Theory Core_DOM_DOM_Components
section ‹Core SC DOM Components›
theory Core_DOM_DOM_Components
imports Core_SC_DOM.Core_DOM
begin
subsection ‹Components›
locale l_get_dom_component⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs =
l_get_root_node_defs get_root_node get_root_node_locs +
l_to_tree_order_defs to_tree_order
for get_root_node :: "(_) object_ptr ⇒ ((_) heap, exception, (_) object_ptr) prog"
and get_root_node_locs :: "((_) heap ⇒ (_) heap ⇒ bool) set"
and to_tree_order :: "(_) object_ptr ⇒ ((_) heap, exception, (_) object_ptr list) prog"
begin
definition a_get_dom_component :: "(_) object_ptr ⇒ (_, (_) object_ptr list) dom_prog"
where
"a_get_dom_component ptr = do {
root ← get_root_node ptr;
to_tree_order root
}"
definition a_is_strongly_dom_component_safe ::
"(_) object_ptr set ⇒ (_) object_ptr set ⇒ (_) heap ⇒ (_) heap ⇒ bool"
where
"a_is_strongly_dom_component_safe S⇩a⇩r⇩g S⇩r⇩e⇩s⇩u⇩l⇩t h h' = (
let removed_pointers = fset (object_ptr_kinds h) - fset (object_ptr_kinds h') in
let added_pointers = fset (object_ptr_kinds h') - fset (object_ptr_kinds h) in
let arg_components =
(⋃ptr ∈ (⋃ptr ∈ S⇩a⇩r⇩g. set |h ⊢ a_get_dom_component ptr|⇩r) ∩
fset (object_ptr_kinds h). set |h ⊢ a_get_dom_component ptr|⇩r) in
let arg_components' =
(⋃ptr ∈ (⋃ptr ∈ S⇩a⇩r⇩g. set |h ⊢ a_get_dom_component ptr|⇩r) ∩
fset (object_ptr_kinds h'). set |h' ⊢ a_get_dom_component ptr|⇩r) in
removed_pointers ⊆ arg_components ∧
added_pointers ⊆ arg_components' ∧
S⇩r⇩e⇩s⇩u⇩l⇩t ⊆ arg_components' ∧
(∀outside_ptr ∈ fset (object_ptr_kinds h) ∩ fset (object_ptr_kinds h') -
(⋃ptr ∈ S⇩a⇩r⇩g. set |h ⊢ a_get_dom_component ptr|⇩r). preserved (get_M outside_ptr id) h h'))"
definition a_is_weakly_dom_component_safe ::
"(_) object_ptr set ⇒ (_) object_ptr set ⇒ (_) heap ⇒ (_) heap ⇒ bool"
where
"a_is_weakly_dom_component_safe S⇩a⇩r⇩g S⇩r⇩e⇩s⇩u⇩l⇩t h h' = (
let removed_pointers = fset (object_ptr_kinds h) - fset (object_ptr_kinds h') in
let added_pointers = fset (object_ptr_kinds h') - fset (object_ptr_kinds h) in
let arg_components =
(⋃ptr ∈ (⋃ptr ∈ S⇩a⇩r⇩g. set |h ⊢ a_get_dom_component ptr|⇩r) ∩
fset (object_ptr_kinds h). set |h ⊢ a_get_dom_component ptr|⇩r) in
let arg_components' =
(⋃ptr ∈ (⋃ptr ∈ S⇩a⇩r⇩g. set |h ⊢ a_get_dom_component ptr|⇩r) ∩
fset (object_ptr_kinds h'). set |h' ⊢ a_get_dom_component ptr|⇩r) in
removed_pointers ⊆ arg_components ∧
S⇩r⇩e⇩s⇩u⇩l⇩t ⊆ arg_components' ∪ added_pointers ∧
(∀outside_ptr ∈ fset (object_ptr_kinds h) ∩ fset (object_ptr_kinds h') -
(⋃ptr ∈ S⇩a⇩r⇩g. set |h ⊢ a_get_dom_component ptr|⇩r). preserved (get_M outside_ptr id) h h'))"
lemma "a_is_strongly_dom_component_safe S⇩a⇩r⇩g S⇩r⇩e⇩s⇩u⇩l⇩t h h' ⟹ a_is_weakly_dom_component_safe S⇩a⇩r⇩g S⇩r⇩e⇩s⇩u⇩l⇩t h h'"
by(auto simp add: a_is_strongly_dom_component_safe_def a_is_weakly_dom_component_safe_def Let_def)
definition is_document_component :: "(_) object_ptr list ⇒ bool"
where
"is_document_component c = is_document_ptr_kind (hd c)"
definition is_disconnected_component :: "(_) object_ptr list ⇒ bool"
where
"is_disconnected_component c = is_node_ptr_kind (hd c)"
end
global_interpretation l_get_dom_component⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs get_root_node get_root_node_locs to_tree_order
defines get_dom_component = a_get_dom_component
and is_strongly_dom_component_safe = a_is_strongly_dom_component_safe
and is_weakly_dom_component_safe = a_is_weakly_dom_component_safe
.
locale l_get_dom_component_defs =
fixes get_dom_component :: "(_) object_ptr ⇒ (_, (_) object_ptr list) dom_prog"
fixes is_strongly_dom_component_safe ::
"(_) object_ptr set ⇒ (_) object_ptr set ⇒ (_) heap ⇒ (_) heap ⇒ bool"
fixes is_weakly_dom_component_safe ::
"(_) object_ptr set ⇒ (_) object_ptr set ⇒ (_) heap ⇒ (_) heap ⇒ bool"
locale l_get_dom_component⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_to_tree_order_wf +
l_get_dom_component_defs +
l_get_dom_component⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_defs +
l_get_ancestors +
l_get_ancestors_wf +
l_get_root_node +
l_get_root_node_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_get_parent +
l_get_parent_wf +
l_get_element_by +
l_to_tree_order_wf_get_root_node_wf +
assumes get_dom_component_impl: "get_dom_component = a_get_dom_component"
assumes is_strongly_dom_component_safe_impl:
"is_strongly_dom_component_safe = a_is_strongly_dom_component_safe"
assumes is_weakly_dom_component_safe_impl:
"is_weakly_dom_component_safe = a_is_weakly_dom_component_safe"
begin
lemmas get_dom_component_def = a_get_dom_component_def[folded get_dom_component_impl]
lemmas is_strongly_dom_component_safe_def =
a_is_strongly_dom_component_safe_def[folded is_strongly_dom_component_safe_impl]
lemmas is_weakly_dom_component_safe_def =
a_is_weakly_dom_component_safe_def[folded is_weakly_dom_component_safe_impl]
lemma get_dom_component_ptr_in_heap:
assumes "h ⊢ ok (get_dom_component ptr)"
shows "ptr |∈| object_ptr_kinds h"
using assms get_root_node_ptr_in_heap
by(auto simp add: get_dom_component_def)
lemma get_dom_component_ok:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "ptr |∈| object_ptr_kinds h"
shows "h ⊢ ok (get_dom_component ptr)"
using assms
apply(auto simp add: get_dom_component_def a_get_root_node_def intro!: bind_is_OK_pure_I)[1]
using get_root_node_ok to_tree_order_ok get_root_node_ptr_in_heap
apply blast
by (simp add: local.get_root_node_root_in_heap local.to_tree_order_ok)
lemma get_dom_component_ptr:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_dom_component ptr →⇩r c"
shows "ptr ∈ set c"
proof(insert assms(1) assms(4), induct ptr rule: heap_wellformed_induct_rev )
case (step child)
then show ?case
proof (cases "is_node_ptr_kind child")
case True
obtain node_ptr where
node_ptr: "cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r node_ptr = child"
using ‹is_node_ptr_kind child› node_ptr_casts_commute3 by blast
have "child |∈| object_ptr_kinds h"
using ‹h ⊢ get_dom_component child →⇩r c› get_dom_component_ptr_in_heap by fast
with node_ptr have "node_ptr |∈| node_ptr_kinds h"
by auto
then obtain parent_opt where
parent: "h ⊢ get_parent node_ptr →⇩r parent_opt"
using get_parent_ok ‹type_wf h› ‹known_ptrs h›
by fast
then show ?thesis
proof (induct parent_opt)
case None
then have "h ⊢ get_root_node (cast node_ptr) →⇩r cast node_ptr"
by (simp add: local.get_root_node_no_parent)
then show ?case
using ‹type_wf h› ‹known_ptrs h› node_ptr step(2)
apply(auto simp add: get_dom_component_def a_get_root_node_def elim!: bind_returns_result_E2)[1]
using to_tree_order_ptr_in_result returns_result_eq by fastforce
next
case (Some parent_ptr)
then have "h ⊢ get_dom_component parent_ptr →⇩r c"
using step(2) node_ptr ‹type_wf h› ‹known_ptrs h› ‹heap_is_wellformed h›
get_root_node_parent_same
by(auto simp add: get_dom_component_def elim!: bind_returns_result_E2
intro!: bind_pure_returns_result_I)
then have "parent_ptr ∈ set c"
using step node_ptr Some by blast
then show ?case
using ‹type_wf h› ‹known_ptrs h› ‹heap_is_wellformed h› step(2) node_ptr Some
apply(auto simp add: get_dom_component_def elim!: bind_returns_result_E2)[1]
using to_tree_order_parent by blast
qed
next
case False
then show ?thesis
using ‹type_wf h› ‹known_ptrs h› step(2)
apply(auto simp add: get_dom_component_def elim!: bind_returns_result_E2)[1]
by (metis is_OK_returns_result_I local.get_root_node_not_node_same
local.get_root_node_ptr_in_heap local.to_tree_order_ptr_in_result returns_result_eq)
qed
qed
lemma get_dom_component_parent_inside:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_dom_component ptr →⇩r c"
assumes "cast node_ptr ∈ set c"
assumes "h ⊢ get_parent node_ptr →⇩r Some parent"
shows "parent ∈ set c"
proof -
have "parent |∈| object_ptr_kinds h"
using assms(6) get_parent_parent_in_heap by blast
obtain root_ptr where root_ptr: "h ⊢ get_root_node ptr →⇩r root_ptr" and c: "h ⊢ to_tree_order root_ptr →⇩r c"
using assms(4)
by (metis (no_types, hide_lams) bind_returns_result_E2 get_dom_component_def get_root_node_pure)
then have "h ⊢ get_root_node (cast node_ptr) →⇩r root_ptr"
using assms(1) assms(2) assms(3) assms(5) to_tree_order_same_root by blast
then have "h ⊢ get_root_node parent →⇩r root_ptr"
using assms(6) get_root_node_parent_same by blast
then have "h ⊢ get_dom_component parent →⇩r c"
using c get_dom_component_def by auto
then show ?thesis
using assms(1) assms(2) assms(3) get_dom_component_ptr by blast
qed
lemma get_dom_component_subset:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_dom_component ptr →⇩r c"
assumes "ptr' ∈ set c"
shows "h ⊢ get_dom_component ptr' →⇩r c"
proof(insert assms(1) assms(5), induct ptr' rule: heap_wellformed_induct_rev )
case (step child)
then show ?case
proof (cases "is_node_ptr_kind child")
case True
obtain node_ptr where
node_ptr: "cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r node_ptr = child"
using ‹is_node_ptr_kind child› node_ptr_casts_commute3 by blast
have "child |∈| object_ptr_kinds h"
using to_tree_order_ptrs_in_heap assms(1) assms(2) assms(3) assms(4) step(2)
unfolding get_dom_component_def
by (meson bind_returns_result_E2 get_root_node_pure)
with node_ptr have "node_ptr |∈| node_ptr_kinds h"
by auto
then obtain parent_opt where
parent: "h ⊢ get_parent node_ptr →⇩r parent_opt"
using get_parent_ok ‹type_wf h› ‹known_ptrs h›
by fast
then show ?thesis
proof (induct parent_opt)
case None
then have "h ⊢ get_root_node child →⇩r child"
using assms(1) get_root_node_no_parent node_ptr by blast
then show ?case
using ‹type_wf h› ‹known_ptrs h› node_ptr step(2) assms(4) assms(1)
by (metis (no_types) bind_pure_returns_result_I2 bind_returns_result_E2
get_dom_component_def get_root_node_pure is_OK_returns_result_I returns_result_eq
to_tree_order_same_root)
next
case (Some parent_ptr)
then have "h ⊢ get_dom_component parent_ptr →⇩r c"
using step get_dom_component_parent_inside assms node_ptr by blast
then show ?case
using Some node_ptr
apply(auto simp add: get_dom_component_def elim!: bind_returns_result_E2
del: bind_pure_returns_result_I intro!: bind_pure_returns_result_I)[1]
using get_root_node_parent_same by blast
qed
next
case False
then have "child |∈| object_ptr_kinds h"
using assms(1) assms(4) step(2)
by (metis (no_types, lifting) assms(2) assms(3) bind_returns_result_E2 get_root_node_pure
get_dom_component_def to_tree_order_ptrs_in_heap)
then have "h ⊢ get_root_node child →⇩r child"
using assms(1) False get_root_node_not_node_same by blast
then show ?thesis
using assms(1) assms(2) assms(3) assms(4) step.prems
by (metis (no_types) False ‹child |∈| object_ptr_kinds h› bind_pure_returns_result_I2
bind_returns_result_E2 get_dom_component_def get_root_node_ok get_root_node_pure returns_result_eq
to_tree_order_node_ptrs)
qed
qed
lemma get_dom_component_to_tree_order_subset:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ to_tree_order ptr →⇩r nodes"
assumes "h ⊢ get_dom_component ptr →⇩r c"
shows "set nodes ⊆ set c"
using assms
apply(auto simp add: get_dom_component_def elim!: bind_returns_result_E2)[1]
by (meson to_tree_order_subset assms(5) contra_subsetD get_dom_component_ptr)
lemma get_dom_component_to_tree_order:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_dom_component ptr →⇩r c"
assumes "h ⊢ to_tree_order ptr' →⇩r to"
assumes "ptr ∈ set to"
shows "h ⊢ get_dom_component ptr' →⇩r c"
by (metis (no_types, hide_lams) assms(1) assms(2) assms(3) assms(4) assms(5) assms(6)
get_dom_component_ok get_dom_component_subset get_dom_component_to_tree_order_subset
is_OK_returns_result_E local.to_tree_order_ptr_in_result local.to_tree_order_ptrs_in_heap
select_result_I2 subsetCE)
lemma get_dom_component_root_node_same:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_dom_component ptr →⇩r c"
assumes "h ⊢ get_root_node ptr →⇩r root_ptr"
assumes "x ∈ set c"
shows "h ⊢ get_root_node x →⇩r root_ptr"
proof(insert assms(1) assms(6), induct x rule: heap_wellformed_induct_rev )
case (step child)
then show ?case
proof (cases "is_node_ptr_kind child")
case True
obtain node_ptr where
node_ptr: "cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r node_ptr = child"
using ‹is_node_ptr_kind child› node_ptr_casts_commute3 by blast
have "child |∈| object_ptr_kinds h"
using to_tree_order_ptrs_in_heap assms(1) assms(2) assms(3) assms(4) step(2)
unfolding get_dom_component_def
by (meson bind_returns_result_E2 get_root_node_pure)
with node_ptr have "node_ptr |∈| node_ptr_kinds h"
by auto
then obtain parent_opt where
parent: "h ⊢ get_parent node_ptr →⇩r parent_opt"
using get_parent_ok ‹type_wf h› ‹known_ptrs h›
by fast
then show ?thesis
proof (induct parent_opt)
case None
then have "h ⊢ get_root_node child →⇩r child"
using assms(1) get_root_node_no_parent node_ptr by blast
then show ?case
using ‹type_wf h› ‹known_ptrs h› node_ptr step(2) assms(4) assms(1) assms(5)
by (metis (no_types) ‹child |∈| object_ptr_kinds h› bind_pure_returns_result_I
get_dom_component_def get_dom_component_ptr get_dom_component_subset get_root_node_pure
is_OK_returns_result_E returns_result_eq to_tree_order_ok to_tree_order_same_root)
next
case (Some parent_ptr)
then have "h ⊢ get_dom_component parent_ptr →⇩r c"
using step get_dom_component_parent_inside assms node_ptr
by (meson get_dom_component_subset)
then show ?case
using Some node_ptr
apply(auto simp add: get_dom_component_def elim!: bind_returns_result_E2)[1]
using get_root_node_parent_same
using ‹h ⊢ get_dom_component parent_ptr →⇩r c› assms(1) assms(2) assms(3)
get_dom_component_ptr step.hyps by blast
qed
next
case False
then have "child |∈| object_ptr_kinds h"
using assms(1) assms(4) step(2)
by (metis (no_types, lifting) assms(2) assms(3) bind_returns_result_E2 get_root_node_pure
get_dom_component_def to_tree_order_ptrs_in_heap)
then have "h ⊢ get_root_node child →⇩r child"
using assms(1) False get_root_node_not_node_same by auto
then show ?thesis
using assms(1) assms(2) assms(3) assms(4) step.prems assms(5)
by (metis (no_types, hide_lams) bind_returns_result_E2 get_dom_component_def
get_root_node_pure returns_result_eq to_tree_order_same_root)
qed
qed
lemma get_dom_component_no_overlap:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_dom_component ptr →⇩r c"
assumes "h ⊢ get_dom_component ptr' →⇩r c'"
shows "set c ∩ set c' = {} ∨ c = c'"
proof (rule ccontr, auto)
fix x
assume 1: "c ≠ c'" and 2: "x ∈ set c" and 3: "x ∈ set c'"
obtain root_ptr where root_ptr: "h ⊢ get_root_node ptr →⇩r root_ptr"
using assms(4) unfolding get_dom_component_def
by (meson bind_is_OK_E is_OK_returns_result_I)
moreover obtain root_ptr' where root_ptr': "h ⊢ get_root_node ptr' →⇩r root_ptr'"
using assms(5) unfolding get_dom_component_def
by (meson bind_is_OK_E is_OK_returns_result_I)
ultimately have "root_ptr ≠ root_ptr'"
using 1 assms
unfolding get_dom_component_def
by (meson bind_returns_result_E3 get_root_node_pure returns_result_eq)
moreover have "h ⊢ get_root_node x →⇩r root_ptr"
using 2 root_ptr get_dom_component_root_node_same assms by blast
moreover have "h ⊢ get_root_node x →⇩r root_ptr'"
using 3 root_ptr' get_dom_component_root_node_same assms by blast
ultimately show False
using select_result_I2 by force
qed
lemma get_dom_component_separates_tree_order:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_dom_component ptr →⇩r c"
assumes "h ⊢ to_tree_order ptr →⇩r to"
assumes "h ⊢ get_dom_component ptr' →⇩r c'"
assumes "ptr' ∉ set c"
shows "set to ∩ set c' = {}"
proof -
have "c ≠ c'"
using assms get_dom_component_ptr by blast
then have "set c ∩ set c' = {}"
using assms get_dom_component_no_overlap by blast
moreover have "set to ⊆ set c"
using assms get_dom_component_to_tree_order_subset by blast
ultimately show ?thesis
by blast
qed
lemma get_dom_component_separates_tree_order_general:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_dom_component ptr →⇩r c"
assumes "h ⊢ to_tree_order ptr'' →⇩r to''"
assumes "ptr'' ∈ set c"
assumes "h ⊢ get_dom_component ptr' →⇩r c'"
assumes "ptr' ∉ set c"
shows "set to'' ∩ set c' = {}"
proof -
obtain root_ptr where root_ptr: "h ⊢ get_root_node ptr →⇩r root_ptr"
using assms(4)
by (metis bind_is_OK_E get_dom_component_def is_OK_returns_result_I)
then have c: "h ⊢ to_tree_order root_ptr →⇩r c"
using assms(4) unfolding get_dom_component_def
by (simp add: bind_returns_result_E3)
with root_ptr show ?thesis
using assms get_dom_component_separates_tree_order get_dom_component_subset
by meson
qed
end
interpretation i_get_dom_component?: l_get_dom_component⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent
get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe
is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs
get_disconnected_nodes get_disconnected_nodes_locs get_element_by_id get_elements_by_class_name
get_elements_by_tag_name
by(auto simp add: l_get_dom_component⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def l_get_dom_component⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms_def
get_dom_component_def is_strongly_dom_component_safe_def is_weakly_dom_component_safe_def instances)
declare l_get_dom_component⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms [instances]
subsubsection ‹get\_child\_nodes›
locale l_get_dom_component_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_get_dom_component⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_get_element_by⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
begin
lemma get_child_nodes_is_strongly_dom_component_safe:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_dom_component ptr →⇩r c"
assumes "h ⊢ get_child_nodes ptr' →⇩r children"
assumes "child ∈ set children"
shows "cast child ∈ set c ⟷ ptr' ∈ set c"
proof
assume 1: "cast child ∈ set c"
obtain root_ptr where
root_ptr: "h ⊢ get_root_node ptr →⇩r root_ptr"
by (metis assms(4) bind_is_OK_E get_dom_component_def is_OK_returns_result_I)
have "h ⊢ get_root_node (cast child) →⇩r root_ptr"
using "1" assms(1) assms(2) assms(3) assms(4) get_dom_component_root_node_same root_ptr
by blast
then have "h ⊢ get_root_node ptr' →⇩r root_ptr"
using assms(1) assms(2) assms(3) assms(5) assms(6) local.child_parent_dual
local.get_root_node_parent_same by blast
then have "h ⊢ get_dom_component ptr' →⇩r c"
using "1" assms(1) assms(2) assms(3) assms(4) assms(5) assms(6) local.child_parent_dual
local.get_dom_component_parent_inside local.get_dom_component_subset by blast
then show "ptr' ∈ set c"
using assms(1) assms(2) assms(3) get_dom_component_ptr by blast
next
assume 1: "ptr' ∈ set c"
obtain root_ptr where
root_ptr: "h ⊢ get_root_node ptr →⇩r root_ptr"
by (metis assms(4) bind_is_OK_E get_dom_component_def is_OK_returns_result_I)
have "h ⊢ get_root_node ptr' →⇩r root_ptr"
using "1" assms(1) assms(2) assms(3) assms(4) get_dom_component_root_node_same root_ptr
by blast
then have "h ⊢ get_root_node (cast child) →⇩r root_ptr"
using assms(1) assms(2) assms(3) assms(5) assms(6) local.child_parent_dual
local.get_root_node_parent_same by blast
then have "h ⊢ get_dom_component ptr' →⇩r c"
using "1" assms(1) assms(2) assms(3) assms(4) assms(5) assms(6) local.child_parent_dual
local.get_dom_component_parent_inside local.get_dom_component_subset by blast
then show "cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r child ∈ set c"
by (smt ‹h ⊢ get_root_node (cast⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r child) →⇩r root_ptr› assms(1) assms(2) assms(3)
assms(5) assms(6) disjoint_iff_not_equal is_OK_returns_result_E is_OK_returns_result_I
l_get_dom_component⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M.get_dom_component_no_overlap local.child_parent_dual local.get_dom_component_ok
local.get_dom_component_parent_inside local.get_dom_component_ptr local.get_root_node_ptr_in_heap
local.l_get_dom_component⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms)
qed
lemma get_child_nodes_get_dom_component:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_dom_component ptr →⇩r c"
assumes "h ⊢ get_child_nodes ptr →⇩r children"
shows "cast ` set children ⊆ set c"
using assms get_child_nodes_is_strongly_dom_component_safe
using local.get_dom_component_ptr by auto
lemma
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_child_nodes ptr →⇩r children"
assumes "h ⊢ get_child_nodes ptr →⇩h h'"
shows "is_strongly_dom_component_safe {ptr} (cast ` set children) h h'"
proof -
have "h = h'"
using assms(5)
by (meson local.get_child_nodes_pure pure_returns_heap_eq)
then show ?thesis
using assms
apply(auto simp add: is_strongly_dom_component_safe_def Let_def preserved_def)[1]
by (smt IntI finite_set_in get_child_nodes_is_strongly_dom_component_safe is_OK_returns_result_E
is_OK_returns_result_I local.get_child_nodes_ptr_in_heap local.get_dom_component_impl
local.get_dom_component_ok local.get_dom_component_ptr select_result_I2)
qed
end
interpretation i_get_dom_component_get_child_nodes?: l_get_dom_component_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent
get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe
is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs
get_disconnected_nodes get_disconnected_nodes_locs get_element_by_id get_elements_by_class_name
get_elements_by_tag_name first_in_tree_order get_attribute get_attribute_locs
by(auto simp add: l_get_dom_component_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def instances)
declare l_get_dom_component_get_child_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms [instances]
subsubsection ‹get\_parent›
locale l_get_dom_component_get_parent⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_get_dom_component⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_get_parent⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_get_element_by⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
begin
lemma get_parent_is_strongly_dom_component_safe_step:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_dom_component ptr →⇩r c"
assumes "h ⊢ get_parent ptr' →⇩r Some parent"
shows "parent ∈ set c ⟷ cast ptr' ∈ set c"
by (meson assms(1) assms(2) assms(3) assms(4) assms(5) is_OK_returns_result_E
l_to_tree_order_wf.to_tree_order_parent local.get_dom_component_parent_inside
local.get_dom_component_subset local.get_dom_component_to_tree_order_subset
local.get_parent_parent_in_heap local.l_to_tree_order_wf_axioms local.to_tree_order_ok
local.to_tree_order_ptr_in_result subsetCE)
lemma get_parent_is_strongly_dom_component_safe:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_parent node_ptr →⇩r Some parent"
assumes "h ⊢ get_parent node_ptr →⇩h h'"
shows "is_strongly_dom_component_safe {cast node_ptr} {parent} h h'"
proof -
have "h = h'"
using assms(5)
by (meson local.get_parent_pure pure_returns_heap_eq)
then show ?thesis
using assms
apply(auto simp add: is_strongly_dom_component_safe_def Let_def preserved_def)[1]
by (metis IntI finite_set_in local.get_dom_component_impl local.get_dom_component_ok
local.get_dom_component_parent_inside local.get_dom_component_ptr local.get_parent_parent_in_heap
local.known_ptrs_known_ptr local.parent_child_rel_child_in_heap local.parent_child_rel_parent
returns_result_select_result)
qed
end
interpretation i_get_dom_component_get_parent?: l_get_dom_component_get_parent⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent
get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe
is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs
get_disconnected_nodes get_disconnected_nodes_locs get_element_by_id get_elements_by_class_name
get_elements_by_tag_name first_in_tree_order get_attribute get_attribute_locs
by(auto simp add: l_get_dom_component_get_parent⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def instances)
declare l_get_dom_component_get_parent⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms [instances]
subsubsection ‹get\_root\_node›
locale l_get_dom_component_get_root_node⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_get_dom_component⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_get_root_node⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_get_element_by⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
begin
lemma get_root_node_is_strongly_dom_component_safe_step:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_dom_component ptr →⇩r c"
assumes "h ⊢ get_root_node ptr' →⇩r root"
shows "root ∈ set c ⟷ ptr' ∈ set c"
proof
assume 1: "root ∈ set c"
obtain root_ptr where
root_ptr: "h ⊢ get_root_node ptr →⇩r root_ptr"
by (metis assms(4) bind_is_OK_E get_dom_component_def is_OK_returns_result_I)
have "h ⊢ get_root_node root →⇩r root_ptr"
using "1" assms(1) assms(2) assms(3) assms(4) get_dom_component_root_node_same root_ptr
by blast
moreover have "h ⊢ get_root_node ptr' →⇩r root_ptr"
by (metis (no_types, lifting) calculation assms(1) assms(2) assms(3) assms(5)
is_OK_returns_result_E local.get_root_node_root_in_heap local.to_tree_order_ok
local.to_tree_order_ptr_in_result local.to_tree_order_same_root select_result_I2)
ultimately have "h ⊢ get_dom_component ptr' →⇩r c"
apply(auto simp add: get_dom_component_def)[1]
by (smt "1" assms(1) assms(2) assms(3) assms(4) bind_pure_returns_result_I bind_returns_result_E3
local.get_dom_component_def local.get_dom_component_subset local.get_root_node_pure)
then show "ptr' ∈ set c"
using assms(1) assms(2) assms(3) get_dom_component_ptr by blast
next
assume 1: "ptr' ∈ set c"
obtain root_ptr where
root_ptr: "h ⊢ get_root_node ptr →⇩r root_ptr"
by (metis assms(4) bind_is_OK_E get_dom_component_def is_OK_returns_result_I)
have "h ⊢ get_root_node ptr' →⇩r root_ptr"
using "1" assms(1) assms(2) assms(3) assms(4) get_dom_component_root_node_same root_ptr
by blast
then have "h ⊢ get_root_node root →⇩r root_ptr"
by (metis assms(1) assms(2) assms(3) assms(5) is_OK_returns_result_E
local.get_root_node_root_in_heap local.to_tree_order_ok local.to_tree_order_ptr_in_result
local.to_tree_order_same_root returns_result_eq)
then have "h ⊢ get_dom_component ptr' →⇩r c"
using "1" assms(1) assms(2) assms(3) assms(4) local.get_dom_component_subset by blast
then show "root ∈ set c"
using assms(5) bind_returns_result_E3 local.get_dom_component_def local.to_tree_order_ptr_in_result
by fastforce
qed
lemma get_root_node_is_strongly_dom_component_safe:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_root_node ptr →⇩r root"
assumes "h ⊢ get_root_node ptr →⇩h h'"
shows "is_strongly_dom_component_safe {ptr} {root} h h'"
proof -
have "h = h'"
using assms(5)
by (meson local.get_root_node_pure pure_returns_heap_eq)
then show ?thesis
using assms
apply(auto simp add: is_strongly_dom_component_safe_def Let_def preserved_def)[1]
by (metis (no_types, lifting) IntI finite_set_in get_root_node_is_strongly_dom_component_safe_step
is_OK_returns_result_I local.get_dom_component_impl local.get_dom_component_ok
local.get_dom_component_ptr local.get_root_node_ptr_in_heap returns_result_select_result)
qed
end
interpretation i_get_dom_component_get_root_node?: l_get_dom_component_get_root_node⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent
get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe
is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs
get_disconnected_nodes get_disconnected_nodes_locs get_element_by_id get_elements_by_class_name
get_elements_by_tag_name first_in_tree_order get_attribute get_attribute_locs
by(auto simp add: l_get_dom_component_get_root_node⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def instances)
declare l_get_dom_component_get_root_node⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms [instances]
subsubsection ‹get\_element\_by\_id›
locale l_get_dom_component_get_element_by_id⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_get_dom_component⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_first_in_tree_order⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_to_tree_order⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_get_element_by⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
begin
lemma get_element_by_id_is_strongly_dom_component_safe_step:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_dom_component ptr →⇩r c"
assumes "h ⊢ get_element_by_id ptr' idd →⇩r Some result"
shows "cast result ∈ set c ⟷ ptr' ∈ set c"
proof
assume 1: "cast result ∈ set c"
obtain root_ptr where
root_ptr: "h ⊢ get_root_node ptr →⇩r root_ptr"
by (metis assms(4) bind_is_OK_E get_dom_component_def is_OK_returns_result_I)
then have "h ⊢ to_tree_order root_ptr →⇩r c"
using ‹h ⊢ get_dom_component ptr →⇩r c›
by (simp add: get_dom_component_def bind_returns_result_E3)
obtain to' where to': "h ⊢ to_tree_order ptr' →⇩r to'"
using ‹h ⊢ get_element_by_id ptr' idd →⇩r Some result›
apply(simp add: get_element_by_id_def first_in_tree_order_def)
by (meson bind_is_OK_E is_OK_returns_result_I)
then have "cast result ∈ set to'"
using ‹h ⊢ get_element_by_id ptr' idd →⇩r Some result› get_element_by_id_result_in_tree_order
by blast
have "h ⊢ get_root_node (cast result) →⇩r root_ptr"
using "1" assms(1) assms(2) assms(3) assms(4) get_dom_component_root_node_same root_ptr
by blast
then have "h ⊢ get_root_node ptr' →⇩r root_ptr"
using ‹cast result ∈ set to'› ‹h ⊢ to_tree_order ptr' →⇩r to'›
using "1" assms(1) assms(2) assms(3) assms(4) get_dom_component_ptr get_dom_component_root_node_same
get_dom_component_subset get_dom_component_to_tree_order
by blast
then have "h ⊢ get_dom_component ptr' →⇩r c"
using ‹h ⊢ to_tree_order root_ptr →⇩r c›
using get_dom_component_def by auto
then show "ptr' ∈ set c"
using assms(1) assms(2) assms(3) get_dom_component_ptr by blast
next
assume "ptr' ∈ set c"
moreover obtain to' where to': "h ⊢ to_tree_order ptr' →⇩r to'"
by (meson assms(1) assms(2) assms(3) assms(4) calculation get_dom_component_ptr_in_heap
get_dom_component_subset is_OK_returns_result_E is_OK_returns_result_I to_tree_order_ok)
ultimately have "set to' ⊆ set c"
using assms(1) assms(2) assms(3) assms(4) get_dom_component_subset
get_dom_component_to_tree_order_subset
by blast
moreover have "cast result ∈ set to'"
using assms(5) get_element_by_id_result_in_tree_order to' by blast
ultimately show "cast result ∈ set c"
by blast
qed
lemma get_element_by_id_is_strongly_dom_component_safe:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_element_by_id ptr idd →⇩r Some result"
assumes "h ⊢ get_element_by_id ptr idd →⇩h h'"
shows "is_strongly_dom_component_safe {ptr} {cast result} h h'"
proof -
have "h = h'"
using assms(5)
by(auto simp add: preserved_def get_element_by_id_def first_in_tree_order_def
elim!: bind_returns_heap_E2 intro!: map_filter_M_pure bind_pure_I
split: option.splits list.splits)
have "ptr |∈| object_ptr_kinds h"
using assms(4)
apply(auto simp add: get_element_by_id_def)[1]
by (metis (no_types, lifting) assms(1) assms(2) assms(3) bind_is_OK_E is_OK_returns_result_I
local.first_in_tree_order_def local.to_tree_order_ptr_in_result local.to_tree_order_ptrs_in_heap)
obtain to where to: "h ⊢ to_tree_order ptr →⇩r to"
by (meson ‹ptr |∈| object_ptr_kinds h› assms(1) assms(2) assms(3) is_OK_returns_result_E
local.to_tree_order_ok)
then have "cast result ∈ set to"
using assms(4) local.get_element_by_id_result_in_tree_order by auto
obtain c where c: "h ⊢ a_get_dom_component ptr →⇩r c"
using ‹ptr |∈| object_ptr_kinds h› assms(1) assms(2) assms(3) local.get_dom_component_impl
local.get_dom_component_ok by blast
then show ?thesis
using assms ‹h = h'›
apply(auto simp add: is_strongly_dom_component_safe_def Let_def preserved_def get_element_by_id_def
first_in_tree_order_def elim!: bind_returns_result_E2 intro!: map_filter_M_pure bind_pure_I
split: option.splits list.splits)[1]
by (metis (no_types, lifting) Int_iff ‹ptr |∈| object_ptr_kinds h› assms(4) finite_set_in
get_element_by_id_is_strongly_dom_component_safe_step local.get_dom_component_impl
local.get_dom_component_ptr select_result_I2)
qed
end
interpretation i_get_dom_component_get_element_by_id?: l_get_dom_component_get_element_by_id⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent
get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe
is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs
get_disconnected_nodes get_disconnected_nodes_locs get_element_by_id get_elements_by_class_name
get_elements_by_tag_name first_in_tree_order get_attribute get_attribute_locs
by(auto simp add: l_get_dom_component_get_element_by_id⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def instances)
declare l_get_dom_component_get_element_by_id⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms [instances]
subsubsection ‹get\_elements\_by\_class\_name›
locale l_get_dom_component_get_elements_by_class_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_get_dom_component⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_first_in_tree_order⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_to_tree_order⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_get_element_by⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
begin
lemma get_elements_by_class_name_is_strongly_dom_component_safe_step:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_dom_component ptr →⇩r c"
assumes "h ⊢ get_elements_by_class_name ptr' idd →⇩r results"
assumes "result ∈ set results"
shows "cast result ∈ set c ⟷ ptr' ∈ set c"
proof
assume 1: "cast result ∈ set c"
obtain root_ptr where
root_ptr: "h ⊢ get_root_node ptr →⇩r root_ptr"
by (metis assms(4) bind_is_OK_E get_dom_component_def is_OK_returns_result_I)
then have "h ⊢ to_tree_order root_ptr →⇩r c"
using ‹h ⊢ get_dom_component ptr →⇩r c›
by (simp add: get_dom_component_def bind_returns_result_E3)
obtain to' where to': "h ⊢ to_tree_order ptr' →⇩r to'"
using assms(5)
apply(simp add: get_elements_by_class_name_def first_in_tree_order_def)
by (meson bind_is_OK_E is_OK_returns_result_I)
then have "cast result ∈ set to'"
using assms get_elements_by_class_name_result_in_tree_order by blast
have "h ⊢ get_root_node (cast result) →⇩r root_ptr"
using "1" assms(1) assms(2) assms(3) assms(4) get_dom_component_root_node_same root_ptr
by blast
then have "h ⊢ get_root_node ptr' →⇩r root_ptr"
using ‹cast result ∈ set to'› ‹h ⊢ to_tree_order ptr' →⇩r to'›
using "1" assms(1) assms(2) assms(3) assms(4) get_dom_component_ptr get_dom_component_root_node_same
get_dom_component_subset get_dom_component_to_tree_order
by blast
then have "h ⊢ get_dom_component ptr' →⇩r c"
using ‹h ⊢ to_tree_order root_ptr →⇩r c›
using get_dom_component_def by auto
then show "ptr' ∈ set c"
using assms(1) assms(2) assms(3) get_dom_component_ptr by blast
next
assume "ptr' ∈ set c"
moreover obtain to' where to': "h ⊢ to_tree_order ptr' →⇩r to'"
by (meson assms(1) assms(2) assms(3) assms(4) calculation get_dom_component_ptr_in_heap
get_dom_component_subset is_OK_returns_result_E is_OK_returns_result_I to_tree_order_ok)
ultimately have "set to' ⊆ set c"
using assms(1) assms(2) assms(3) assms(4) get_dom_component_subset
get_dom_component_to_tree_order_subset
by blast
moreover have "cast result ∈ set to'"
using assms get_elements_by_class_name_result_in_tree_order to' by blast
ultimately show "cast result ∈ set c"
by blast
qed
lemma get_elements_by_class_name_pure [simp]:
"pure (get_elements_by_class_name ptr name) h"
by(auto simp add: get_elements_by_class_name_def intro!: bind_pure_I map_filter_M_pure
split: option.splits)
lemma get_elements_by_class_name_is_strongly_dom_component_safe:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_elements_by_class_name ptr name →⇩r results"
assumes "h ⊢ get_elements_by_class_name ptr name →⇩h h'"
shows "is_strongly_dom_component_safe {ptr} (cast ` set results) h h'"
proof -
have "h = h'"
using assms(5)
by (meson get_elements_by_class_name_pure pure_returns_heap_eq)
have "ptr |∈| object_ptr_kinds h"
using assms(4)
apply(auto simp add: get_elements_by_class_name_def)[1]
by (metis (no_types, lifting) assms(1) assms(2) assms(3) bind_is_OK_E is_OK_returns_result_I
local.first_in_tree_order_def local.to_tree_order_ptr_in_result local.to_tree_order_ptrs_in_heap)
obtain to where to: "h ⊢ to_tree_order ptr →⇩r to"
by (meson ‹ptr |∈| object_ptr_kinds h› assms(1) assms(2) assms(3) is_OK_returns_result_E
local.to_tree_order_ok)
then have "cast ` set results ⊆ set to"
using assms(4) local.get_elements_by_class_name_result_in_tree_order by auto
obtain c where c: "h ⊢ a_get_dom_component ptr →⇩r c"
using ‹ptr |∈| object_ptr_kinds h› assms(1) assms(2) assms(3) local.get_dom_component_impl
local.get_dom_component_ok by blast
then show ?thesis
using assms ‹h = h'›
apply(auto simp add: is_strongly_dom_component_safe_def Let_def preserved_def
get_elements_by_class_name_def first_in_tree_order_def elim!: bind_returns_result_E2
intro!: map_filter_M_pure bind_pure_I split: option.splits list.splits)[1]
by (metis (no_types, lifting) Int_iff ‹ptr |∈| object_ptr_kinds h› assms(4) finite_set_in
get_elements_by_class_name_is_strongly_dom_component_safe_step local.get_dom_component_impl
local.get_dom_component_ptr select_result_I2)
qed
end
interpretation i_get_dom_component_get_elements_by_class_name?:
l_get_dom_component_get_elements_by_class_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent
get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe
is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs
get_disconnected_nodes get_disconnected_nodes_locs get_element_by_id get_elements_by_class_name
get_elements_by_tag_name first_in_tree_order get_attribute get_attribute_locs
by(auto simp add: l_get_dom_component_get_elements_by_class_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def instances)
declare l_get_dom_component_get_elements_by_class_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms [instances]
subsubsection ‹get\_elements\_by\_tag\_name›
locale l_get_dom_component_get_elements_by_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_get_dom_component⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_first_in_tree_order⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_to_tree_order⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M +
l_get_element_by⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
begin
lemma get_elements_by_tag_name_is_strongly_dom_component_safe_step:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_dom_component ptr →⇩r c"
assumes "h ⊢ get_elements_by_tag_name ptr' idd →⇩r results"
assumes "result ∈ set results"
shows "cast result ∈ set c ⟷ ptr' ∈ set c"
proof
assume 1: "cast result ∈ set c"
obtain root_ptr where
root_ptr: "h ⊢ get_root_node ptr →⇩r root_ptr"
by (metis assms(4) bind_is_OK_E get_dom_component_def is_OK_returns_result_I)
then have "h ⊢ to_tree_order root_ptr →⇩r c"
using ‹h ⊢ get_dom_component ptr →⇩r c›
by (simp add: get_dom_component_def bind_returns_result_E3)
obtain to' where to': "h ⊢ to_tree_order ptr' →⇩r to'"
using assms(5)
apply(simp add: get_elements_by_tag_name_def first_in_tree_order_def)
by (meson bind_is_OK_E is_OK_returns_result_I)
then have "cast result ∈ set to'"
using assms get_elements_by_tag_name_result_in_tree_order by blast
have "h ⊢ get_root_node (cast result) →⇩r root_ptr"
using "1" assms(1) assms(2) assms(3) assms(4) get_dom_component_root_node_same root_ptr
by blast
then have "h ⊢ get_root_node ptr' →⇩r root_ptr"
using ‹cast result ∈ set to'› ‹h ⊢ to_tree_order ptr' →⇩r to'›
using "1" assms(1) assms(2) assms(3) assms(4) get_dom_component_ptr
get_dom_component_root_node_same get_dom_component_subset get_dom_component_to_tree_order
by blast
then have "h ⊢ get_dom_component ptr' →⇩r c"
using ‹h ⊢ to_tree_order root_ptr →⇩r c›
using get_dom_component_def by auto
then show "ptr' ∈ set c"
using assms(1) assms(2) assms(3) get_dom_component_ptr by blast
next
assume "ptr' ∈ set c"
moreover obtain to' where to': "h ⊢ to_tree_order ptr' →⇩r to'"
by (meson assms(1) assms(2) assms(3) assms(4) calculation get_dom_component_ptr_in_heap
get_dom_component_subset is_OK_returns_result_E is_OK_returns_result_I to_tree_order_ok)
ultimately have "set to' ⊆ set c"
using assms(1) assms(2) assms(3) assms(4) get_dom_component_subset
get_dom_component_to_tree_order_subset
by blast
moreover have "cast result ∈ set to'"
using assms get_elements_by_tag_name_result_in_tree_order to' by blast
ultimately show "cast result ∈ set c"
by blast
qed
lemma get_elements_by_tag_name_is_strongly_dom_component_safe:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ get_elements_by_tag_name ptr name →⇩r results"
assumes "h ⊢ get_elements_by_tag_name ptr name →⇩h h'"
shows "is_strongly_dom_component_safe {ptr} (cast ` set results) h h'"
proof -
have "h = h'"
using assms(5)
by (meson get_elements_by_tag_name_pure pure_returns_heap_eq)
have "ptr |∈| object_ptr_kinds h"
using assms(4)
apply(auto simp add: get_elements_by_tag_name_def)[1]
by (metis (no_types, lifting) assms(1) assms(2) assms(3) bind_is_OK_E is_OK_returns_result_I
local.first_in_tree_order_def local.to_tree_order_ptr_in_result local.to_tree_order_ptrs_in_heap)
obtain to where to: "h ⊢ to_tree_order ptr →⇩r to"
by (meson ‹ptr |∈| object_ptr_kinds h› assms(1) assms(2) assms(3) is_OK_returns_result_E
local.to_tree_order_ok)
then have "cast ` set results ⊆ set to"
using assms(4) local.get_elements_by_tag_name_result_in_tree_order by auto
obtain c where c: "h ⊢ a_get_dom_component ptr →⇩r c"
using ‹ptr |∈| object_ptr_kinds h› assms(1) assms(2) assms(3) local.get_dom_component_impl
local.get_dom_component_ok by blast
then show ?thesis
using assms ‹h = h'›
apply(auto simp add: is_strongly_dom_component_safe_def Let_def preserved_def
get_elements_by_class_name_def first_in_tree_order_def elim!: bind_returns_result_E2
intro!: map_filter_M_pure bind_pure_I split: option.splits list.splits)[1]
by (metis (no_types, lifting) IntI ‹ptr |∈| object_ptr_kinds h› finite_set_in
get_elements_by_tag_name_is_strongly_dom_component_safe_step local.get_dom_component_impl
local.get_dom_component_ptr select_result_I2)
qed
end
interpretation i_get_dom_component_get_elements_by_tag_name?:
l_get_dom_component_get_elements_by_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent
get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe
is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs
get_disconnected_nodes get_disconnected_nodes_locs get_element_by_id get_elements_by_class_name
get_elements_by_tag_name first_in_tree_order get_attribute get_attribute_locs
by(auto simp add: l_get_dom_component_get_elements_by_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def instances)
declare l_get_dom_component_get_elements_by_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms [instances]
subsubsection ‹remove\_child›
lemma remove_child_unsafe: "¬(∀(h
:: ('object_ptr::{equal,linorder}, 'node_ptr::{equal,linorder}, 'element_ptr::{equal,linorder},
'character_data_ptr::{equal,linorder}, 'document_ptr::{equal,linorder}, 'shadow_root_ptr::{equal,linorder},
'Object::{equal,linorder}, 'Node::{equal,linorder}, 'Element::{equal,linorder},
'CharacterData::{equal,linorder}, 'Document::{equal,linorder}) heap
) h' ptr child. heap_is_wellformed h ⟶ type_wf h ⟶ known_ptrs h ⟶
h ⊢ remove_child ptr child →⇩h h' ⟶ is_weakly_dom_component_safe {ptr, cast child} {} h h')"
proof -
obtain h document_ptr e1 e2 where h: "Inr ((document_ptr, e1, e2), h) = (Heap (fmempty)
:: ('object_ptr::{equal,linorder}, 'node_ptr::{equal,linorder}, 'element_ptr::{equal,linorder},
'character_data_ptr::{equal,linorder}, 'document_ptr::{equal,linorder}, 'shadow_root_ptr::{equal,linorder},
'Object::{equal,linorder}, 'Node::{equal,linorder}, 'Element::{equal,linorder},
'CharacterData::{equal,linorder}, 'Document::{equal,linorder}) heap
) ⊢ (do {
document_ptr ← create_document;
e1 ← create_element document_ptr ''div'';
e2 ← create_element document_ptr ''div'';
append_child (cast e1) (cast e2);
return (document_ptr, e1, e2)
})"
by(code_simp, auto simp add: equal_eq List.member_def)+
then obtain h' where
"heap_is_wellformed h" and "type_wf h" and "known_ptrs h" and
h': "h ⊢ remove_child (cast e1) (cast e2) →⇩h h'" and
"¬(is_weakly_dom_component_safe {cast e1, cast e2} {} h h')"
apply(code_simp)
apply(clarify)
by(code_simp, auto simp add: equal_eq List.member_def)+
then show ?thesis
by auto
qed
subsubsection ‹adopt\_node›
lemma adopt_node_unsafe: "¬(∀(h
:: ('object_ptr::{equal,linorder}, 'node_ptr::{equal,linorder}, 'element_ptr::{equal,linorder},
'character_data_ptr::{equal,linorder}, 'document_ptr::{equal,linorder}, 'shadow_root_ptr::{equal,linorder},
'Object::{equal,linorder}, 'Node::{equal,linorder}, 'Element::{equal,linorder},
'CharacterData::{equal,linorder}, 'Document::{equal,linorder}) heap
) h' document_ptr child. heap_is_wellformed h ⟶ type_wf h ⟶ known_ptrs h ⟶ h ⊢ adopt_node document_ptr child →⇩h h' ⟶ is_weakly_dom_component_safe {cast document_ptr, cast child} {} h h')"
proof -
obtain h document_ptr document_ptr2 e1 where h: "Inr ((document_ptr, document_ptr2, e1), h) = (Heap (fmempty)
:: ('object_ptr::{equal,linorder}, 'node_ptr::{equal,linorder}, 'element_ptr::{equal,linorder},
'character_data_ptr::{equal,linorder}, 'document_ptr::{equal,linorder}, 'shadow_root_ptr::{equal,linorder},
'Object::{equal,linorder}, 'Node::{equal,linorder}, 'Element::{equal,linorder},
'CharacterData::{equal,linorder}, 'Document::{equal,linorder}) heap
) ⊢ (do {
document_ptr ← create_document;
document_ptr2 ← create_document;
e1 ← create_element document_ptr ''div'';
adopt_node document_ptr2 (cast e1);
return (document_ptr, document_ptr2, e1)
})"
by(code_simp, auto simp add: equal_eq List.member_def)+
then obtain h' where
"heap_is_wellformed h" and "type_wf h" and "known_ptrs h" and
h': "h ⊢ adopt_node document_ptr (cast e1) →⇩h h'" and
"¬(is_weakly_dom_component_safe {cast document_ptr, cast e1} {} h h')"
apply(code_simp)
apply(clarify)
by(code_simp, auto simp add: equal_eq List.member_def)+
then show ?thesis
by auto
qed
subsubsection ‹create\_element›
lemma create_element_not_strongly_dom_component_safe:
obtains
h :: "('object_ptr::{equal,linorder}, 'node_ptr::{equal,linorder}, 'element_ptr::{equal,linorder},
'character_data_ptr::{equal,linorder}, 'document_ptr::{equal,linorder}, 'shadow_root_ptr::{equal,linorder},
'Object::{equal,linorder}, 'Node::{equal,linorder}, 'Element::{equal,linorder},
'CharacterData::{equal,linorder}, 'Document::{equal,linorder}) heap" and
h' and document_ptr and new_element_ptr and tag where
"heap_is_wellformed h" and "type_wf h" and "known_ptrs h" and
"h ⊢ create_element document_ptr tag →⇩r new_element_ptr →⇩h h'" and
"¬ is_strongly_dom_component_safe {cast document_ptr} {cast new_element_ptr} h h'"
proof -
let ?h0 = "Heap fmempty ::('object_ptr::{equal,linorder}, 'node_ptr::{equal,linorder},
'element_ptr::{equal,linorder}, 'character_data_ptr::{equal,linorder}, 'document_ptr::{equal,linorder},
'shadow_root_ptr::{equal,linorder}, 'Object::{equal,linorder}, 'Node::{equal,linorder},
'Element::{equal,linorder}, 'CharacterData::{equal,linorder}, 'Document::{equal,linorder}) heap"
let ?P = "create_document"
let ?h1 = "|?h0 ⊢ ?P|⇩h"
let ?document_ptr = "|?h0 ⊢ ?P|⇩r"
show thesis
apply(rule that[where h="?h1" and document_ptr="?document_ptr"])
by code_simp+
qed
locale l_get_dom_component_create_element⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_get_dom_component⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs
to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component
is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs
get_ancestors get_ancestors_locs get_disconnected_nodes get_disconnected_nodes_locs get_element_by_id
get_elements_by_class_name get_elements_by_tag_name +
l_create_element⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes
set_disconnected_nodes_locs set_tag_name set_tag_name_locs type_wf create_element known_ptr +
l_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf get_disconnected_nodes get_disconnected_nodes_locs +
l_set_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf set_disconnected_nodes set_disconnected_nodes_locs +
l_set_tag_name⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf set_tag_name set_tag_name_locs +
l_new_element_get_disconnected_nodes get_disconnected_nodes get_disconnected_nodes_locs +
l_new_element_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs +
l_set_tag_name_get_child_nodes type_wf set_tag_name set_tag_name_locs known_ptr
get_child_nodes get_child_nodes_locs +
l_create_element_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M known_ptr known_ptrs type_wf get_child_nodes get_child_nodes_locs
get_disconnected_nodes get_disconnected_nodes_locs heap_is_wellformed parent_child_rel set_tag_name
set_tag_name_locs
set_disconnected_nodes set_disconnected_nodes_locs create_element
for known_ptr :: "(_::linorder) object_ptr ⇒ bool"
and heap_is_wellformed :: "(_) heap ⇒ bool"
and parent_child_rel :: "(_) heap ⇒ ((_) object_ptr × (_) object_ptr) set"
and type_wf :: "(_) heap ⇒ bool"
and known_ptrs :: "(_) heap ⇒ bool"
and to_tree_order :: "(_) object_ptr ⇒ ((_) heap, exception, (_) object_ptr list) prog"
and get_parent :: "(_) node_ptr ⇒ ((_) heap, exception, (_) object_ptr option) prog"
and get_parent_locs :: "((_) heap ⇒ (_) heap ⇒ bool) set"
and get_child_nodes :: "(_) object_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_child_nodes_locs :: "(_) object_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and get_dom_component :: "(_) object_ptr ⇒ ((_) heap, exception, (_) object_ptr list) prog"
and is_strongly_dom_component_safe :: "(_) object_ptr set ⇒ (_) object_ptr set ⇒ (_) heap ⇒ (_) heap ⇒ bool"
and is_weakly_dom_component_safe :: "(_) object_ptr set ⇒ (_) object_ptr set ⇒ (_) heap ⇒ (_) heap ⇒ bool"
and get_root_node :: "(_) object_ptr ⇒ ((_) heap, exception, (_) object_ptr) prog"
and get_root_node_locs :: "((_) heap ⇒ (_) heap ⇒ bool) set"
and get_ancestors :: "(_) object_ptr ⇒ ((_) heap, exception, (_) object_ptr list) prog"
and get_ancestors_locs :: "((_) heap ⇒ (_) heap ⇒ bool) set"
and get_element_by_id :: "(_) object_ptr ⇒ char list ⇒ ((_) heap, exception, (_) element_ptr option) prog"
and get_elements_by_class_name :: "(_) object_ptr ⇒ char list ⇒ ((_) heap, exception, (_) element_ptr list) prog"
and get_elements_by_tag_name :: "(_) object_ptr ⇒ char list ⇒ ((_) heap, exception, (_) element_ptr list) prog"
and get_disconnected_nodes :: "(_) document_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and set_disconnected_nodes :: "(_) document_ptr ⇒ (_) node_ptr list ⇒ ((_) heap, exception, unit) prog"
and set_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap, exception, unit) prog set"
and set_tag_name :: "(_) element_ptr ⇒ char list ⇒ ((_) heap, exception, unit) prog"
and set_tag_name_locs :: "(_) element_ptr ⇒ ((_) heap, exception, unit) prog set"
and create_element :: "(_) document_ptr ⇒ char list ⇒ ((_) heap, exception, (_) element_ptr) prog"
begin
lemma create_element_is_weakly_dom_component_safe_step:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ create_element document_ptr tag →⇩h h'"
assumes "ptr ∉ set |h ⊢ get_dom_component (cast document_ptr)|⇩r"
assumes "ptr ≠ cast |h ⊢ create_element document_ptr tag|⇩r"
shows "preserved (get_M ptr getter) h h'"
proof -
obtain new_element_ptr h2 h3 disc_nodes where
new_element_ptr: "h ⊢ new_element →⇩r new_element_ptr" and
h2: "h ⊢ new_element →⇩h h2" and
h3: "h2 ⊢set_tag_name new_element_ptr tag →⇩h h3" and
disc_nodes: "h3 ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes" and
h': "h3 ⊢ set_disconnected_nodes document_ptr (cast new_element_ptr # disc_nodes) →⇩h h'"
using assms(4)
by(auto simp add: create_element_def
elim!: bind_returns_heap_E bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated])
have "h ⊢ create_element document_ptr tag →⇩r new_element_ptr"
using new_element_ptr h2 h3 disc_nodes h'
apply(auto simp add: create_element_def intro!: bind_returns_result_I
bind_pure_returns_result_I[OF get_disconnected_nodes_pure])[1]
apply (metis is_OK_returns_heap_I is_OK_returns_result_E old.unit.exhaust)
by (metis is_OK_returns_heap_I is_OK_returns_result_E old.unit.exhaust)
have "preserved (get_M ptr getter) h h2"
using h2 new_element_ptr
apply(rule new_element_get_M⇩O⇩b⇩j⇩e⇩c⇩t)
using new_element_ptr assms(6) ‹h ⊢ create_element document_ptr tag →⇩r new_element_ptr›
by simp
have "preserved (get_M ptr getter) h2 h3"
using set_tag_name_writes h3
apply(rule reads_writes_preserved2)
apply(auto simp add: set_tag_name_locs_impl a_set_tag_name_locs_def all_args_def)[1]
by (metis (no_types, lifting) ‹h ⊢ create_element document_ptr tag →⇩r new_element_ptr› assms(6)
get_M_Element_preserved8 select_result_I2)
have "document_ptr |∈| document_ptr_kinds h"
using create_element_document_in_heap
using assms(4)
by blast
then
have "ptr ≠ (cast document_ptr)"
using assms(5) assms(1) assms(2) assms(3) local.get_dom_component_ok local.get_dom_component_ptr
by auto
have "preserved (get_M ptr getter) h3 h'"
using set_disconnected_nodes_writes h'
apply(rule reads_writes_preserved2)
apply(auto simp add: set_disconnected_nodes_locs_def all_args_def)[1]
by (metis ‹ptr ≠ cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r document_ptr› get_M_Mdocument_preserved3)
show ?thesis
using ‹preserved (get_M ptr getter) h h2› ‹preserved (get_M ptr getter) h2 h3›
‹preserved (get_M ptr getter) h3 h'›
by(auto simp add: preserved_def)
qed
lemma create_element_is_weakly_dom_component_safe:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ create_element document_ptr tag →⇩r result"
assumes "h ⊢ create_element document_ptr tag →⇩h h'"
shows "is_weakly_dom_component_safe {cast document_ptr} {cast result} h h'"
proof -
obtain new_element_ptr h2 h3 disc_nodes_h3 where
new_element_ptr: "h ⊢ new_element →⇩r new_element_ptr" and
h2: "h ⊢ new_element →⇩h h2" and
h3: "h2 ⊢ set_tag_name new_element_ptr tag →⇩h h3" and
disc_nodes_h3: "h3 ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes_h3" and
h': "h3 ⊢ set_disconnected_nodes document_ptr (cast new_element_ptr # disc_nodes_h3) →⇩h h'"
using assms(5)
by(auto simp add: create_element_def
elim!: bind_returns_heap_E
bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated] )
then have "h ⊢ create_element document_ptr tag →⇩r new_element_ptr"
apply(auto simp add: create_element_def intro!: bind_returns_result_I)[1]
apply (metis is_OK_returns_heap_I is_OK_returns_result_E old.unit.exhaust)
apply (metis is_OK_returns_heap_E is_OK_returns_result_I local.get_disconnected_nodes_pure
pure_returns_heap_eq)
by (metis is_OK_returns_heap_I is_OK_returns_result_E old.unit.exhaust)
have "new_element_ptr ∉ set |h ⊢ element_ptr_kinds_M|⇩r"
using new_element_ptr ElementMonad.ptr_kinds_ptr_kinds_M h2
using new_element_ptr_not_in_heap by blast
then have "cast new_element_ptr ∉ set |h ⊢ node_ptr_kinds_M|⇩r"
by simp
then have "cast new_element_ptr ∉ set |h ⊢ object_ptr_kinds_M|⇩r"
by simp
have object_ptr_kinds_eq_h: "object_ptr_kinds h2 = object_ptr_kinds h |∪| {|cast new_element_ptr|}"
using new_element_new_ptr h2 new_element_ptr by blast
then have node_ptr_kinds_eq_h: "node_ptr_kinds h2 = node_ptr_kinds h |∪| {|cast new_element_ptr|}"
apply(simp add: node_ptr_kinds_def)
by force
then have element_ptr_kinds_eq_h: "element_ptr_kinds h2 = element_ptr_kinds h |∪| {|new_element_ptr|}"
apply(simp add: element_ptr_kinds_def)
by force
have character_data_ptr_kinds_eq_h: "character_data_ptr_kinds h2 = character_data_ptr_kinds h"
using object_ptr_kinds_eq_h
by(auto simp add: node_ptr_kinds_def character_data_ptr_kinds_def)
have document_ptr_kinds_eq_h: "document_ptr_kinds h2 = document_ptr_kinds h"
using object_ptr_kinds_eq_h
by(auto simp add: document_ptr_kinds_def)
have object_ptr_kinds_eq_h2: "object_ptr_kinds h3 = object_ptr_kinds h2"
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h' = object_ptr_kinds h",
OF set_tag_name_writes h3])
using set_tag_name_pointers_preserved
by (auto simp add: reflp_def transp_def)
then have document_ptr_kinds_eq_h2: "document_ptr_kinds h3 = document_ptr_kinds h2"
by (auto simp add: document_ptr_kinds_def)
have node_ptr_kinds_eq_h2: "node_ptr_kinds h3 = node_ptr_kinds h2"
using object_ptr_kinds_eq_h2
by(auto simp add: node_ptr_kinds_def)
have object_ptr_kinds_eq_h3: "object_ptr_kinds h' = object_ptr_kinds h3"
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h' = object_ptr_kinds h",
OF set_disconnected_nodes_writes h'])
using set_disconnected_nodes_pointers_preserved
by (auto simp add: reflp_def transp_def)
then have document_ptr_kinds_eq_h3: "document_ptr_kinds h' = document_ptr_kinds h3"
by (auto simp add: document_ptr_kinds_def)
have node_ptr_kinds_eq_h3: "node_ptr_kinds h' = node_ptr_kinds h3"
using object_ptr_kinds_eq_h3
by(auto simp add: node_ptr_kinds_def)
have "known_ptr (cast new_element_ptr)"
using ‹h ⊢ create_element document_ptr tag →⇩r new_element_ptr› local.create_element_known_ptr
by blast
then
have "known_ptrs h2"
using known_ptrs_new_ptr object_ptr_kinds_eq_h ‹known_ptrs h› h2
by blast
then
have "known_ptrs h3"
using known_ptrs_preserved object_ptr_kinds_eq_h2 by blast
then
have "known_ptrs h'"
using known_ptrs_preserved object_ptr_kinds_eq_h3 by blast
have "document_ptr |∈| document_ptr_kinds h"
using disc_nodes_h3 document_ptr_kinds_eq_h object_ptr_kinds_eq_h2
get_disconnected_nodes_ptr_in_heap ‹type_wf h› document_ptr_kinds_def
by (metis is_OK_returns_result_I)
have children_eq_h: "⋀(ptr'::(_) object_ptr) children. ptr' ≠ cast new_element_ptr
⟹ h ⊢ get_child_nodes ptr' →⇩r children = h2 ⊢ get_child_nodes ptr' →⇩r children"
using get_child_nodes_reads h2 get_child_nodes_new_element[rotated, OF new_element_ptr h2]
apply(auto simp add: reads_def reflp_def transp_def preserved_def)[1]
by blast+
then have children_eq2_h: "⋀ptr'. ptr' ≠ cast new_element_ptr
⟹ |h ⊢ get_child_nodes ptr'|⇩r = |h2 ⊢ get_child_nodes ptr'|⇩r"
using select_result_eq by force
have "h2 ⊢ get_child_nodes (cast new_element_ptr) →⇩r []"
using new_element_ptr h2 new_element_ptr_in_heap[OF h2 new_element_ptr]
new_element_is_element_ptr[OF new_element_ptr] new_element_no_child_nodes
by blast
have disconnected_nodes_eq_h:
"⋀doc_ptr disc_nodes. h ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes
= h2 ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes"
using get_disconnected_nodes_reads h2 get_disconnected_nodes_new_element[OF new_element_ptr h2]
apply(auto simp add: reads_def reflp_def transp_def preserved_def)[1]
by blast+
then have disconnected_nodes_eq2_h:
"⋀doc_ptr. |h ⊢ get_disconnected_nodes doc_ptr|⇩r = |h2 ⊢ get_disconnected_nodes doc_ptr|⇩r"
using select_result_eq by force
have children_eq_h2:
"⋀ptr' children. h2 ⊢ get_child_nodes ptr' →⇩r children = h3 ⊢ get_child_nodes ptr' →⇩r children"
using get_child_nodes_reads set_tag_name_writes h3
apply(rule reads_writes_preserved)
by(auto simp add: set_tag_name_get_child_nodes)
then have children_eq2_h2: "⋀ptr'. |h2 ⊢ get_child_nodes ptr'|⇩r = |h3 ⊢ get_child_nodes ptr'|⇩r"
using select_result_eq by force
have disconnected_nodes_eq_h2:
"⋀doc_ptr disc_nodes. h2 ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes
= h3 ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes"
using get_disconnected_nodes_reads set_tag_name_writes h3
apply(rule reads_writes_preserved)
by(auto simp add: set_tag_name_get_disconnected_nodes)
then have disconnected_nodes_eq2_h2:
"⋀doc_ptr. |h2 ⊢ get_disconnected_nodes doc_ptr|⇩r = |h3 ⊢ get_disconnected_nodes doc_ptr|⇩r"
using select_result_eq by force
have "type_wf h2"
using ‹type_wf h› new_element_types_preserved h2 by blast
then have "type_wf h3"
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'", OF set_tag_name_writes h3]
using set_tag_name_types_preserved
by(auto simp add: reflp_def transp_def)
then have "type_wf h'"
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'", OF set_disconnected_nodes_writes h']
using set_disconnected_nodes_types_preserved
by(auto simp add: reflp_def transp_def)
have children_eq_h3:
"⋀ptr' children. h3 ⊢ get_child_nodes ptr' →⇩r children = h' ⊢ get_child_nodes ptr' →⇩r children"
using get_child_nodes_reads set_disconnected_nodes_writes h'
apply(rule reads_writes_preserved)
by(auto simp add: set_disconnected_nodes_get_child_nodes)
then have children_eq2_h3: "⋀ptr'. |h3 ⊢ get_child_nodes ptr'|⇩r = |h' ⊢ get_child_nodes ptr'|⇩r"
using select_result_eq by force
have disconnected_nodes_eq_h3:
"⋀doc_ptr disc_nodes. document_ptr ≠ doc_ptr
⟹ h3 ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes
= h' ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes"
using get_disconnected_nodes_reads set_disconnected_nodes_writes h'
apply(rule reads_writes_preserved)
by(auto simp add: set_disconnected_nodes_get_disconnected_nodes_different_pointers)
then have disconnected_nodes_eq2_h3:
"⋀doc_ptr. document_ptr ≠ doc_ptr
⟹ |h3 ⊢ get_disconnected_nodes doc_ptr|⇩r = |h' ⊢ get_disconnected_nodes doc_ptr|⇩r"
using select_result_eq by force
have disc_nodes_document_ptr_h2: "h2 ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes_h3"
using disconnected_nodes_eq_h2 disc_nodes_h3 by auto
then have disc_nodes_document_ptr_h: "h ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes_h3"
using disconnected_nodes_eq_h by auto
then have "cast new_element_ptr ∉ set disc_nodes_h3"
using ‹heap_is_wellformed h›
using ‹cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r new_element_ptr ∉ set |h ⊢ node_ptr_kinds_M|⇩r›
a_all_ptrs_in_heap_def heap_is_wellformed_def
using NodeMonad.ptr_kinds_ptr_kinds_M local.heap_is_wellformed_disc_nodes_in_heap by blast
have "acyclic (parent_child_rel h)"
using ‹heap_is_wellformed h›
by (simp add: heap_is_wellformed_def acyclic_heap_def)
also have "parent_child_rel h = parent_child_rel h2"
proof(auto simp add: parent_child_rel_def)[1]
fix a x
assume 0: "a |∈| object_ptr_kinds h"
and 1: "x ∈ set |h ⊢ get_child_nodes a|⇩r"
then show "a |∈| object_ptr_kinds h2"
by (simp add: object_ptr_kinds_eq_h)
next
fix a x
assume 0: "a |∈| object_ptr_kinds h"
and 1: "x ∈ set |h ⊢ get_child_nodes a|⇩r"
then show "x ∈ set |h2 ⊢ get_child_nodes a|⇩r"
by (metis ObjectMonad.ptr_kinds_ptr_kinds_M
‹cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r new_element_ptr ∉ set |h ⊢ object_ptr_kinds_M|⇩r› children_eq2_h)
next
fix a x
assume 0: "a |∈| object_ptr_kinds h2"
and 1: "x ∈ set |h2 ⊢ get_child_nodes a|⇩r"
then show "a |∈| object_ptr_kinds h"
using object_ptr_kinds_eq_h ‹h2 ⊢ get_child_nodes (cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r new_element_ptr) →⇩r []›
by(auto)
next
fix a x
assume 0: "a |∈| object_ptr_kinds h2"
and 1: "x ∈ set |h2 ⊢ get_child_nodes a|⇩r"
then show "x ∈ set |h ⊢ get_child_nodes a|⇩r"
by (metis (no_types, lifting)
‹h2 ⊢ get_child_nodes (cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r new_element_ptr) →⇩r []›
children_eq2_h empty_iff empty_set image_eqI select_result_I2)
qed
also have "… = parent_child_rel h3"
by(auto simp add: parent_child_rel_def object_ptr_kinds_eq_h2 children_eq2_h2)
also have "… = parent_child_rel h'"
by(auto simp add: parent_child_rel_def object_ptr_kinds_eq_h3 children_eq2_h3)
finally have "a_acyclic_heap h'"
by (simp add: acyclic_heap_def)
have "a_all_ptrs_in_heap h"
using ‹heap_is_wellformed h› by (simp add: heap_is_wellformed_def)
then have "a_all_ptrs_in_heap h2"
apply(auto simp add: a_all_ptrs_in_heap_def)[1]
apply (metis ‹known_ptrs h2› ‹parent_child_rel h = parent_child_rel h2› ‹type_wf h2› assms
funion_iff local.get_child_nodes_ok local.known_ptrs_known_ptr local.parent_child_rel_child_in_heap
local.parent_child_rel_child_nodes2 node_ptr_kinds_commutes node_ptr_kinds_eq_h
returns_result_select_result)
by (metis assms disconnected_nodes_eq2_h document_ptr_kinds_eq_h funion_iff
local.get_disconnected_nodes_ok local.heap_is_wellformed_disc_nodes_in_heap node_ptr_kinds_eq_h
returns_result_select_result)
then have "a_all_ptrs_in_heap h3"
by (simp add: children_eq2_h2 disconnected_nodes_eq2_h2 document_ptr_kinds_eq_h2
local.a_all_ptrs_in_heap_def node_ptr_kinds_eq_h2 object_ptr_kinds_eq_h2)
then have "a_all_ptrs_in_heap h'"
by (smt ‹h2 ⊢ get_child_nodes (cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r new_element_ptr) →⇩r []› children_eq2_h3
disc_nodes_document_ptr_h2 disconnected_nodes_eq2_h2 disconnected_nodes_eq2_h3 document_ptr_kinds_eq_h3
finite_set_in h' is_OK_returns_result_I
l_set_disconnected_nodes_get_disconnected_nodes.set_disconnected_nodes_get_disconnected_nodes
local.a_all_ptrs_in_heap_def local.get_child_nodes_ptr_in_heap
local.l_set_disconnected_nodes_get_disconnected_nodes_axioms node_ptr_kinds_commutes
object_ptr_kinds_eq_h2 object_ptr_kinds_eq_h3 select_result_I2 set_ConsD subset_code(1))
have "⋀p. p |∈| object_ptr_kinds h ⟹ cast new_element_ptr ∉ set |h ⊢ get_child_nodes p|⇩r"
using ‹heap_is_wellformed h› ‹cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r new_element_ptr ∉ set |h ⊢ node_ptr_kinds_M|⇩r›
heap_is_wellformed_children_in_heap
by (meson NodeMonad.ptr_kinds_ptr_kinds_M a_all_ptrs_in_heap_def assms fset_mp
fset_of_list_elem get_child_nodes_ok known_ptrs_known_ptr returns_result_select_result)
then have "⋀p. p |∈| object_ptr_kinds h2 ⟹ cast new_element_ptr ∉ set |h2 ⊢ get_child_nodes p|⇩r"
using children_eq2_h
apply(auto simp add: object_ptr_kinds_eq_h)[1]
using ‹h2 ⊢ get_child_nodes (cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r new_element_ptr) →⇩r []› apply auto[1]
by (metis ObjectMonad.ptr_kinds_ptr_kinds_M
‹cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r new_element_ptr ∉ set |h ⊢ object_ptr_kinds_M|⇩r›)
then have "⋀p. p |∈| object_ptr_kinds h3 ⟹ cast new_element_ptr ∉ set |h3 ⊢ get_child_nodes p|⇩r"
using object_ptr_kinds_eq_h2 children_eq2_h2 by auto
then have new_element_ptr_not_in_any_children:
"⋀p. p |∈| object_ptr_kinds h' ⟹ cast new_element_ptr ∉ set |h' ⊢ get_child_nodes p|⇩r"
using object_ptr_kinds_eq_h3 children_eq2_h3 by auto
have "a_distinct_lists h"
using ‹heap_is_wellformed h›
by (simp add: heap_is_wellformed_def)
then have "a_distinct_lists h2"
using ‹h2 ⊢ get_child_nodes (cast new_element_ptr) →⇩r []›
apply(auto simp add: a_distinct_lists_def object_ptr_kinds_eq_h document_ptr_kinds_eq_h
disconnected_nodes_eq2_h intro!: distinct_concat_map_I)[1]
apply (metis distinct_sorted_list_of_set finite_fset sorted_list_of_set_insert)
apply(case_tac "x=cast new_element_ptr")
apply(auto simp add: children_eq2_h[symmetric] insort_split dest: distinct_concat_map_E(2))[1]
apply(auto simp add: children_eq2_h[symmetric] insort_split dest: distinct_concat_map_E(2))[1]
apply(auto simp add: children_eq2_h[symmetric] insort_split dest: distinct_concat_map_E(2))[1]
apply (metis IntI assms empty_iff local.get_child_nodes_ok
local.heap_is_wellformed_one_parent local.known_ptrs_known_ptr returns_result_select_result)
apply(auto simp add: children_eq2_h[symmetric] insort_split dest: distinct_concat_map_E(2))[1]
by (metis ‹local.a_distinct_lists h› ‹type_wf h2› disconnected_nodes_eq_h document_ptr_kinds_eq_h
local.distinct_lists_no_parent local.get_disconnected_nodes_ok returns_result_select_result)
then have "a_distinct_lists h3"
by(auto simp add: a_distinct_lists_def disconnected_nodes_eq2_h2 document_ptr_kinds_eq_h2
children_eq2_h2 object_ptr_kinds_eq_h2)
then have "a_distinct_lists h'"
proof(auto simp add: a_distinct_lists_def disconnected_nodes_eq2_h3 children_eq2_h3
object_ptr_kinds_eq_h3 document_ptr_kinds_eq_h3
intro!: distinct_concat_map_I)[1]
fix x
assume "distinct (concat (map (λdocument_ptr. |h3 ⊢ get_disconnected_nodes document_ptr|⇩r)
(sorted_list_of_set (fset (document_ptr_kinds h3)))))"
and "x |∈| document_ptr_kinds h3"
then show "distinct |h' ⊢ get_disconnected_nodes x|⇩r"
using document_ptr_kinds_eq_h3 disconnected_nodes_eq_h3 h' set_disconnected_nodes_get_disconnected_nodes
by (metis (no_types, lifting) ‹cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r new_element_ptr ∉ set disc_nodes_h3›
‹a_distinct_lists h3› ‹type_wf h'› disc_nodes_h3 distinct.simps(2)
distinct_lists_disconnected_nodes get_disconnected_nodes_ok returns_result_eq
returns_result_select_result)
next
fix x y xa
assume "distinct (concat (map (λdocument_ptr. |h3 ⊢ get_disconnected_nodes document_ptr|⇩r)
(sorted_list_of_set (fset (document_ptr_kinds h3)))))"
and "x |∈| document_ptr_kinds h3"
and "y |∈| document_ptr_kinds h3"
and "x ≠ y"
and "xa ∈ set |h' ⊢ get_disconnected_nodes x|⇩r"
and "xa ∈ set |h' ⊢ get_disconnected_nodes y|⇩r"
moreover have "set |h3 ⊢ get_disconnected_nodes x|⇩r ∩ set |h3 ⊢ get_disconnected_nodes y|⇩r = {}"
using calculation by(auto dest: distinct_concat_map_E(1))
ultimately show "False"
apply(-)
apply(cases "x = document_ptr")
apply (smt NodeMonad.ptr_kinds_ptr_kinds_M
‹cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r new_element_ptr ∉ set |h ⊢ node_ptr_kinds_M|⇩r› ‹local.a_all_ptrs_in_heap h›
disc_nodes_h3 disconnected_nodes_eq2_h disconnected_nodes_eq2_h2 disconnected_nodes_eq2_h3
disjoint_iff_not_equal document_ptr_kinds_eq_h document_ptr_kinds_eq_h2 finite_set_in h'
l_set_disconnected_nodes_get_disconnected_nodes.set_disconnected_nodes_get_disconnected_nodes
local.a_all_ptrs_in_heap_def local.l_set_disconnected_nodes_get_disconnected_nodes_axioms
select_result_I2 set_ConsD subsetD)
by (smt NodeMonad.ptr_kinds_ptr_kinds_M
‹cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r new_element_ptr ∉ set |h ⊢ node_ptr_kinds_M|⇩r› ‹local.a_all_ptrs_in_heap h›
disc_nodes_document_ptr_h2 disconnected_nodes_eq2_h disconnected_nodes_eq2_h2
disconnected_nodes_eq2_h3
disjoint_iff_not_equal document_ptr_kinds_eq_h document_ptr_kinds_eq_h2 finite_set_in h'
l_set_disconnected_nodes_get_disconnected_nodes.set_disconnected_nodes_get_disconnected_nodes
local.a_all_ptrs_in_heap_def local.l_set_disconnected_nodes_get_disconnected_nodes_axioms
select_result_I2 set_ConsD subsetD)
next
fix x xa xb
assume 2: "(⋃x∈fset (object_ptr_kinds h3). set |h' ⊢ get_child_nodes x|⇩r)
∩ (⋃x∈fset (document_ptr_kinds h3). set |h3 ⊢ get_disconnected_nodes x|⇩r) = {}"
and 3: "xa |∈| object_ptr_kinds h3"
and 4: "x ∈ set |h' ⊢ get_child_nodes xa|⇩r"
and 5: "xb |∈| document_ptr_kinds h3"
and 6: "x ∈ set |h' ⊢ get_disconnected_nodes xb|⇩r"
show "False"
using disc_nodes_document_ptr_h disconnected_nodes_eq2_h3
apply -
apply(cases "xb = document_ptr")
apply (metis (no_types, hide_lams) "3" "4" "6"
‹⋀p. p |∈| object_ptr_kinds h3
⟹ cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r new_element_ptr ∉ set |h3 ⊢ get_child_nodes p|⇩r›
‹a_distinct_lists h3› children_eq2_h3 disc_nodes_h3 distinct_lists_no_parent h'
select_result_I2 set_ConsD set_disconnected_nodes_get_disconnected_nodes)
by (metis "3" "4" "5" "6" ‹a_distinct_lists h3› ‹type_wf h3› children_eq2_h3
distinct_lists_no_parent get_disconnected_nodes_ok returns_result_select_result)
qed
have "a_owner_document_valid h"
using ‹heap_is_wellformed h› by (simp add: heap_is_wellformed_def)
then have "a_owner_document_valid h'"
using disc_nodes_h3 ‹document_ptr |∈| document_ptr_kinds h›
apply(auto simp add: a_owner_document_valid_def)[1]
apply(auto simp add: object_ptr_kinds_eq_h object_ptr_kinds_eq_h3 )[1]
apply(auto simp add: object_ptr_kinds_eq_h2)[1]
apply(auto simp add: document_ptr_kinds_eq_h document_ptr_kinds_eq_h3 )[1]
apply(auto simp add: document_ptr_kinds_eq_h2)[1]
apply(auto simp add: node_ptr_kinds_eq_h node_ptr_kinds_eq_h3 )[1]
apply(auto simp add: node_ptr_kinds_eq_h2 node_ptr_kinds_eq_h )[1]
apply(auto simp add: children_eq2_h2[symmetric] children_eq2_h3[symmetric]
disconnected_nodes_eq2_h disconnected_nodes_eq2_h2
disconnected_nodes_eq2_h3)[1]
apply (metis (no_types, lifting) document_ptr_kinds_eq_h h' list.set_intros(1)
local.set_disconnected_nodes_get_disconnected_nodes select_result_I2)
apply(simp add: object_ptr_kinds_eq_h)
by(metis (no_types, lifting) NodeMonad.ptr_kinds_ptr_kinds_M
‹cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r new_element_ptr ∉ set |h ⊢ node_ptr_kinds_M|⇩r› children_eq2_h
children_eq2_h2 children_eq2_h3 disconnected_nodes_eq2_h disconnected_nodes_eq2_h2
disconnected_nodes_eq2_h3 document_ptr_kinds_eq_h finite_set_in h'
l_set_disconnected_nodes_get_disconnected_nodes.set_disconnected_nodes_get_disconnected_nodes
list.set_intros(2) local.l_set_disconnected_nodes_get_disconnected_nodes_axioms
node_ptr_kinds_commutes select_result_I2)
have "parent_child_rel h = parent_child_rel h'"
proof -
have "parent_child_rel h = parent_child_rel h2"
proof(auto simp add: parent_child_rel_def)[1]
fix a x
assume 0: "a |∈| object_ptr_kinds h"
and 1: "x ∈ set |h ⊢ get_child_nodes a|⇩r"
then show "a |∈| object_ptr_kinds h2"
by (simp add: object_ptr_kinds_eq_h)
next
fix a x
assume 0: "a |∈| object_ptr_kinds h"
and 1: "x ∈ set |h ⊢ get_child_nodes a|⇩r"
then show "x ∈ set |h2 ⊢ get_child_nodes a|⇩r"
by (metis ObjectMonad.ptr_kinds_ptr_kinds_M
‹cast new_element_ptr ∉ set |h ⊢ object_ptr_kinds_M|⇩r› children_eq2_h)
next
fix a x
assume 0: "a |∈| object_ptr_kinds h2"
and 1: "x ∈ set |h2 ⊢ get_child_nodes a|⇩r"
then show "a |∈| object_ptr_kinds h"
using object_ptr_kinds_eq_h ‹h2 ⊢ get_child_nodes (cast new_element_ptr) →⇩r []›
by(auto)
next
fix a x
assume 0: "a |∈| object_ptr_kinds h2"
and 1: "x ∈ set |h2 ⊢ get_child_nodes a|⇩r"
then show "x ∈ set |h ⊢ get_child_nodes a|⇩r"
by (metis (no_types, lifting) ‹h2 ⊢ get_child_nodes (cast new_element_ptr) →⇩r []›
children_eq2_h empty_iff empty_set image_eqI select_result_I2)
qed
also have "… = parent_child_rel h3"
by(auto simp add: parent_child_rel_def object_ptr_kinds_eq_h2 children_eq2_h2)
also have "… = parent_child_rel h'"
by(auto simp add: parent_child_rel_def object_ptr_kinds_eq_h3 children_eq2_h3)
finally show ?thesis
by simp
qed
have root: "h ⊢ get_root_node (cast document_ptr) →⇩r cast document_ptr"
by (simp add: ‹document_ptr |∈| document_ptr_kinds h› local.get_root_node_not_node_same)
then
have root': "h' ⊢ get_root_node (cast document_ptr) →⇩r cast document_ptr"
by (simp add: ‹document_ptr |∈| document_ptr_kinds h› document_ptr_kinds_eq_h
local.get_root_node_not_node_same object_ptr_kinds_eq_h2 object_ptr_kinds_eq_h3)
have "heap_is_wellformed h'" and "known_ptrs h'"
using create_element_preserves_wellformedness assms
by blast+
have "cast result |∉| object_ptr_kinds h"
using ‹cast new_element_ptr ∉ set |h ⊢ object_ptr_kinds_M|⇩r›
by (metis (full_types) ObjectMonad.ptr_kinds_ptr_kinds_M
‹h ⊢ create_element document_ptr tag →⇩r new_element_ptr› assms(4) returns_result_eq)
obtain to where to: "h ⊢ to_tree_order (cast document_ptr) →⇩r to"
by (meson ‹document_ptr |∈| document_ptr_kinds h› assms(1) assms(2) assms(3)
document_ptr_kinds_commutes is_OK_returns_result_E local.to_tree_order_ok)
then
have "h ⊢ a_get_dom_component (cast document_ptr) →⇩r to"
using root
by(auto simp add: a_get_dom_component_def)
moreover
obtain to' where to': "h' ⊢ to_tree_order (cast document_ptr) →⇩r to'"
by (meson ‹heap_is_wellformed h'› ‹known_ptrs h'› ‹type_wf h'› is_OK_returns_result_E
local.get_root_node_root_in_heap local.to_tree_order_ok root')
then
have "h' ⊢ a_get_dom_component (cast document_ptr) →⇩r to'"
using root'
by(auto simp add: a_get_dom_component_def)
moreover
have "⋀child. child ∈ set to ⟷ child ∈ set to'"
by (metis ‹heap_is_wellformed h'› ‹known_ptrs h'› ‹parent_child_rel h = parent_child_rel h'›
‹type_wf h'› assms(1) assms(2) assms(3) local.to_tree_order_parent_child_rel to to')
ultimately
have "set |h ⊢ local.a_get_dom_component (cast document_ptr)|⇩r =
set |h' ⊢ local.a_get_dom_component (cast document_ptr)|⇩r"
by(auto simp add: a_get_dom_component_def)
show ?thesis
apply(auto simp add: is_weakly_dom_component_safe_def Let_def)[1]
using ‹h2 ⊢ get_child_nodes (cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r new_element_ptr) →⇩r []› assms(2) assms(3)
children_eq_h local.get_child_nodes_ok local.get_child_nodes_ptr_in_heap local.known_ptrs_known_ptr
object_ptr_kinds_eq_h2 object_ptr_kinds_eq_h3 returns_result_select_result
apply (metis is_OK_returns_result_I)
apply (metis ‹h ⊢ create_element document_ptr tag →⇩r new_element_ptr› assms(4)
element_ptr_kinds_commutes h2 new_element_ptr new_element_ptr_in_heap node_ptr_kinds_eq_h2
node_ptr_kinds_eq_h3 returns_result_eq returns_result_heap_def)
using ‹cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r result |∉| object_ptr_kinds h› element_ptr_kinds_commutes
node_ptr_kinds_commutes apply blast
using assms(1) assms(2) assms(3) ‹h ⊢ create_element document_ptr tag →⇩h h'›
apply(rule create_element_is_weakly_dom_component_safe_step)
apply (simp add: local.get_dom_component_impl)
using ‹cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r new_element_ptr ∉ set |h ⊢ object_ptr_kinds_M|⇩r›
‹h ⊢ create_element document_ptr tag →⇩r new_element_ptr›
by auto
qed
end
interpretation i_get_dom_component_create_element?: l_get_dom_component_create_element⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
known_ptr heap_is_wellformed parent_child_rel type_wf known_ptrs to_tree_order get_parent
get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe
is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs
get_element_by_id get_elements_by_class_name get_elements_by_tag_name get_disconnected_nodes
get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs set_tag_name
set_tag_name_locs create_element
by(auto simp add: l_get_dom_component_create_element⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def instances)
declare l_get_dom_component_create_element⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms [instances]
subsubsection ‹create\_character\_data›
lemma create_character_data_not_strongly_dom_component_safe:
obtains
h :: "('object_ptr::{equal,linorder}, 'node_ptr::{equal,linorder},
'element_ptr::{equal,linorder}, 'character_data_ptr::{equal,linorder},
'document_ptr::{equal,linorder}, 'shadow_root_ptr::{equal,linorder},
'Object::{equal,linorder}, 'Node::{equal,linorder}, 'Element::{equal,linorder},
'CharacterData::{equal,linorder}, 'Document::{equal,linorder}) heap" and
h' and document_ptr and create_character_datanew_character_data_ptr and tag where
"heap_is_wellformed h" and "type_wf h" and "known_ptrs h" and
"h ⊢ create_character_data document_ptr tag →⇩r create_character_datanew_character_data_ptr →⇩h h'" and
"¬ is_strongly_dom_component_safe {cast document_ptr} {cast create_character_datanew_character_data_ptr} h h'"
proof -
let ?h0 = "Heap fmempty ::('object_ptr::{equal,linorder}, 'node_ptr::{equal,linorder},
'element_ptr::{equal,linorder}, 'character_data_ptr::{equal,linorder}, 'document_ptr::{equal,linorder},
'shadow_root_ptr::{equal,linorder}, 'Object::{equal,linorder}, 'Node::{equal,linorder},
'Element::{equal,linorder}, 'CharacterData::{equal,linorder}, 'Document::{equal,linorder}) heap"
let ?P = "create_document"
let ?h1 = "|?h0 ⊢ ?P|⇩h"
let ?document_ptr = "|?h0 ⊢ ?P|⇩r"
show thesis
apply(rule that[where h="?h1" and document_ptr="?document_ptr"])
by code_simp+
qed
locale l_get_dom_component_create_character_data⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_get_dom_component⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs
to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component
is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs
get_ancestors get_ancestors_locs get_disconnected_nodes get_disconnected_nodes_locs
get_element_by_id get_elements_by_class_name get_elements_by_tag_name +
l_create_character_data⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes
set_disconnected_nodes_locs set_val set_val_locs type_wf create_character_data known_ptr +
l_get_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf get_disconnected_nodes get_disconnected_nodes_locs +
l_set_disconnected_nodes⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf set_disconnected_nodes set_disconnected_nodes_locs +
l_set_val⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M type_wf set_val set_val_locs +
l_create_character_data_wf⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M known_ptr type_wf get_child_nodes get_child_nodes_locs
get_disconnected_nodes get_disconnected_nodes_locs heap_is_wellformed parent_child_rel set_val
set_val_locs set_disconnected_nodes set_disconnected_nodes_locs
create_character_data known_ptrs
for known_ptr :: "(_::linorder) object_ptr ⇒ bool"
and heap_is_wellformed :: "(_) heap ⇒ bool"
and parent_child_rel :: "(_) heap ⇒ ((_) object_ptr × (_) object_ptr) set"
and type_wf :: "(_) heap ⇒ bool"
and known_ptrs :: "(_) heap ⇒ bool"
and to_tree_order :: "(_) object_ptr ⇒ ((_) heap, exception, (_) object_ptr list) prog"
and get_parent :: "(_) node_ptr ⇒ ((_) heap, exception, (_) object_ptr option) prog"
and get_parent_locs :: "((_) heap ⇒ (_) heap ⇒ bool) set"
and get_child_nodes :: "(_) object_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_child_nodes_locs :: "(_) object_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and get_dom_component :: "(_) object_ptr ⇒ ((_) heap, exception, (_) object_ptr list) prog"
and is_strongly_dom_component_safe :: "(_) object_ptr set ⇒ (_) object_ptr set ⇒ (_) heap ⇒ (_) heap ⇒ bool"
and is_weakly_dom_component_safe :: "(_) object_ptr set ⇒ (_) object_ptr set ⇒ (_) heap ⇒ (_) heap ⇒ bool"
and get_root_node :: "(_) object_ptr ⇒ ((_) heap, exception, (_) object_ptr) prog"
and get_root_node_locs :: "((_) heap ⇒ (_) heap ⇒ bool) set"
and get_ancestors :: "(_) object_ptr ⇒ ((_) heap, exception, (_) object_ptr list) prog"
and get_ancestors_locs :: "((_) heap ⇒ (_) heap ⇒ bool) set"
and get_element_by_id :: "(_) object_ptr ⇒ char list ⇒ ((_) heap, exception, (_) element_ptr option) prog"
and get_elements_by_class_name :: "(_) object_ptr ⇒ char list ⇒ ((_) heap, exception, (_) element_ptr list) prog"
and get_elements_by_tag_name :: "(_) object_ptr ⇒ char list ⇒ ((_) heap, exception, (_) element_ptr list) prog"
and set_val :: "(_) character_data_ptr ⇒ char list ⇒ ((_) heap, exception, unit) prog"
and set_val_locs :: "(_) character_data_ptr ⇒ ((_) heap, exception, unit) prog set"
and get_disconnected_nodes :: "(_) document_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and set_disconnected_nodes :: "(_) document_ptr ⇒ (_) node_ptr list ⇒ ((_) heap, exception, unit) prog"
and set_disconnected_nodes_locs :: "(_) document_ptr ⇒ ((_) heap, exception, unit) prog set"
and create_character_data :: "(_) document_ptr ⇒ char list ⇒ ((_) heap, exception, (_) character_data_ptr) prog"
begin
lemma create_character_data_is_weakly_dom_component_safe_step:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ create_character_data document_ptr text →⇩h h'"
assumes "ptr ∉ set |h ⊢ get_dom_component (cast document_ptr)|⇩r"
assumes "ptr ≠ cast |h ⊢ create_character_data document_ptr text|⇩r"
shows "preserved (get_M ptr getter) h h'"
proof -
obtain new_character_data_ptr h2 h3 disc_nodes where
new_character_data_ptr: "h ⊢ new_character_data →⇩r new_character_data_ptr" and
h2: "h ⊢ new_character_data →⇩h h2" and
h3: "h2 ⊢ set_val new_character_data_ptr text →⇩h h3" and
disc_nodes: "h3 ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes" and
h': "h3 ⊢ set_disconnected_nodes document_ptr (cast new_character_data_ptr # disc_nodes) →⇩h h'"
using assms(4)
by(auto simp add: create_character_data_def elim!: bind_returns_heap_E
bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated])
have "h ⊢ create_character_data document_ptr text →⇩r new_character_data_ptr"
using new_character_data_ptr h2 h3 disc_nodes h'
apply(auto simp add: create_character_data_def intro!: bind_returns_result_I
bind_pure_returns_result_I[OF get_disconnected_nodes_pure])[1]
apply (metis is_OK_returns_heap_I is_OK_returns_result_E old.unit.exhaust)
by (metis is_OK_returns_heap_I is_OK_returns_result_E old.unit.exhaust)
have "preserved (get_M ptr getter) h h2"
using h2 new_character_data_ptr
apply(rule new_character_data_get_M⇩O⇩b⇩j⇩e⇩c⇩t)
using new_character_data_ptr assms(6)
‹h ⊢ create_character_data document_ptr text →⇩r new_character_data_ptr›
by simp
have "preserved (get_M ptr getter) h2 h3"
using set_val_writes h3
apply(rule reads_writes_preserved2)
apply(auto simp add: set_val_locs_impl a_set_val_locs_def all_args_def)[1]
by (metis (mono_tags) CharacterData_simp11
‹h ⊢ create_character_data document_ptr text →⇩r new_character_data_ptr› assms(4) assms(6)
is_OK_returns_heap_I is_OK_returns_result_E returns_result_eq select_result_I2)
have "document_ptr |∈| document_ptr_kinds h"
using create_character_data_document_in_heap
using assms(4)
by blast
then
have "ptr ≠ (cast document_ptr)"
using assms(5) assms(1) assms(2) assms(3) local.get_dom_component_ok local.get_dom_component_ptr
by auto
have "preserved (get_M ptr getter) h3 h'"
using set_disconnected_nodes_writes h'
apply(rule reads_writes_preserved2)
apply(auto simp add: set_disconnected_nodes_locs_def all_args_def)[1]
by (metis ‹ptr ≠ cast⇩d⇩o⇩c⇩u⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r document_ptr› get_M_Mdocument_preserved3)
show ?thesis
using ‹preserved (get_M ptr getter) h h2› ‹preserved (get_M ptr getter) h2 h3›
‹preserved (get_M ptr getter) h3 h'›
by(auto simp add: preserved_def)
qed
lemma create_character_data_is_weakly_dom_component_safe:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ create_character_data document_ptr text →⇩r result"
assumes "h ⊢ create_character_data document_ptr text →⇩h h'"
shows "is_weakly_dom_component_safe {cast document_ptr} {cast result} h h'"
proof -
obtain new_character_data_ptr h2 h3 disc_nodes_h3 where
new_character_data_ptr: "h ⊢ new_character_data →⇩r new_character_data_ptr" and
h2: "h ⊢ new_character_data →⇩h h2" and
h3: "h2 ⊢ set_val new_character_data_ptr text →⇩h h3" and
disc_nodes_h3: "h3 ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes_h3" and
h': "h3 ⊢ set_disconnected_nodes document_ptr (cast new_character_data_ptr # disc_nodes_h3) →⇩h h'"
using assms(5)
by(auto simp add: create_character_data_def
elim!: bind_returns_heap_E
bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated] )
then
have "h ⊢ create_character_data document_ptr text →⇩r new_character_data_ptr"
apply(auto simp add: create_character_data_def intro!: bind_returns_result_I)[1]
apply (metis is_OK_returns_heap_I is_OK_returns_result_E old.unit.exhaust)
apply (metis is_OK_returns_heap_E is_OK_returns_result_I local.get_disconnected_nodes_pure
pure_returns_heap_eq)
by (metis is_OK_returns_heap_I is_OK_returns_result_E old.unit.exhaust)
have "new_character_data_ptr ∉ set |h ⊢ character_data_ptr_kinds_M|⇩r"
using new_character_data_ptr CharacterDataMonad.ptr_kinds_ptr_kinds_M h2
using new_character_data_ptr_not_in_heap by blast
then have "cast new_character_data_ptr ∉ set |h ⊢ node_ptr_kinds_M|⇩r"
by simp
then have "cast new_character_data_ptr ∉ set |h ⊢ object_ptr_kinds_M|⇩r"
by simp
have object_ptr_kinds_eq_h:
"object_ptr_kinds h2 = object_ptr_kinds h |∪| {|cast new_character_data_ptr|}"
using new_character_data_new_ptr h2 new_character_data_ptr by blast
then have node_ptr_kinds_eq_h:
"node_ptr_kinds h2 = node_ptr_kinds h |∪| {|cast new_character_data_ptr|}"
apply(simp add: node_ptr_kinds_def)
by force
then have character_data_ptr_kinds_eq_h:
"character_data_ptr_kinds h2 = character_data_ptr_kinds h |∪| {|new_character_data_ptr|}"
apply(simp add: character_data_ptr_kinds_def)
by force
have element_ptr_kinds_eq_h: "element_ptr_kinds h2 = element_ptr_kinds h"
using object_ptr_kinds_eq_h
by(auto simp add: node_ptr_kinds_def element_ptr_kinds_def)
have document_ptr_kinds_eq_h: "document_ptr_kinds h2 = document_ptr_kinds h"
using object_ptr_kinds_eq_h
by(auto simp add: document_ptr_kinds_def)
have object_ptr_kinds_eq_h2: "object_ptr_kinds h3 = object_ptr_kinds h2"
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h' = object_ptr_kinds h",
OF set_val_writes h3])
using set_val_pointers_preserved
by (auto simp add: reflp_def transp_def)
then have document_ptr_kinds_eq_h2: "document_ptr_kinds h3 = document_ptr_kinds h2"
by (auto simp add: document_ptr_kinds_def)
have node_ptr_kinds_eq_h2: "node_ptr_kinds h3 = node_ptr_kinds h2"
using object_ptr_kinds_eq_h2
by(auto simp add: node_ptr_kinds_def)
have object_ptr_kinds_eq_h3: "object_ptr_kinds h' = object_ptr_kinds h3"
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h' = object_ptr_kinds h",
OF set_disconnected_nodes_writes h'])
using set_disconnected_nodes_pointers_preserved
by (auto simp add: reflp_def transp_def)
then have document_ptr_kinds_eq_h3: "document_ptr_kinds h' = document_ptr_kinds h3"
by (auto simp add: document_ptr_kinds_def)
have node_ptr_kinds_eq_h3: "node_ptr_kinds h' = node_ptr_kinds h3"
using object_ptr_kinds_eq_h3
by(auto simp add: node_ptr_kinds_def)
have "document_ptr |∈| document_ptr_kinds h"
using disc_nodes_h3 document_ptr_kinds_eq_h object_ptr_kinds_eq_h2
get_disconnected_nodes_ptr_in_heap ‹type_wf h› document_ptr_kinds_def
by (metis is_OK_returns_result_I)
have children_eq_h: "⋀(ptr'::(_) object_ptr) children. ptr' ≠ cast new_character_data_ptr
⟹ h ⊢ get_child_nodes ptr' →⇩r children = h2 ⊢ get_child_nodes ptr' →⇩r children"
using get_child_nodes_reads h2 get_child_nodes_new_character_data[rotated, OF new_character_data_ptr h2]
apply(auto simp add: reads_def reflp_def transp_def preserved_def)[1]
by blast+
then have children_eq2_h:
"⋀ptr'. ptr' ≠ cast new_character_data_ptr
⟹ |h ⊢ get_child_nodes ptr'|⇩r = |h2 ⊢ get_child_nodes ptr'|⇩r"
using select_result_eq by force
have object_ptr_kinds_eq_h:
"object_ptr_kinds h2 = object_ptr_kinds h |∪| {|cast new_character_data_ptr|}"
using new_character_data_new_ptr h2 new_character_data_ptr by blast
then have node_ptr_kinds_eq_h:
"node_ptr_kinds h2 = node_ptr_kinds h |∪| {|cast new_character_data_ptr|}"
apply(simp add: node_ptr_kinds_def)
by force
then have character_data_ptr_kinds_eq_h:
"character_data_ptr_kinds h2 = character_data_ptr_kinds h |∪| {|new_character_data_ptr|}"
apply(simp add: character_data_ptr_kinds_def)
by force
have element_ptr_kinds_eq_h: "element_ptr_kinds h2 = element_ptr_kinds h"
using object_ptr_kinds_eq_h
by(auto simp add: node_ptr_kinds_def element_ptr_kinds_def)
have document_ptr_kinds_eq_h: "document_ptr_kinds h2 = document_ptr_kinds h"
using object_ptr_kinds_eq_h
by(auto simp add: document_ptr_kinds_def)
have object_ptr_kinds_eq_h2: "object_ptr_kinds h3 = object_ptr_kinds h2"
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h' = object_ptr_kinds h",
OF set_val_writes h3])
using set_val_pointers_preserved
by (auto simp add: reflp_def transp_def)
then have document_ptr_kinds_eq_h2: "document_ptr_kinds h3 = document_ptr_kinds h2"
by (auto simp add: document_ptr_kinds_def)
have node_ptr_kinds_eq_h2: "node_ptr_kinds h3 = node_ptr_kinds h2"
using object_ptr_kinds_eq_h2
by(auto simp add: node_ptr_kinds_def)
have object_ptr_kinds_eq_h3: "object_ptr_kinds h' = object_ptr_kinds h3"
apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h' = object_ptr_kinds h",
OF set_disconnected_nodes_writes h'])
using set_disconnected_nodes_pointers_preserved
by (auto simp add: reflp_def transp_def)
then have document_ptr_kinds_eq_h3: "document_ptr_kinds h' = document_ptr_kinds h3"
by (auto simp add: document_ptr_kinds_def)
have node_ptr_kinds_eq_h3: "node_ptr_kinds h' = node_ptr_kinds h3"
using object_ptr_kinds_eq_h3
by(auto simp add: node_ptr_kinds_def)
have "document_ptr |∈| document_ptr_kinds h"
using disc_nodes_h3 document_ptr_kinds_eq_h object_ptr_kinds_eq_h2
get_disconnected_nodes_ptr_in_heap ‹type_wf h› document_ptr_kinds_def
by (metis is_OK_returns_result_I)
have children_eq_h: "⋀(ptr'::(_) object_ptr) children. ptr' ≠ cast new_character_data_ptr
⟹ h ⊢ get_child_nodes ptr' →⇩r children = h2 ⊢ get_child_nodes ptr' →⇩r children"
using get_child_nodes_reads h2 get_child_nodes_new_character_data[rotated, OF new_character_data_ptr h2]
apply(auto simp add: reads_def reflp_def transp_def preserved_def)[1]
by blast+
then have children_eq2_h: "⋀ptr'. ptr' ≠ cast new_character_data_ptr
⟹ |h ⊢ get_child_nodes ptr'|⇩r = |h2 ⊢ get_child_nodes ptr'|⇩r"
using select_result_eq by force
have "h2 ⊢ get_child_nodes (cast new_character_data_ptr) →⇩r []"
using new_character_data_ptr h2 new_character_data_ptr_in_heap[OF h2 new_character_data_ptr]
new_character_data_is_character_data_ptr[OF new_character_data_ptr]
new_character_data_no_child_nodes
by blast
have disconnected_nodes_eq_h:
"⋀doc_ptr disc_nodes. h ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes
= h2 ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes"
using get_disconnected_nodes_reads h2
get_disconnected_nodes_new_character_data[OF new_character_data_ptr h2]
apply(auto simp add: reads_def reflp_def transp_def preserved_def)[1]
by blast+
then have disconnected_nodes_eq2_h:
"⋀doc_ptr. |h ⊢ get_disconnected_nodes doc_ptr|⇩r = |h2 ⊢ get_disconnected_nodes doc_ptr|⇩r"
using select_result_eq by force
have children_eq_h2:
"⋀ptr' children. h2 ⊢ get_child_nodes ptr' →⇩r children = h3 ⊢ get_child_nodes ptr' →⇩r children"
using get_child_nodes_reads set_val_writes h3
apply(rule reads_writes_preserved)
by(auto simp add: set_val_get_child_nodes)
then have children_eq2_h2:
"⋀ptr'. |h2 ⊢ get_child_nodes ptr'|⇩r = |h3 ⊢ get_child_nodes ptr'|⇩r"
using select_result_eq by force
have disconnected_nodes_eq_h2:
"⋀doc_ptr disc_nodes. h2 ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes
= h3 ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes"
using get_disconnected_nodes_reads set_val_writes h3
apply(rule reads_writes_preserved)
by(auto simp add: set_val_get_disconnected_nodes)
then have disconnected_nodes_eq2_h2:
"⋀doc_ptr. |h2 ⊢ get_disconnected_nodes doc_ptr|⇩r = |h3 ⊢ get_disconnected_nodes doc_ptr|⇩r"
using select_result_eq by force
have "type_wf h2"
using ‹type_wf h› new_character_data_types_preserved h2 by blast
then have "type_wf h3"
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'", OF set_val_writes h3]
using set_val_types_preserved
by(auto simp add: reflp_def transp_def)
then have "type_wf h'"
using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'", OF set_disconnected_nodes_writes h']
using set_disconnected_nodes_types_preserved
by(auto simp add: reflp_def transp_def)
have children_eq_h3:
"⋀ptr' children. h3 ⊢ get_child_nodes ptr' →⇩r children = h' ⊢ get_child_nodes ptr' →⇩r children"
using get_child_nodes_reads set_disconnected_nodes_writes h'
apply(rule reads_writes_preserved)
by(auto simp add: set_disconnected_nodes_get_child_nodes)
then have children_eq2_h3:
" ⋀ptr'. |h3 ⊢ get_child_nodes ptr'|⇩r = |h' ⊢ get_child_nodes ptr'|⇩r"
using select_result_eq by force
have disconnected_nodes_eq_h3: "⋀doc_ptr disc_nodes. document_ptr ≠ doc_ptr
⟹ h3 ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes
= h' ⊢ get_disconnected_nodes doc_ptr →⇩r disc_nodes"
using get_disconnected_nodes_reads set_disconnected_nodes_writes h'
apply(rule reads_writes_preserved)
by(auto simp add: set_disconnected_nodes_get_disconnected_nodes_different_pointers)
then have disconnected_nodes_eq2_h3: "⋀doc_ptr. document_ptr ≠ doc_ptr
⟹ |h3 ⊢ get_disconnected_nodes doc_ptr|⇩r = |h' ⊢ get_disconnected_nodes doc_ptr|⇩r"
using select_result_eq by force
have disc_nodes_document_ptr_h2: "h2 ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes_h3"
using disconnected_nodes_eq_h2 disc_nodes_h3 by auto
then have disc_nodes_document_ptr_h: "h ⊢ get_disconnected_nodes document_ptr →⇩r disc_nodes_h3"
using disconnected_nodes_eq_h by auto
then have "cast new_character_data_ptr ∉ set disc_nodes_h3"
using ‹heap_is_wellformed h› using ‹cast new_character_data_ptr ∉ set |h ⊢ node_ptr_kinds_M|⇩r›
a_all_ptrs_in_heap_def heap_is_wellformed_def
using NodeMonad.ptr_kinds_ptr_kinds_M local.heap_is_wellformed_disc_nodes_in_heap by blast
have "parent_child_rel h = parent_child_rel h'"
proof -
have "parent_child_rel h = parent_child_rel h2"
proof(auto simp add: parent_child_rel_def)[1]
fix a x
assume 0: "a |∈| object_ptr_kinds h"
and 1: "x ∈ set |h ⊢ get_child_nodes a|⇩r"
then show "a |∈| object_ptr_kinds h2"
by (simp add: object_ptr_kinds_eq_h)
next
fix a x
assume 0: "a |∈| object_ptr_kinds h"
and 1: "x ∈ set |h ⊢ get_child_nodes a|⇩r"
then show "x ∈ set |h2 ⊢ get_child_nodes a|⇩r"
by (metis ObjectMonad.ptr_kinds_ptr_kinds_M
‹cast new_character_data_ptr ∉ set |h ⊢ object_ptr_kinds_M|⇩r› children_eq2_h)
next
fix a x
assume 0: "a |∈| object_ptr_kinds h2"
and 1: "x ∈ set |h2 ⊢ get_child_nodes a|⇩r"
then show "a |∈| object_ptr_kinds h"
using object_ptr_kinds_eq_h ‹h2 ⊢ get_child_nodes (cast new_character_data_ptr) →⇩r []›
by(auto)
next
fix a x
assume 0: "a |∈| object_ptr_kinds h2"
and 1: "x ∈ set |h2 ⊢ get_child_nodes a|⇩r"
then show "x ∈ set |h ⊢ get_child_nodes a|⇩r"
by (metis (no_types, lifting) ‹h2 ⊢ get_child_nodes (cast new_character_data_ptr) →⇩r []›
children_eq2_h empty_iff empty_set image_eqI select_result_I2)
qed
also have "… = parent_child_rel h3"
by(auto simp add: parent_child_rel_def object_ptr_kinds_eq_h2 children_eq2_h2)
also have "… = parent_child_rel h'"
by(auto simp add: parent_child_rel_def object_ptr_kinds_eq_h3 children_eq2_h3)
finally show ?thesis
by simp
qed
have root: "h ⊢ get_root_node (cast document_ptr) →⇩r cast document_ptr"
by (simp add: ‹document_ptr |∈| document_ptr_kinds h› local.get_root_node_not_node_same)
then
have root': "h' ⊢ get_root_node (cast document_ptr) →⇩r cast document_ptr"
by (simp add: ‹document_ptr |∈| document_ptr_kinds h› document_ptr_kinds_eq_h
local.get_root_node_not_node_same object_ptr_kinds_eq_h2 object_ptr_kinds_eq_h3)
have "heap_is_wellformed h'" and "known_ptrs h'"
using create_character_data_preserves_wellformedness assms
by blast+
have "cast result |∉| object_ptr_kinds h"
using ‹cast new_character_data_ptr ∉ set |h ⊢ object_ptr_kinds_M|⇩r›
by (metis (full_types) ObjectMonad.ptr_kinds_ptr_kinds_M
‹h ⊢ create_character_data document_ptr text →⇩r new_character_data_ptr› assms(4) returns_result_eq)
obtain to where to: "h ⊢ to_tree_order (cast document_ptr) →⇩r to"
by (meson ‹document_ptr |∈| document_ptr_kinds h› assms(1) assms(2) assms(3)
document_ptr_kinds_commutes is_OK_returns_result_E local.to_tree_order_ok)
then
have "h ⊢ a_get_dom_component (cast document_ptr) →⇩r to"
using root
by(auto simp add: a_get_dom_component_def)
moreover
obtain to' where to': "h' ⊢ to_tree_order (cast document_ptr) →⇩r to'"
by (meson ‹heap_is_wellformed h'› ‹known_ptrs h'› ‹type_wf h'› is_OK_returns_result_E
local.get_root_node_root_in_heap local.to_tree_order_ok root')
then
have "h' ⊢ a_get_dom_component (cast document_ptr) →⇩r to'"
using root'
by(auto simp add: a_get_dom_component_def)
moreover
have "⋀child. child ∈ set to ⟷ child ∈ set to'"
by (metis ‹heap_is_wellformed h'› ‹known_ptrs h'› ‹parent_child_rel h = parent_child_rel h'›
‹type_wf h'› assms(1) assms(2) assms(3) local.to_tree_order_parent_child_rel to to')
ultimately
have "set |h ⊢ local.a_get_dom_component (cast document_ptr)|⇩r =
set |h' ⊢ local.a_get_dom_component (cast document_ptr)|⇩r"
by(auto simp add: a_get_dom_component_def)
show ?thesis
apply(auto simp add: is_weakly_dom_component_safe_def Let_def)[1]
using assms(2) assms(3) children_eq_h local.get_child_nodes_ok
local.get_child_nodes_ptr_in_heap local.known_ptrs_known_ptr object_ptr_kinds_eq_h2
object_ptr_kinds_eq_h3 returns_result_select_result
apply (metis ‹h2 ⊢ get_child_nodes (cast⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r new_character_data_ptr) →⇩r []›
is_OK_returns_result_I)
apply (metis ‹h ⊢ create_character_data document_ptr text →⇩r new_character_data_ptr› assms(4)
character_data_ptr_kinds_commutes h2 new_character_data_ptr new_character_data_ptr_in_heap
node_ptr_kinds_eq_h2 node_ptr_kinds_eq_h3 returns_result_eq)
using ‹h ⊢ create_character_data document_ptr text →⇩r new_character_data_ptr›
‹new_character_data_ptr ∉ set |h ⊢ character_data_ptr_kinds_M|⇩r› assms(4) returns_result_eq
apply fastforce
using assms(2) assms(3) children_eq_h local.get_child_nodes_ok local.get_child_nodes_ptr_in_heap
local.known_ptrs_known_ptr object_ptr_kinds_eq_h2 object_ptr_kinds_eq_h3 returns_result_select_result
apply (smt ObjectMonad.ptr_kinds_ptr_kinds_M
‹cast⇩c⇩h⇩a⇩r⇩a⇩c⇩t⇩e⇩r⇩_⇩d⇩a⇩t⇩a⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r new_character_data_ptr ∉ set |h ⊢ object_ptr_kinds_M|⇩r›
‹h ⊢ create_character_data document_ptr text →⇩r new_character_data_ptr› assms(1) assms(5)
create_character_data_is_weakly_dom_component_safe_step local.get_dom_component_impl select_result_I2)
done
qed
end
interpretation i_get_dom_component_create_character_data?: l_get_dom_component_create_character_data⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
known_ptr heap_is_wellformed parent_child_rel type_wf known_ptrs to_tree_order get_parent
get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe
is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs
get_element_by_id get_elements_by_class_name get_elements_by_tag_name set_val set_val_locs
get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs
create_character_data
by(auto simp add: l_get_dom_component_create_character_data⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def instances)
declare l_get_dom_component_create_character_data⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms [instances]
subsubsection ‹create\_document›
lemma create_document_unsafe: "¬(∀(h
:: ('object_ptr::{equal,linorder}, 'node_ptr::{equal,linorder}, 'element_ptr::{equal,linorder},
'character_data_ptr::{equal,linorder}, 'document_ptr::{equal,linorder}, 'shadow_root_ptr::{equal,linorder},
'Object::{equal,linorder}, 'Node::{equal,linorder}, 'Element::{equal,linorder},
'CharacterData::{equal,linorder}, 'Document::{equal,linorder}) heap
) h' new_document_ptr. heap_is_wellformed h ⟶ type_wf h ⟶ known_ptrs h ⟶
h ⊢ create_document →⇩r new_document_ptr ⟶ h ⊢ create_document →⇩h h' ⟶
is_strongly_dom_component_safe {} {cast new_document_ptr} h h')"
proof -
obtain h document_ptr where h: "Inr (document_ptr, h) = (Heap (fmempty)
:: ('object_ptr::{equal,linorder}, 'node_ptr::{equal,linorder}, 'element_ptr::{equal,linorder},
'character_data_ptr::{equal,linorder}, 'document_ptr::{equal,linorder}, 'shadow_root_ptr::{equal,linorder},
'Object::{equal,linorder}, 'Node::{equal,linorder}, 'Element::{equal,linorder},
'CharacterData::{equal,linorder}, 'Document::{equal,linorder}) heap
) ⊢ (do {
document_ptr ← create_document;
return (document_ptr)
})"
by(code_simp, auto simp add: equal_eq List.member_def)+
then obtain h' new_document_ptr where
"heap_is_wellformed h" and "type_wf h" and "known_ptrs h" and
h': "h ⊢ create_document →⇩r new_document_ptr" and
h': "h ⊢ create_document →⇩h h'" and
"¬(is_strongly_dom_component_safe {} {cast new_document_ptr} h h')"
by(code_simp, auto simp add: equal_eq List.member_def)+
then show ?thesis
by blast
qed
locale l_get_dom_component_create_document⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M =
l_get_dom_component⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs
to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component
is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs
get_ancestors get_ancestors_locs get_disconnected_nodes get_disconnected_nodes_locs get_element_by_id
get_elements_by_class_name get_elements_by_tag_name +
l_create_document⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M create_document
for known_ptr :: "(_::linorder) object_ptr ⇒ bool"
and heap_is_wellformed :: "(_) heap ⇒ bool"
and parent_child_rel :: "(_) heap ⇒ ((_) object_ptr × (_) object_ptr) set"
and type_wf :: "(_) heap ⇒ bool"
and known_ptrs :: "(_) heap ⇒ bool"
and to_tree_order :: "(_) object_ptr ⇒ ((_) heap, exception, (_) object_ptr list) prog"
and get_parent :: "(_) node_ptr ⇒ ((_) heap, exception, (_) object_ptr option) prog"
and get_parent_locs :: "((_) heap ⇒ (_) heap ⇒ bool) set"
and get_child_nodes :: "(_) object_ptr ⇒ ((_) heap, exception, (_) node_ptr list) prog"
and get_child_nodes_locs :: "(_) object_ptr ⇒ ((_) heap ⇒ (_) heap ⇒ bool) set"
and get_dom_component :: "(_) object_ptr ⇒ ((_) heap, exception, (_) object_ptr list) prog"
and is_strongly_dom_component_safe ::
"(_) object_ptr set ⇒ (_) object_ptr set ⇒ (_) heap ⇒ (_) heap ⇒ bool"
and is_weakly_dom_component_safe ::
"(_) object_ptr set ⇒ (_) object_ptr set ⇒ (_) heap ⇒ (_) heap ⇒ bool"
and get_root_node :: "(_) object_ptr ⇒ ((_) heap, exception, (_) object_ptr) prog"
and get_root_node_locs :: "((_) heap ⇒ (_) heap ⇒ bool) set"
and get_ancestors :: "(_) object_ptr ⇒ ((_) heap, exception, (_) object_ptr list) prog"
and get_ancestors_locs :: "((_) heap ⇒ (_) heap ⇒ bool) set"
and get_element_by_id ::
"(_) object_ptr ⇒ char list ⇒ ((_) heap, exception, (_) element_ptr option) prog"
and get_elements_by_class_name ::
"(_) object_ptr ⇒ char list ⇒ ((_) heap, exception, (_) element_ptr list) prog"
and get_elements_by_tag_name ::
"(_) object_ptr ⇒ char list ⇒ ((_) heap, exception, (_) element_ptr list) prog"
and create_document :: "((_) heap, exception, (_) document_ptr) prog"
begin
lemma create_document_is_weakly_dom_component_safe_step:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ create_document →⇩h h'"
assumes "ptr ≠ cast |h ⊢ create_document|⇩r"
shows "preserved (get_M⇩O⇩b⇩j⇩e⇩c⇩t ptr getter) h h'"
using assms
apply(auto simp add: create_document_def)[1]
by (metis assms(4) assms(5) is_OK_returns_heap_I local.create_document_def new_document_get_M⇩O⇩b⇩j⇩e⇩c⇩t
select_result_I)
lemma create_document_is_weakly_dom_component_safe:
assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
assumes "h ⊢ create_document →⇩r result"
assumes "h ⊢ create_document →⇩h h'"
shows "is_weakly_dom_component_safe {} {cast result} h h'"
proof -
have "object_ptr_kinds h' = object_ptr_kinds h |∪| {|cast result|}"
using assms(4) assms(5) local.create_document_def new_document_new_ptr by auto
moreover have "result |∉| document_ptr_kinds h"
using assms(4) assms(5) local.create_document_def new_document_ptr_not_in_heap by auto
ultimately show ?thesis
using assms
apply(auto simp add: is_weakly_dom_component_safe_def Let_def local.create_document_def
new_document_ptr_not_in_heap)[1]
by (metis ‹result |∉| document_ptr_kinds h› document_ptr_kinds_commutes new_document_get_M⇩O⇩b⇩j⇩e⇩c⇩t)
qed
end
interpretation i_get_dom_component_create_document?: l_get_dom_component_create_document⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M
known_ptr heap_is_wellformed parent_child_rel type_wf known_ptrs to_tree_order get_parent
get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe
is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs
get_element_by_id get_elements_by_class_name get_elements_by_tag_name create_document
by(auto simp add: l_get_dom_component_create_document⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_def instances)
declare l_get_dom_component_create_document⇩C⇩o⇩r⇩e⇩_⇩D⇩O⇩M_axioms [instances]
subsubsection ‹insert\_before›
lemma insert_before_unsafe: "¬(∀(h
:: ('object_ptr::{equal,linorder}, 'node_ptr::{equal,linorder}, 'element_ptr::{equal,linorder},
'character_data_ptr::{equal,linorder}, 'document_ptr::{equal,linorder}, 'shadow_root_ptr::{equal,linorder},
'Object::{equal,linorder}, 'Node::{equal,linorder}, 'Element::{equal,linorder},
'CharacterData::{equal,linorder}, 'Document::{equal,linorder}) heap
) h' ptr child. heap_is_wellformed h ⟶ type_wf h ⟶ known_ptrs h ⟶
h ⊢ insert_before ptr child None →⇩h h' ⟶ is_weakly_dom_component_safe {ptr, cast child} {} h h')"
proof -
obtain h document_ptr e1 e2 where h: "Inr ((document_ptr, e1, e2), h) = (Heap (fmempty)
:: ('object_ptr::{equal,linorder}, 'node_ptr::{equal,linorder}, 'element_ptr::{equal,linorder},
'character_data_ptr::{equal,linorder}, 'document_ptr::{equal,linorder}, 'shadow_root_ptr::{equal,linorder},
'Object::{equal,linorder}, 'Node::{equal,linorder}, 'Element::{equal,linorder},
'CharacterData::{equal,linorder}, 'Document::{equal,linorder}) heap
) ⊢ (do {
document_ptr ← create_document;
e1 ← create_element document_ptr ''div'';
e2 ← create_element document_ptr ''div'';
return (document_ptr, e1, e2)
})"
by(code_simp, auto simp add: equal_eq List.member_def)+
then obtain h' where
"heap_is_wellformed h" and "type_wf h" and "known_ptrs h" and
h': "h ⊢ insert_before (cast e1) (cast e2) None →⇩h h'" and
"¬(is_weakly_dom_component_safe {cast e1, cast e2} {} h h')"
by(code_simp, auto simp add: equal_eq List.member_def)+
then show ?thesis
by auto
qed
lemma insert_before_unsafe2: "¬(∀(h
:: ('object_ptr::{equal,linorder}, 'node_ptr::{equal,linorder}, 'element_ptr::{equal,linorder},
'character_data_ptr::{equal,linorder}, 'document_ptr::{equal,linorder}, 'shadow_root_ptr::{equal,linorder},
'Object::{equal,linorder}, 'Node::{equal,linorder}, 'Element::{equal,linorder},
'CharacterData::{equal,linorder}, 'Document::{equal,linorder}) heap
) h' ptr child ref. heap_is_wellformed h ⟶ type_wf h ⟶ known_ptrs h ⟶
h ⊢ insert_before ptr child (Some ref) →⇩h h' ⟶
is_weakly_dom_component_safe {ptr, cast child, cast ref} {} h h')"
proof -
obtain h document_ptr e1 e2 e3 where h: "Inr ((document_ptr, e1, e2, e3), h) = (Heap (fmempty)
:: ('object_ptr::{equal,linorder}, 'node_ptr::{equal,linorder}, 'element_ptr::{equal,linorder},
'character_data_ptr::{equal,linorder}, 'document_ptr::{equal,linorder}, 'shadow_root_ptr::{equal,linorder},
'Object::{equal,linorder}, 'Node::{equal,linorder}, 'Element::{equal,linorder},
'CharacterData::{equal,linorder}, 'Document::{equal,linorder}) heap
) ⊢ (do {
document_ptr ← create_document;
e1 ← create_element document_ptr ''div'';
e2 ← create_element document_ptr ''div'';
e3 ← create_element document_ptr ''div'';
append_child (cast e1) (cast e2);
return (document_ptr, e1, e2, e3)
})"
by(code_simp, auto simp add: equal_eq List.member_def)+
then obtain h' where
"heap_is_wellformed h" and "type_wf h" and "known_ptrs h" and
h': "h ⊢ insert_before (cast e1) (cast e3) (Some (cast e2)) →⇩h h'" and
"¬(is_weakly_dom_component_safe {cast e1, cast e3, cast e2} {} h h')"
apply(code_simp)
apply(clarify)
by(code_simp, auto simp add: equal_eq List.member_def)+
then show ?thesis
by fast
qed
lemma append_child_unsafe:
obtains
h :: "('object_ptr::{equal,linorder}, 'node_ptr::{equal,linorder}, 'element_ptr::{equal,linorder},
'character_data_ptr::{equal,linorder}, 'document_ptr::{equal,linorder}, 'shadow_root_ptr::{equal,linorder},
'Object::{equal,linorder}, 'Node::{equal,linorder}, 'Element::{equal,linorder},
'CharacterData::{equal,linorder}, 'Document::{equal,linorder}) heap" and
h' and ptr and child where
"heap_is_wellformed h" and "type_wf h" and "known_ptrs h" and
"h ⊢ append_child ptr child →⇩h h'" and
"¬ is_weakly_dom_component_safe {ptr, cast child} {} h h'"
proof -
let ?h0 = "Heap fmempty ::('object_ptr::{equal,linorder}, 'node_ptr::{equal,linorder},
'element_ptr::{equal,linorder}, 'character_data_ptr::{equal,linorder}, 'document_ptr::{equal,linorder},
'shadow_root_ptr::{equal,linorder}, 'Object::{equal,linorder}, 'Node::{equal,linorder},
'Element::{equal,linorder}, 'CharacterData::{equal,linorder}, 'Document::{equal,linorder}) heap"
let ?P = "do {
document_ptr ← create_document;
e1 ← create_element document_ptr ''div'';
e2 ← create_element document_ptr ''div'';
return (e1, e2)
}"
let ?h1 = "|?h0 ⊢ ?P|⇩h"
let ?e1 = "fst |?h0 ⊢ ?P|⇩r"
let ?e2 = "snd |?h0 ⊢ ?P|⇩r"
show thesis
apply(rule that[where h="?h1" and ptr="cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r ?e1" and child="cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩n⇩o⇩d⇩e⇩_⇩p⇩t⇩r ?e2"])
by code_simp+
qed
subsubsection ‹get\_owner\_document›
lemma get_owner_document_unsafe:
obtains
h :: "('object_ptr::{equal,linorder}, 'node_ptr::{equal,linorder}, 'element_ptr::{equal,linorder},
'character_data_ptr::{equal,linorder}, 'document_ptr::{equal,linorder}, 'shadow_root_ptr::{equal,linorder},
'Object::{equal,linorder}, 'Node::{equal,linorder}, 'Element::{equal,linorder},
'CharacterData::{equal,linorder}, 'Document::{equal,linorder}) heap" and
h' and ptr and owner_document where
"heap_is_wellformed h" and "type_wf h" and "known_ptrs h" and
"h ⊢ get_owner_document ptr →⇩r owner_document →⇩h h'" and
"¬is_weakly_dom_component_safe {ptr} {cast owner_document} h h'"
proof -
let ?h0 = "Heap fmempty ::('object_ptr::{equal,linorder}, 'node_ptr::{equal,linorder},
'element_ptr::{equal,linorder}, 'character_data_ptr::{equal,linorder}, 'document_ptr::{equal,linorder},
'shadow_root_ptr::{equal,linorder}, 'Object::{equal,linorder}, 'Node::{equal,linorder},
'Element::{equal,linorder}, 'CharacterData::{equal,linorder}, 'Document::{equal,linorder}) heap"
let ?P = "do {
document_ptr ← create_document;
e1 ← create_element document_ptr ''div'';
return (document_ptr, e1)
}"
let ?h1 = "|?h0 ⊢ ?P|⇩h"
let ?document_ptr = "fst |?h0 ⊢ ?P|⇩r"
let ?e1 = "snd |?h0 ⊢ ?P|⇩r"
show thesis
apply(rule that[where h="?h1" and ptr="cast⇩e⇩l⇩e⇩m⇩e⇩n⇩t⇩_⇩p⇩t⇩r⇩2⇩o⇩b⇩j⇩e⇩c⇩t⇩_⇩p⇩t⇩r ?e1" and owner_document="?document_ptr"])
by code_simp+
qed
end