (simp add: delete_eq removeAll_filter_not_eq filter_map split_def comp_def)
lemma distinct_delete:
assumes "distinct (map fst al)"
shows "distinct (map fst (delete k al))"
using assms by (simp add: delete_keys distinct_removeAll)
lemma delete_id [simp]: "k \ fst ` set al \ delete k al = al"
by (auto simp add: image_iff delete_eq filter_id_conv)
lemma delete_idem: "delete k (delete k al) = delete k al"
by (simp add: delete_eq)
lemma map_of_delete [simp]:
"k' \ k \ map_of (delete k al) k' = map_of al k'"
by (simp add: delete_conv')
lemma delete_notin_dom: "k \ fst ` set (delete k al)"
by (auto simp add: delete_eq)
lemma dom_delete_subset: "fst ` set (delete k al) \ fst ` set al"
by (auto simp add: delete_eq)
lemma delete_update_same:
"delete k (update k v al) = delete k al"
by (induct al) simp_all
lemma delete_update:
"k \ l \ delete l (update k v al) = update k v (delete l al)"
by (induct al) simp_all
lemma delete_twist: "delete x (delete y al) = delete y (delete x al)"
by (simp add: delete_eq conj_commute)
lemma length_delete_le: "length (delete k al) \ length al"
by (simp add: delete_eq)
subsection {* @{text restrict} *}
definition restrict :: "'key set \ ('key \ 'val) list \ ('key \ 'val) list" where
restrict_eq: "restrict A = filter (\(k, v). k \ A)"
lemma restr_simps [simp]:
"restrict A [] = []"
"restrict A (p#ps) = (if fst p \ A then p # restrict A ps else restrict A ps)"
by (auto simp add: restrict_eq)
lemma restr_conv': "map_of (restrict A al) = ((map_of al)|` A)"
proof
fix k
show "map_of (restrict A al) k = ((map_of al)|` A) k"
by (induct al) (simp, cases "k \ A", auto)
qed
corollary restr_conv: "map_of (restrict A al) k = ((map_of al)|` A) k"
by (simp add: restr_conv')
lemma distinct_restr:
"distinct (map fst al) \ distinct (map fst (restrict A al))"
by (induct al) (auto simp add: restrict_eq)
lemma restr_empty [simp]:
"restrict {} al = []"
"restrict A [] = []"
by (induct al) (auto simp add: restrict_eq)
lemma restr_in [simp]: "x \ A \ map_of (restrict A al) x = map_of al x"
by (simp add: restr_conv')
lemma restr_out [simp]: "x \ A \ map_of (restrict A al) x = None"
by (simp add: restr_conv')
lemma dom_restr [simp]: "fst ` set (restrict A al) = fst ` set al \ A"
by (induct al) (auto simp add: restrict_eq)
lemma restr_upd_same [simp]: "restrict (-{x}) (update x y al) = restrict (-{x}) al"
by (induct al) (auto simp add: restrict_eq)
lemma restr_restr [simp]: "restrict A (restrict B al) = restrict (A\B) al"
by (induct al) (auto simp add: restrict_eq)
lemma restr_update[simp]:
"map_of (restrict D (update x y al)) =
map_of ((if x \ D then (update x y (restrict (D-{x}) al)) else restrict D al))"
by (simp add: restr_conv' update_conv')
lemma restr_delete [simp]:
"(delete x (restrict D al)) =
(if x \ D then restrict (D - {x}) al else restrict D al)"
apply (simp add: delete_eq restrict_eq)
apply (auto simp add: split_def)
proof -
have "\y. y \ x \ x \ y" by auto
then show "[p \ al. fst p \ D \ x \ fst p] = [p \ al. fst p \ D \ fst p \ x]"
by simp
assume "x \ D"
then have "\y. y \ D \ y \ D \ x \ y" by auto
then show "[p \ al . fst p \ D \ x \ fst p] = [p \ al . fst p \ D]"
by simp
qed
lemma update_restr:
"map_of (update x y (restrict D al)) = map_of (update x y (restrict (D-{x}) al))"
by (simp add: update_conv' restr_conv') (rule fun_upd_restrict)
lemma upate_restr_conv [simp]:
"x \ D \
map_of (update x y (restrict D al)) = map_of (update x y (restrict (D-{x}) al))"
by (simp add: update_conv' restr_conv')
lemma restr_updates [simp]: "
\ length xs = length ys; set xs \ D \
\ map_of (restrict D (updates xs ys al)) =
map_of (updates xs ys (restrict (D - set xs) al))"
by (simp add: updates_conv' restr_conv')
lemma restr_delete_twist: "(restrict A (delete a ps)) = delete a (restrict A ps)"
by (induct ps) auto
subsection {* @{text clearjunk} *}
function clearjunk :: "('key \ 'val) list \ ('key \ 'val) list" where
"clearjunk [] = []"
| "clearjunk (p#ps) = p # clearjunk (delete (fst p) ps)"
by pat_completeness auto
termination by (relation "measure length")
(simp_all add: less_Suc_eq_le length_delete_le)
lemma map_of_clearjunk:
"map_of (clearjunk al) = map_of al"
by (induct al rule: clearjunk.induct)
(simp_all add: expand_fun_eq)
lemma clearjunk_keys_set:
"set (map fst (clearjunk al)) = set (map fst al)"
by (induct al rule: clearjunk.induct)
(simp_all add: delete_keys)
lemma dom_clearjunk:
"fst ` set (clearjunk al) = fst ` set al"
using clearjunk_keys_set by simp
lemma distinct_clearjunk [simp]:
"distinct (map fst (clearjunk al))"
by (induct al rule: clearjunk.induct)
(simp_all del: set_map add: clearjunk_keys_set delete_keys)
lemma ran_clearjunk:
"ran (map_of (clearjunk al)) = ran (map_of al)"
by (simp add: map_of_clearjunk)
lemma ran_map_of:
"ran (map_of al) = snd ` set (clearjunk al)"
proof -
have "ran (map_of al) = ran (map_of (clearjunk al))"
by (simp add: ran_clearjunk)
also have "\ = snd ` set (clearjunk al)"
by (simp add: ran_distinct)
finally show ?thesis .
qed
lemma clearjunk_update:
"clearjunk (update k v al) = update k v (clearjunk al)"
by (induct al rule: clearjunk.induct)
(simp_all add: delete_update)
lemma clearjunk_updates:
"clearjunk (updates ks vs al) = updates ks vs (clearjunk al)"
proof -
have "foldl (\al (k, v). update k v al) (clearjunk al) (zip ks vs) =
clearjunk (foldl (\al (k, v). update k v al) al (zip ks vs))"
by (rule foldl_apply) (simp add: clearjunk_update expand_fun_eq split_def)
then show ?thesis by (simp add: updates_def)
qed
lemma clearjunk_delete:
"clearjunk (delete x al) = delete x (clearjunk al)"
by (induct al rule: clearjunk.induct) (auto simp add: delete_idem delete_twist)
lemma clearjunk_restrict:
"clearjunk (restrict A al) = restrict A (clearjunk al)"
by (induct al rule: clearjunk.induct) (auto simp add: restr_delete_twist)
lemma distinct_clearjunk_id [simp]:
"distinct (map fst al) \ clearjunk al = al"
by (induct al rule: clearjunk.induct) auto
lemma clearjunk_idem:
"clearjunk (clearjunk al) = clearjunk al"
by simp
lemma length_clearjunk:
"length (clearjunk al) \ length al"
proof (induct al rule: clearjunk.induct [case_names Nil Cons])
case Nil then show ?case by simp
next
case (Cons kv al)
moreover have "length (delete (fst kv) al) \ length al" by (fact length_delete_le)
ultimately have "length (clearjunk (delete (fst kv) al)) \ length al" by (rule order_trans)
then show ?case by simp
qed
lemma delete_map:
assumes "\kv. fst (f kv) = fst kv"
shows "delete k (map f ps) = map f (delete k ps)"
by (simp add: delete_eq filter_map comp_def split_def assms)
lemma clearjunk_map:
assumes "\kv. fst (f kv) = fst kv"
shows "clearjunk (map f ps) = map f (clearjunk ps)"
by (induct ps rule: clearjunk.induct [case_names Nil Cons])
(simp_all add: clearjunk_delete delete_map assms)
subsection {* @{text map_ran} *}
definition map_ran :: "('key \ 'val \ 'val) \ ('key \ 'val) list \ ('key \ 'val) list" where
"map_ran f = map (\(k, v). (k, f k v))"
lemma map_ran_simps [simp]:
"map_ran f [] = []"
"map_ran f ((k, v) # ps) = (k, f k v) # map_ran f ps"
by (simp_all add: map_ran_def)
lemma dom_map_ran:
"fst ` set (map_ran f al) = fst ` set al"
by (simp add: map_ran_def image_image split_def)
lemma map_ran_conv:
"map_of (map_ran f al) k = Option.map (f k) (map_of al k)"
by (induct al) auto
lemma distinct_map_ran:
"distinct (map fst al) \ distinct (map fst (map_ran f al))"
by (simp add: map_ran_def split_def comp_def)
lemma map_ran_filter:
"map_ran f [p\ps. fst p \ a] = [p\map_ran f ps. fst p \ a]"
by (simp add: map_ran_def filter_map split_def comp_def)
lemma clearjunk_map_ran:
"clearjunk (map_ran f al) = map_ran f (clearjunk al)"
by (simp add: map_ran_def split_def clearjunk_map)
subsection {* @{text merge} *}
definition merge :: "('key \ 'val) list \ ('key \ 'val) list \ ('key \ 'val) list" where
"merge qs ps = foldr (\(k, v). update k v) ps qs"
lemma merge_simps [simp]:
"merge qs [] = qs"
"merge qs (p#ps) = update (fst p) (snd p) (merge qs ps)"
by (simp_all add: merge_def split_def)
lemma merge_updates:
"merge qs ps = updates (rev (map fst ps)) (rev (map snd ps)) qs"
by (simp add: merge_def updates_def split_def
foldr_foldl zip_rev zip_map_fst_snd)
lemma dom_merge: "fst ` set (merge xs ys) = fst ` set xs \ fst ` set ys"
by (induct ys arbitrary: xs) (auto simp add: dom_update)
lemma distinct_merge:
assumes "distinct (map fst xs)"
shows "distinct (map fst (merge xs ys))"
using assms by (simp add: merge_updates distinct_updates)
lemma clearjunk_merge:
"clearjunk (merge xs ys) = merge (clearjunk xs) ys"
by (simp add: merge_updates clearjunk_updates)
lemma merge_conv':
"map_of (merge xs ys) = map_of xs ++ map_of ys"
proof -
have "foldl (\m (k, v). m(k \ v)) (map_of xs) (rev ys) =
map_of (foldl (\xs (k, v). update k v xs) xs (rev ys)) "
by (rule foldl_apply) (simp add: expand_fun_eq split_def update_conv')
then show ?thesis
by (simp add: merge_def map_add_map_of_foldr foldr_foldl split_def)
qed
corollary merge_conv:
"map_of (merge xs ys) k = (map_of xs ++ map_of ys) k"
by (simp add: merge_conv')
lemma merge_empty: "map_of (merge [] ys) = map_of ys"
by (simp add: merge_conv')
lemma merge_assoc[simp]: "map_of (merge m1 (merge m2 m3)) =
map_of (merge (merge m1 m2) m3)"
by (simp add: merge_conv')
lemma merge_Some_iff:
"(map_of (merge m n) k = Some x) =
(map_of n k = Some x \ map_of n k = None \ map_of m k = Some x)"
by (simp add: merge_conv' map_add_Some_iff)
lemmas merge_SomeD [dest!] = merge_Some_iff [THEN iffD1, standard]
lemma merge_find_right[simp]: "map_of n k = Some v \ map_of (merge m n) k = Some v"
by (simp add: merge_conv')
lemma merge_None [iff]:
"(map_of (merge m n) k = None) = (map_of n k = None \ map_of m k = None)"
by (simp add: merge_conv')
lemma merge_upd[simp]:
"map_of (merge m (update k v n)) = map_of (update k v (merge m n))"
by (simp add: update_conv' merge_conv')
lemma merge_updatess[simp]:
"map_of (merge m (updates xs ys n)) = map_of (updates xs ys (merge m n))"
by (simp add: updates_conv' merge_conv')
lemma merge_append: "map_of (xs@ys) = map_of (merge ys xs)"
by (simp add: merge_conv')
subsection {* @{text compose} *}
function compose :: "('key \ 'a) list \ ('a \ 'b) list \ ('key \ 'b) list" where
"compose [] ys = []"
| "compose (x#xs) ys = (case map_of ys (snd x)
of None \ compose (delete (fst x) xs) ys
| Some v \ (fst x, v) # compose xs ys)"
by pat_completeness auto
termination by (relation "measure (length \ fst)")
(simp_all add: less_Suc_eq_le length_delete_le)
lemma compose_first_None [simp]:
assumes "map_of xs k = None"
shows "map_of (compose xs ys) k = None"
using assms by (induct xs ys rule: compose.induct)
(auto split: option.splits split_if_asm)
lemma compose_conv:
shows "map_of (compose xs ys) k = (map_of ys \\<^sub>m map_of xs) k"
proof (induct xs ys rule: compose.induct)
case 1 then show ?case by simp
next
case (2 x xs ys) show ?case
proof (cases "map_of ys (snd x)")
case None with 2
have hyp: "map_of (compose (delete (fst x) xs) ys) k =
(map_of ys \\<^sub>m map_of (delete (fst x) xs)) k"
by simp
show ?thesis
proof (cases "fst x = k")
case True
from True delete_notin_dom [of k xs]
have "map_of (delete (fst x) xs) k = None"
by (simp add: map_of_eq_None_iff)
with hyp show ?thesis
using True None
by simp
next
case False
from False have "map_of (delete (fst x) xs) k = map_of xs k"
by simp
with hyp show ?thesis
using False None
by (simp add: map_comp_def)
qed
next
case (Some v)
with 2
have "map_of (compose xs ys) k = (map_of ys \\<^sub>m map_of xs) k"
by simp
with Some show ?thesis
by (auto simp add: map_comp_def)
qed
qed
lemma compose_conv':
shows "map_of (compose xs ys) = (map_of ys \\<^sub>m map_of xs)"
by (rule ext) (rule compose_conv)
lemma compose_first_Some [simp]:
assumes "map_of xs k = Some v"
shows "map_of (compose xs ys) k = map_of ys v"
using assms by (simp add: compose_conv)
lemma dom_compose: "fst ` set (compose xs ys) \ fst ` set xs"
proof (induct xs ys rule: compose.induct)
case 1 thus ?case by simp
next
case (2 x xs ys)
show ?case
proof (cases "map_of ys (snd x)")
case None
with "2.hyps"
have "fst ` set (compose (delete (fst x) xs) ys) \ fst ` set (delete (fst x) xs)"
by simp
also
have "\ \ fst ` set xs"
by (rule dom_delete_subset)
finally show ?thesis
using None
by auto
next
case (Some v)
with "2.hyps"
have "fst ` set (compose xs ys) \ fst ` set xs"
by simp
with Some show ?thesis
by auto
qed
qed
lemma distinct_compose:
assumes "distinct (map fst xs)"
shows "distinct (map fst (compose xs ys))"
using assms
proof (induct xs ys rule: compose.induct)
case 1 thus ?case by simp
next
case (2 x xs ys)
show ?case
proof (cases "map_of ys (snd x)")
case None
with 2 show ?thesis by simp
next
case (Some v)
with 2 dom_compose [of xs ys] show ?thesis
by (auto)
qed
qed
lemma compose_delete_twist: "(compose (delete k xs) ys) = delete k (compose xs ys)"
proof (induct xs ys rule: compose.induct)
case 1 thus ?case by simp
next
case (2 x xs ys)
show ?case
proof (cases "map_of ys (snd x)")
case None
with 2 have
hyp: "compose (delete k (delete (fst x) xs)) ys =
delete k (compose (delete (fst x) xs) ys)"
by simp
show ?thesis
proof (cases "fst x = k")
case True
with None hyp
show ?thesis
by (simp add: delete_idem)
next
case False
from None False hyp
show ?thesis
by (simp add: delete_twist)
qed
next
case (Some v)
with 2 have hyp: "compose (delete k xs) ys = delete k (compose xs ys)" by simp
with Some show ?thesis
by simp
qed
qed
lemma compose_clearjunk: "compose xs (clearjunk ys) = compose xs ys"
by (induct xs ys rule: compose.induct)
(auto simp add: map_of_clearjunk split: option.splits)
lemma clearjunk_compose: "clearjunk (compose xs ys) = compose (clearjunk xs) ys"
by (induct xs rule: clearjunk.induct)
(auto split: option.splits simp add: clearjunk_delete delete_idem
compose_delete_twist)
lemma compose_empty [simp]:
"compose xs [] = []"
by (induct xs) (auto simp add: compose_delete_twist)
lemma compose_Some_iff:
"(map_of (compose xs ys) k = Some v) =
(\k'. map_of xs k = Some k' \ map_of ys k' = Some v)"
by (simp add: compose_conv map_comp_Some_iff)
lemma map_comp_None_iff:
"(map_of (compose xs ys) k = None) =
(map_of xs k = None \ (\k'. map_of xs k = Some k' \ map_of ys k' = None)) "
by (simp add: compose_conv map_comp_None_iff)
subsection {* Implementation of mappings *}
definition Mapping :: "('a \ 'b) list \ ('a, 'b) mapping" where
"Mapping xs = Mapping.Mapping (map_of xs)"
code_datatype Mapping
lemma lookup_Mapping [simp, code]:
"Mapping.lookup (Mapping xs) = map_of xs"
by (simp add: Mapping_def)
lemma keys_Mapping [simp, code]:
"Mapping.keys (Mapping xs) = set (map fst xs)"
by (simp add: keys_def dom_map_of_conv_image_fst)
lemma empty_Mapping [code]:
"Mapping.empty = Mapping []"
by (rule mapping_eqI) simp
lemma is_empty_Mapping [code]:
"Mapping.is_empty (Mapping xs) \ null xs"
by (cases xs) (simp_all add: is_empty_def)
lemma update_Mapping [code]:
"Mapping.update k v (Mapping xs) = Mapping (update k v xs)"
by (rule mapping_eqI) (simp add: update_conv')
lemma delete_Mapping [code]:
"Mapping.delete k (Mapping xs) = Mapping (delete k xs)"
by (rule mapping_eqI) (simp add: delete_conv')
lemma ordered_keys_Mapping [code]:
"Mapping.ordered_keys (Mapping xs) = sort (remdups (map fst xs))"
by (simp only: ordered_keys_def keys_Mapping sorted_list_of_set_sort_remdups) simp
lemma size_Mapping [code]:
"Mapping.size (Mapping xs) = length (remdups (map fst xs))"
by (simp add: size_def length_remdups_card_conv dom_map_of_conv_image_fst)
lemma tabulate_Mapping [code]:
"Mapping.tabulate ks f = Mapping (map (\k. (k, f k)) ks)"
by (rule mapping_eqI) (simp add: map_of_map_restrict)
lemma bulkload_Mapping [code]:
"Mapping.bulkload vs = Mapping (map (\n. (n, vs ! n)) [0.. x = y" by (fact eq_equals) (*FIXME*)
end