Eine aufbereitete Darstellung der Quelle
Stufen
Anforderungen
|
Konzepte
|
Entwurf
|
Entwicklung
|
Qualitätssicherung
|
Lebenszyklus
|
Steuerung
Ziele
Untersuchung
mit Columbo
Integrität von
Datenbanken
Interaktion und
Portierbarkeit
Ergonomie der
Schnittstellen
Angebot
Produkte
Projekt
Beratung
Mittel
Analytik
Modellierung
Sprachen
Algebra
Logik
Hardware
Denken
Kreativität
Zusammenhänge
Gesellschaft
Wirtschaft
Branche
Firma
Benutzer
products
/
Sources
/
formale Sprachen
/
C
/
Gnome
/
testsuite
/
svg
/
animation
/ (
Linux Kernel
Version 6.17.9
©
) Datei vom 30.5.2026 mit Größe 470 B
Bilddatei
Trie.thy
products/Sources/formale Sprachen/Isabelle/Archive-of-Formal-Proofs/thys/Trie/Trie.thy
(* Author: Andreas Lochbihler and Tobias Nipkow *) section "Trie" theory Trie imports "HOL-Library.AList" begin (* A truly generic Trie would be parameterized by the data type for mappings in each node. For this purpose the mapping type needs to become a bnf. import "HOL-Library.Mapping" lift_definition all_mapping :: "('b \
bool) \
('a, 'b) mapping \
bool" is "\
p m. \
a b. m a = Some b \
p b" . lemma all_mapping_cong [fundef_cong]: "m = n \
(\
x y. Mapping.lookup n x = Some y \
p y = q y) \
all_mapping p m = all_mapping q n" by transfer auto lift_definition set_mapping :: "('a,'b)mapping \
'b set" is "\
f. Union(set_option ` range f)" . lift_definition rel_mapping :: "('b \
'c \
bool) \
('a,'b)mapping \
('a,'c)mapping \
bool" is "\
r. rel_fun (=) (rel_option r)" . bnf "('a,'b)mapping" map: "Mapping.map id" sets: set_mapping bd: "BNF_Cardinal_Arithmetic.csum (card_of (UNIV::'a set)) natLeq" wits: Mapping.empty rel: rel_mapping proof- case goal1 show ?case by(rule Mapping.map.id) next case goal2 show ?case by transfer (simp add: fun_eq_iff option.map_comp) next case goal3 thus ?case by transfer (auto simp: fun_eq_iff intro: option.map_cong) next case goal4 thus ?case by transfer (auto simp: fun_eq_iff) next case goal5 thus ?case by (simp add: Field_card_of Field_natLeq card_of_card_order_on csum_def) next case goal6 thus ?case by (simp add: cinfinite_csum natLeq_cinfinite) next case goal7 thus ?case apply transfer apply(rule ordLeq_transitive[OF comp_single_set_bd[where fset=set_option and gset=range, of natLeq "BNF_Cardinal_Arithmetic.csum (card_of (UNIV::'d set)) natLeq"]]) apply (rule natLeq_Card_order) apply(rename_tac opt) apply(case_tac opt) apply (simp add: card_of_empty1 natLeq_Well_order) using Card_order_singl_ordLeq Field_natLeq natLeq_Card_order apply fastforce apply(rule ordLeq_transitive[OF card_of_image]) apply (simp add: card_of_Card_order ordLeq_csum1) apply(rule cprod_cinfinite_bound) apply(rule ordLeq_refl) apply (rule Card_order_csum) apply (simp add: natLeq_Card_order ordLeq_csum2) apply (rule Card_order_csum) apply (rule natLeq_Card_order) using Cinfinite_csum natLeq_Card_order natLeq_cinfinite by blast next case goal8 thus ?case apply(rule predicate2I) apply transfer by(auto simp: option.rel_compp rel_fun_def) next case goal9 show ?case unfolding OO_Grp_alt fun_eq_iff apply transfer apply safe apply(rename_tac f g) apply(rule_tac x = "\
x. case (f x, g x) of (Some u,Some v) \
Some(u,v) | _ \
None" in exI) apply(auto simp: fun_eq_iff rel_fun_def elim!: Option.option.rel_cases split: option.splits) apply (metis option.rel_inject(2)) apply (metis rel_option_None1) apply(auto simp add: option.rel_map subset_eq intro!: rel_option_reflI split: prod.splits) done next case goal10 thus ?case by transfer simp qed *) datatype ('k, 'v) trie = Trie "'v option" "('k * ('k,'v)trie) list" lemma trie_induct [case_names Trie, induct type]: "(\
vo kvs. (\
k t. (k, t) \
set kvs \
P t) \
P (Trie vo kvs)) \
P t" by induction_schema (pat_completeness, lexicographic_order) definition empty_trie :: "('k, 'v) trie" where "empty_trie = Trie None []" fun is_empty_trie :: "('k, 'v) trie \
bool" where "is_empty_trie (Trie v m) = (v = None \
m = [])" fun lookup_trie :: "('k, 'v) trie \
'k list \
'v option" where "lookup_trie (Trie v m) [] = v" | "lookup_trie (Trie v m) (k#ks) = (case map_of m k of None \
None | Some st \
lookup_trie st ks)" fun update_with_trie :: "'k list \
('v option \
'v) \
('k, 'v) trie \
('k, 'v) trie" where "update_with_trie [] f (Trie v ps) = Trie (Some(f v)) ps" | "update_with_trie (k#ks) f (Trie v ps) = Trie v (AList.update_with_aux empty_trie k (update_with_trie ks f) ps)" text\
The function argument \
f\
of @{const update_with_trie} does not return an optional value because @{const None} could break the invariant that no empty tries are contained in a trie because @{const AList.update_with_aux} cannot recognise and remove empty tries. Therefore the delete function is implemented separately rather than via @{const update_with_trie}. Do not use @{const update_with_trie} if most of the calls do not change the entry (because of the garbage this creates); use @{const lookup_trie} possibly followed by \
update_trie\
. This shortcoming could be addressed if \
f\
indicated that the entry is unchanged, eg by @{const None}.\
definition update_trie :: "'k list \
'v \
('k, 'v) trie \
('k, 'v) trie" where "update_trie ks v = update_with_trie ks (%_. v)" lemma update_trie_induct: "\
\
v ps. P [] (Trie v ps); \
k ks v ps. (\
x. P ks x) \
P (k#ks) (Trie v ps)\
\
P xs t" by induction_schema (pat_completeness, lexicographic_order) lemma update_trie_Nil[simp]: "update_trie [] v (Trie vo ts) = Trie (Some v) ts" by(simp add: update_trie_def) lemma update_trie_Cons[simp]: "update_trie (k#ks) v (Trie vo ts) = Trie vo (AList.update_with_aux (Trie None []) k (update_trie ks v) ts)" by(simp add: update_trie_def empty_trie_def) (* A simple implementation of delete; does not shrink the trie! definition delete_trie :: "'k list \
('k, 'v) trie \
('k, 'v) trie" where "delete_trie ks t = update_with_trie ks (\
_. None) t" *) fun delete_trie :: "'key list \
('key, 'val) trie \
('key, 'val) trie" where "delete_trie [] (Trie vo ts) = Trie None ts" | "delete_trie (k#ks) (Trie vo ts) = (case map_of ts k of None \
Trie vo ts | Some t \
let t' = delete_trie ks t in if is_empty_trie t' then Trie vo (AList.delete_aux k ts) else Trie vo (AList.update k t' ts))" fun all_trie :: "('v \
bool) \
('k, 'v) trie \
bool" where "all_trie p (Trie v ps) = ((case v of None \
True | Some v \
p v) \
(\
(k,t) \
set ps. all_trie p t))" fun invar_trie :: "('key, 'val) trie \
bool" where "invar_trie (Trie vo kts) = (distinct (map fst kts) \
(\
(k, t) \
set kts. \
is_empty_trie t \
invar_trie t))" subsection \
Empty trie\
lemma invar_empty [simp]: "invar_trie empty_trie" by(simp add: empty_trie_def) lemma is_empty_conv: "is_empty_trie ts \
ts = Trie None []" by(cases ts)(simp) subsection \
@{const lookup_trie}\
lemma lookup_empty [simp]: "lookup_trie empty_trie = Map.empty" proof fix ks show "lookup_trie empty_trie ks = Map.empty ks" by(cases ks)(auto simp add: empty_trie_def) qed lemma lookup_empty' [simp]: "lookup_trie (Trie None []) ks = None" by(simp add: lookup_empty[unfolded empty_trie_def]) lemma lookup_update: "lookup_trie (update_trie ks v t) ks' = (if ks = ks' then Some v else lookup_trie t ks')" proof(induct ks t arbitrary: ks' rule: update_trie_induct) case (1 vo ts ks') show ?case by(fastforce simp add: neq_Nil_conv dest: not_sym) next case (2 k ks vo ts ks') show ?case by(cases ks')(auto simp add: map_of_update_with_aux 2 split: option.split) qed lemma lookup_update': "lookup_trie (update_trie ks v t) = (lookup_trie t)(ks \
v)" by(rule ext)(simp add: lookup_update) lemma lookup_eq_Some_iff: assumes invar: "invar_trie ((Trie vo kvs) :: ('key, 'val) trie)" shows "lookup_trie (Trie vo kvs) ks = Some v \
(ks = [] \
vo = Some v) \
(\
k t ks'. ks = k # ks' \
(k, t) \
set kvs \
lookup_trie t ks' = Some v)" proof (cases ks) case Nil thus ?thesis by simp next case (Cons k ks') note ks_eq[simp] = Cons show ?thesis proof (cases "map_of kvs k") case None thus ?thesis apply (simp) apply (auto simp add: map_of_eq_None_iff image_iff Ball_def) done next case (Some t') note map_eq = this from invar have dist_kvs: "distinct (map fst kvs)" by simp from map_of_eq_Some_iff[OF dist_kvs, of k] map_eq show ?thesis by simp metis qed qed lemma lookup_eq_None_iff: assumes invar: "invar_trie ((Trie vo kvs) :: ('key, 'val) trie)" shows "lookup_trie (Trie vo kvs) ks = None \
(ks = [] \
vo = None) \
(\
k ks'. ks = k # ks' \
(\
t. (k, t) \
set kvs \
lookup_trie t ks' = None))" using lookup_eq_Some_iff[of vo kvs ks, OF invar] apply (cases ks) apply auto[] apply (auto split: option.split) apply (metis option.simps(3) weak_map_of_SomeI) apply (metis option.exhaust) apply (metis option.exhaust) done lemma update_not_empty: "\
is_empty_trie (update_trie ks v t)" apply(cases t) apply(rename_tac kvs) apply(cases ks) apply(case_tac [2] kvs) apply (auto) done lemma invar_trie_update: "invar_trie t \
invar_trie (update_trie ks v t)" by(induct ks t rule: update_trie_induct)(auto simp add: set_update_with_aux update_not_empty split: option.splits) lemma is_empty_lookup_empty: "invar_trie t \
is_empty_trie t \
lookup_trie t = Map.empty" proof(induct t) case (Trie vo kvs) thus ?case apply(cases kvs) apply(auto simp add: fun_eq_iff elim: allE[where x="[]"]) apply(erule meta_allE)+ apply(erule meta_impE) apply(rule disjI1) apply(fastforce intro: exI[where x="a # b" for a b])+ done qed lemma lookup_update_with_trie: "lookup_trie (update_with_trie ks f t) ks' = (if ks' = ks then Some(f(lookup_trie t ks')) else lookup_trie t ks')" proof(induction ks t arbitrary: ks' rule: update_trie_induct) case 1 thus ?case by(auto simp add: neq_Nil_conv) next have *: "\
xs y ys. (xs \
y#ys) = (xs = [] \
(\
x zs. xs = x#zs \
(x \
y \
zs \
ys)))" by auto (metis neq_Nil_conv) case 2 thus ?case by(auto simp: * map_of_update_with_aux split: option.split) qed subsection \
@{const delete_trie}\
lemma delete_eq_empty_lookup_other_fail: "\
delete_trie ks t = Trie None []; ks' \
ks \
\
lookup_trie t ks' = None" proof(induct ks t arbitrary: ks' rule: delete_trie.induct) case 1 thus ?case by(auto simp add: neq_Nil_conv) next case (2 k ks vo ts) show ?case proof(cases "map_of ts k") case (Some t) show ?thesis proof(cases ks') case (Cons k' ks'') show ?thesis proof(cases "k' = k") case False from Some Cons "2.prems"(1) have "AList.delete_aux k ts = []" by(clarsimp simp add: Let_def split: if_split_asm) with False have "map_of ts k' = None" by(cases "map_of ts k'")(auto dest: map_of_SomeD simp add: delete_aux_eq_Nil_conv) thus ?thesis using False Some Cons "2.prems"(1) by simp next case True with Some "2.prems" Cons show ?thesis by(clarsimp simp add: "2.hyps" Let_def is_empty_conv split: if_split_asm) qed qed(insert Some "2.prems"(1), simp add: Let_def split: if_split_asm) next case None thus ?thesis using "2.prems"(1) by simp qed qed lemma lookup_delete: "invar_trie t \
lookup_trie (delete_trie ks t) ks' = (if ks = ks' then None else lookup_trie t ks')" proof(induct ks t arbitrary: ks' rule: delete_trie.induct) case 1 show ?case by(fastforce dest: not_sym simp add: neq_Nil_conv) next case (2 k ks vo ts) show ?case proof(cases ks') case Nil thus ?thesis by(simp split: option.split add: Let_def) next case [simp]: (Cons k' ks'') show ?thesis proof(cases "k' = k") case False thus ?thesis using "2.prems" by(auto simp add: Let_def update_conv' map_of_delete_aux split: option.split) next case [simp]: True show ?thesis proof(cases "map_of ts k") case None thus ?thesis by simp next case (Some t) thus ?thesis proof(cases "is_empty_trie (delete_trie ks t)") case True with Some "2.prems" show ?thesis by(auto simp add: map_of_delete_aux is_empty_conv dest: delete_eq_empty_lookup_other_fail) next case False thus ?thesis using Some 2 by(auto simp add: update_conv') qed qed qed qed qed lemma lookup_delete': "invar_trie t \
lookup_trie (delete_trie ks t) = (lookup_trie t)(ks := None)" by(rule ext)(simp add: lookup_delete) lemma invar_trie_delete: "invar_trie t \
invar_trie (delete_trie ks t)" proof(induct ks t rule: delete_trie.induct) case 1 thus ?case by simp next case (2 k ks vo ts) show ?case proof(cases "map_of ts k") case None thus ?thesis using "2.prems" by simp next case (Some t) with "2.prems" have "invar_trie t" by auto with Some have "invar_trie (delete_trie ks t)" by(rule 2) from "2.prems" Some have distinct: "distinct (map fst ts)" "\
is_empty_trie t" by auto show ?thesis proof(cases "is_empty_trie (delete_trie ks t)") case True { fix k' t' assume k't': "(k', t') \
set (AList.delete_aux k ts)" with distinct have "map_of (AList.delete_aux k ts) k' = Some t'" by simp hence "map_of ts k' = Some t'" using distinct by (auto simp del: map_of_eq_Some_iff simp add: map_of_delete_aux split: if_split_asm) with "2.prems" have "\
is_empty_trie t' \
invar_trie t'" by auto } with "2.prems" have "invar_trie (Trie vo (AList.delete_aux k ts))" by auto thus ?thesis using True Some by(simp) next case False { fix k' t' assume k't':"(k', t') \
set (AList.update k (delete_trie ks t) ts)" hence "map_of (AList.update k (delete_trie ks t) ts) k' = Some t'" using "2.prems" by(auto simp add: distinct_update) hence eq: "((map_of ts)(k \
delete_trie ks t)) k' = Some t'" unfolding update_conv . have "\
is_empty_trie t' \
invar_trie t'" proof(cases "k' = k") case True with eq have "t' = delete_trie ks t" by simp with \
invar_trie (delete_trie ks t)\
False show ?thesis by simp next case False with eq distinct have "(k', t') \
set ts" by simp with "2.prems" show ?thesis by auto qed } thus ?thesis using Some "2.prems" False by(auto simp add: distinct_update) qed qed qed subsection \
@{const update_with_trie}\
(* FIXME mv *) lemma nonempty_update_with_aux: "AList.update_with_aux v k f ps \
[]" by (induction ps) auto lemma nonempty_update_with_trie: "\
is_empty_trie (update_with_trie ks f t)" by(induction ks t rule: update_trie_induct) (auto simp: nonempty_update_with_aux) lemma invar_update_with_trie: "invar_trie t \
invar_trie (update_with_trie ks f t)" by(induction ks f t rule: update_with_trie.induct) (auto simp: set_update_with_aux nonempty_update_with_trie split: option.split prod.splits) subsection \
Domain of a trie\
lemma dom_lookup: "dom (lookup_trie (Trie vo kts)) = (\
k\
dom (map_of kts). Cons k ` dom (lookup_trie (the (map_of kts k)))) \
(if vo = None then {} else {[]})" unfolding dom_def apply(rule sym) apply(safe) apply simp apply(clarsimp simp add: if_split_asm) apply(case_tac x) apply(auto split: option.split_asm) done lemma finite_dom_lookup: "finite (dom (lookup_trie t))" proof(induct t) case (Trie vo kts) have "finite (\
k\
dom (map_of kts). Cons k ` dom (lookup_trie (the (map_of kts k))))" proof(rule finite_UN_I) show "finite (dom (map_of kts))" by(rule finite_dom_map_of) next fix k assume "k \
dom (map_of kts)" then obtain v where "(k, v) \
set kts" "map_of kts k = Some v" by(auto dest: map_of_SomeD) hence "finite (dom (lookup_trie (the (map_of kts k))))" by simp(rule Trie) thus "finite (Cons k ` dom (lookup_trie (the (map_of kts k))))" by(rule finite_imageI) qed thus ?case by(simp add: dom_lookup) qed lemma dom_lookup_empty_conv: "invar_trie t \
dom (lookup_trie t) = {} \
is_empty_trie t" proof(induct t) case (Trie vo kvs) show ?case proof assume dom: "dom (lookup_trie (Trie vo kvs)) = {}" have "vo = None" proof(cases vo) case (Some v) hence "[] \
dom (lookup_trie (Trie vo kvs))" by auto with dom have False by simp thus ?thesis .. qed moreover have "kvs = []" proof(cases kvs) case (Cons kt kvs') with \
invar_trie (Trie vo kvs)\
have "\
is_empty_trie (snd kt)" "invar_trie (snd kt)" by auto from Cons have "(fst kt, snd kt) \
set kvs" by simp hence "dom (lookup_trie (snd kt)) = {} \
is_empty_trie (snd kt)" using \
invar_trie (snd kt)\
by(rule Trie) with \
\
is_empty_trie (snd kt)\
have "dom (lookup_trie (snd kt)) \
{}" by simp with dom Cons have False by(auto simp add: dom_lookup) thus ?thesis .. qed ultimately show "is_empty_trie (Trie vo kvs)" by simp next assume "is_empty_trie (Trie vo kvs)" thus "dom (lookup_trie (Trie vo kvs)) = {}" by(simp add: lookup_empty[unfolded empty_trie_def]) qed qed subsection \
Range of a trie\
lemma ran_lookup_Trie: "invar_trie (Trie vo ps) \
ran (lookup_trie (Trie vo ps)) = (case vo of None \
{} | Some v \
{v}) \
(UN (k,t) : set ps. ran(lookup_trie t))" by(auto simp add: ran_def lookup_eq_Some_iff split: prod.splits option.split) lemma all_trie_eq_ran: "invar_trie t \
all_trie P t = (\
x \
ran(lookup_trie t). P x)" by(induction P t rule: all_trie.induct) (auto simp add: ran_lookup_Trie split: prod.splits option.split) end
2026-10-10
Neuigkeiten
Aktuelles
Motto des Tages
Open Source Software
Quellcodebibliothek
Eigene Quellcodes
Fremde Quellcodes
Suchen
Jenseits des Üblichen ....
Besucherstatistik
Statistik
#Sources=1126864
#Domains=1897691
Impressum
|
Ethik und Gesetz
|
Haftungsausschluß
|
Download des
|
zur Elbe Produktseite wechseln
|
©
2026 JDD
|
