Ziele Untersuchung
mit Columbo Integrität von
Datenbanken Interaktion und
Portierbarkeit Ergonomie der
Schnittstellen

Angebot Produkte Projekt Beratung

Mittel Analytik Modellierung Sprachen Algebra Logik Hardware Denken Kreativität

Zusammenhänge Gesellschaft Wirtschaft Branche Firma

Benutzer


products/Sources/formale Sprachen/Isabelle/Archive-of-Formal-Proofs/thys/Shadow_DOM/ (Sammlung formaler Beweise Version 2026-5^©) Datei vom 29.4.2026 mit Größe 637 kB

Quelle Shadow_DOM.thy

Sprache: Isabelle

(***********************************************************************************
* Copyright (c) 2016-2020 The University of Sheffield, UK
*               2019-2020 University of Exeter, UK
*
* All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions are met:
*
* * Redistributions of source code must retain the above copyright notice, this
*   list of conditions and the following disclaimer.
*
* * Redistributions in binary form must reproduce the above copyright notice,
*   this list of conditions and the following disclaimer in the documentation
*   and/or other materials provided with the distribution.
*
* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
* AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
* DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
* SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
* CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
* OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
* OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*
* SPDX-License-Identifier: BSD-2-Clause
***********************************************************************************)

section‹The Shadow DOM›
theory Shadow_DOM
  imports
    "monads/ShadowRootMonad"
    Core_DOM.Core_DOM
begin

abbreviation "safe_shadow_root_element_types ≡ {''article'', ''aside'', ''blockquote'', ''body'',
  ''div'', ''footer'', ''h1'', ''h2'', ''h3'', ''h4'', ''h5'', ''h6'', ''header'', ''main'',
  ''nav'', ''p'', ''section'', ''span''}"

subsection ‹Function Definitions›

subsubsection ‹get\_child\_nodes›

locale l_get_child_nodes_S_h_a_d_o_w_{_}_D_O_{M_defs} =
  CD: l_get_child_nodes_C_o_r_e_{_}_D_O_{M_defs}
begin
definition get_child_nodes_s_h_a_d_o_w_{_}_r_o_o_t_{_}_p_t_r :: "(_) shadow_root_ptr ==> unit
    ==> (_, (_) node_ptr list) dom_prog" where
  "get_child_nodes_s_h_a_d_o_w_{_}_r_o_o_t_{_}_p_t_r shadow_root_ptr _ = get_M shadow_root_ptr RShadowRoot.child_nodes"

definition a_get_child_nodes_tups :: "(((_) object_ptr ==> bool) × ((_) object_ptr ==> unit
    ==> (_, (_) node_ptr list) dom_prog)) list" where
  "a_get_child_nodes_tups ≡ [(is_shadow_root_ptr_o_b_j_e_c_t_{_}_p_t_r, get_child_nodes_s_h_a_d_o_w_{_}_r_o_o_t_{_}_p_t_r ∘ the ∘ cast)]"

definition a_get_child_nodes :: "(_) object_ptr ==> (_, (_) node_ptr list) dom_prog" where
  "a_get_child_nodes ptr = invoke (CD.a_get_child_nodes_tups @ a_get_child_nodes_tups) ptr ()"

definition a_get_child_nodes_locs :: "(_) object_ptr ==> ((_) heap ==> (_) heap ==> bool) set" where
  "a_get_child_nodes_locs ptr ≡
    (if is_shadow_root_ptr_kind ptr
    then {preserved (get_M (the (cast ptr)) RShadowRoot.child_nodes)} else {}) ∪
    CD.a_get_child_nodes_locs ptr"

definition first_child :: "(_) object_ptr ==> (_, (_) node_ptr option) dom_prog"
  where
    "first_child ptr = do {
      children ← a_get_child_nodes ptr;
      return (case children of [] ==> None | child#_ ==> Some child)}"
end

global_interpretation l_get_child_nodes_S_h_a_d_o_w_{_}_D_O_{M_defs} defines
  get_child_nodes = l_get_child_nodes_S_h_a_d_o_w_{_}_D_O_{M_defs}.a_get_child_nodes and
  get_child_nodes_locs = l_get_child_nodes_S_h_a_d_o_w_{_}_D_O_{M_defs}.a_get_child_nodes_locs
  .

locale l_get_child_nodes_S_h_a_d_o_w_{_}_D_O_M =
  l_type_wf type_wf +
  l_known_ptr known_ptr +
  l_get_child_nodes_S_h_a_d_o_w_{_}_D_O_{M_defs} +
  l_get_child_nodes_defs get_child_nodes get_child_nodes_locs +
  CD: l_get_child_nodes_C_o_r_e_{_}_D_O_M type_wf_C_o_r_e_{_}_D_O_M known_ptr_C_o_r_e_{_}_D_O_M get_child_nodes_C_o_r_e_{_}_D_O_M
  get_child_nodes_locs_C_o_r_e_{_}_D_O_M
  for type_wf :: "(_) heap ==> bool"
    and known_ptr :: "(_) object_ptr ==> bool"
    and type_wf_C_o_r_e_{_}_D_O_M :: "(_) heap ==> bool"
    and known_ptr_C_o_r_e_{_}_D_O_M :: "(_) object_ptr ==> bool"
    and get_child_nodes :: "(_) object_ptr ==> (_, (_) node_ptr list) dom_prog"
    and get_child_nodes_locs :: "(_) object_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
    and get_child_nodes_C_o_r_e_{_}_D_O_M :: "(_) object_ptr ==> (_, (_) node_ptr list) dom_prog"
    and get_child_nodes_locs_C_o_r_e_{_}_D_O_M :: "(_) object_ptr ==> ((_) heap ==> (_) heap ==> bool) set" +
  assumes known_ptr_impl: "known_ptr = ShadowRootClass.known_ptr"
  assumes type_wf_impl: "type_wf = ShadowRootClass.type_wf"
  assumes get_child_nodes_impl: "get_child_nodes = a_get_child_nodes"
  assumes get_child_nodes_locs_impl: "get_child_nodes_locs = a_get_child_nodes_locs"
begin
lemmas get_child_nodes_def = get_child_nodes_impl[unfolded a_get_child_nodes_def get_child_nodes_def]
lemmas get_child_nodes_locs_def = get_child_nodes_locs_impl[unfolded a_get_child_nodes_locs_def
    get_child_nodes_locs_def, folded CD.get_child_nodes_locs_impl]

lemma get_child_nodes_ok:
  assumes "known_ptr ptr"
  assumes "type_wf h"
  assumes "ptr |∈| object_ptr_kinds h"
  shows "h ⊨ ok (get_child_nodes ptr)"
  using assms[unfolded known_ptr_impl type_wf_impl]
  apply(auto simp add: get_child_nodes_def)[1]
  apply(split CD.get_child_nodes_splits, rule conjI)+
  using ShadowRootClass.type_wf_D_o_c_u_m_e_n_t CD.get_child_nodes_ok CD.known_ptr_impl CD.type_wf_impl
   apply blast
  apply(auto simp add: CD.known_ptr_impl a_get_child_nodes_tups_def get_child_nodes_s_h_a_d_o_w_{_}_r_o_o_t_{_}_p_t_{r_def}
      get_M_S_h_a_d_o_w_R_o_o_{t_ok}
      dest!: known_ptr_new_shadow_root_ptr intro!: bind_is_OK_I2)[1]
  by (metis is_shadow_root_ptr_kind_none l_get_M_S_h_a_d_o_w_R_o_o_{t_lemmas}.get_M_S_h_a_d_o_w_R_o_o_{t_ok}
      l_get_M_S_h_a_d_o_w_R_o_o_{t_lemmas_axioms} option.case_eq_if shadow_root_ptr_casts_commute3
      shadow_root_ptr_kinds_commutes)

lemma get_child_nodes_ptr_in_heap:
  assumes "h ⊨ get_child_nodes ptr →_r children"
  shows "ptr |∈| object_ptr_kinds h"
  using assms
  by(auto simp add: get_child_nodes_def invoke_ptr_in_heap dest: is_OK_returns_result_I)

lemma get_child_nodes_pure [simp]:
  "pure (get_child_nodes ptr) h"
  unfolding get_child_nodes_def a_get_child_nodes_tups_def
proof(split CD.get_child_nodes_splits, rule conjI; clarify)
  assume "known_ptr_C_o_r_e_{_}_D_O_M ptr"
  then show "pure (get_child_nodes_C_o_r_e_{_}_D_O_M ptr) h"
    by simp
next
  assume "¬ known_ptr_C_o_r_e_{_}_D_O_M ptr"
  then show "pure (invoke [(is_shadow_root_ptr_o_b_j_e_c_t_{_}_p_t_r,
       get_child_nodes_s_h_a_d_o_w_{_}_r_o_o_t_{_}_p_t_r ∘ the ∘ cast_o_b_j_e_c_t_{_}_p_t_r₂_s_h_a_d_o_w_{_}_r_o_o_t_{_}_p_t_r)]
     ptr ()) h"
    by(auto simp add: get_child_nodes_s_h_a_d_o_w_{_}_r_o_o_t_{_}_p_t_{r_def} intro: bind_pure_I split: invoke_splits)
qed

lemma get_child_nodes_reads: "reads (get_child_nodes_locs ptr) (get_child_nodes ptr) h h'"
  apply (simp add: get_child_nodes_def a_get_child_nodes_tups_def get_child_nodes_locs_def
      CD.get_child_nodes_locs_def)
  apply(split CD.get_child_nodes_splits, rule conjI)+
   apply(auto intro!: reads_subset[OF CD.get_child_nodes_reads[unfolded CD.get_child_nodes_locs_def]]
      split: if_splits)[1]
  apply(split invoke_splits, rule conjI)+
   apply(auto)[1]
  apply(auto simp add: get_child_nodes_s_h_a_d_o_w_{_}_r_o_o_t_{_}_p_t_{r_def}
      intro: reads_subset[OF reads_singleton] reads_subset[OF check_in_heap_reads]
      intro!: reads_bind_pure reads_subset[OF return_reads] split: option.splits)[1]
  done
end

interpretation i_get_child_nodes?: l_get_child_nodes_S_h_a_d_o_w_{_}_D_O_M type_wf known_ptr DocumentClass.type_wf
  DocumentClass.known_ptr get_child_nodes get_child_nodes_locs Core_DOM_Functions.get_child_nodes
  Core_DOM_Functions.get_child_nodes_locs
  by(simp add: l_get_child_nodes_S_h_a_d_o_w_{_}_D_O_{M_def} l_get_child_nodes_S_h_a_d_o_w_{_}_D_O_{M_axioms_def} instances)
declare l_get_child_nodes_S_h_a_d_o_w_{_}_D_O_{M_axioms} [instances]

lemma get_child_nodes_is_l_get_child_nodes [instances]: "l_get_child_nodes type_wf known_ptr
    get_child_nodes get_child_nodes_locs"
  apply(auto simp add: l_get_child_nodes_def instances)[1]
  using get_child_nodes_reads get_child_nodes_ok get_child_nodes_ptr_in_heap get_child_nodes_pure
  by blast+

paragraph ‹new\_document›

locale l_new_document_get_child_nodes_S_h_a_d_o_w_{_}_D_O_M =
  CD: l_new_document_get_child_nodes_C_o_r_e_{_}_D_O_M type_wf_C_o_r_e_{_}_D_O_M known_ptr_C_o_r_e_{_}_D_O_M get_child_nodes_C_o_r_e_{_}_D_O_M
  get_child_nodes_locs_C_o_r_e_{_}_D_O_M
  + l_get_child_nodes_S_h_a_d_o_w_{_}_D_O_M type_wf known_ptr type_wf_C_o_r_e_{_}_D_O_M known_ptr_C_o_r_e_{_}_D_O_M get_child_nodes
  get_child_nodes_locs get_child_nodes_C_o_r_e_{_}_D_O_M get_child_nodes_locs_C_o_r_e_{_}_D_O_M
  for type_wf :: "(_) heap ==> bool"
    and known_ptr :: "(_) object_ptr ==> bool"
    and get_child_nodes :: "(_) object_ptr ==> ((_) heap, exception, (_) node_ptr list) prog"
    and get_child_nodes_locs :: "(_) object_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
    and type_wf_C_o_r_e_{_}_D_O_M :: "(_) heap ==> bool"
    and known_ptr_C_o_r_e_{_}_D_O_M :: "(_) object_ptr ==> bool"
    and get_child_nodes_C_o_r_e_{_}_D_O_M :: "(_) object_ptr ==> ((_) heap, exception, (_) node_ptr list) prog"
    and get_child_nodes_locs_C_o_r_e_{_}_D_O_M :: "(_) object_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
begin
lemma get_child_nodes_new_document:
  "ptr' ≠ cast new_document_ptr ==> h ⊨ new_document →_r new_document_ptr
  ==> h ⊨ new_document →_h h' ==> r ∈ get_child_nodes_locs ptr' ==> r h h'"
  apply(auto simp add: get_child_nodes_locs_def)[1]
  using CD.get_child_nodes_new_document
   apply (metis document_ptr_casts_commute3 empty_iff is_document_ptr_kind_none
      new_document_get_M_S_h_a_d_o_w_R_o_o_t option.case_eq_if shadow_root_ptr_casts_commute3
      singletonD)
  by (simp add: CD.get_child_nodes_new_document)

lemma new_document_no_child_nodes:
  "h ⊨ new_document →_r new_document_ptr ==> h ⊨ new_document →_h h'
     ==> h' ⊨ get_child_nodes (cast new_document_ptr) →_r []"
  apply(auto simp add: get_child_nodes_def)[1]
  apply(split CD.get_child_nodes_splits, rule conjI)+
  using CD.new_document_no_child_nodes apply auto[1]
  by(auto simp add: DocumentClass.a_known_ptr_def CD.known_ptr_impl known_ptr_def
      dest!: new_document_is_document_ptr)
end
interpretation i_new_document_get_child_nodes?:
  l_new_document_get_child_nodes_S_h_a_d_o_w_{_}_D_O_M type_wf known_ptr get_child_nodes get_child_nodes_locs
  DocumentClass.type_wf DocumentClass.known_ptr Core_DOM_Functions.get_child_nodes
  Core_DOM_Functions.get_child_nodes_locs
  by(unfold_locales)
declare l_new_document_get_child_nodes_C_o_r_e_{_}_D_O_{M_axioms}[instances]

lemma new_document_get_child_nodes_is_l_new_document_get_child_nodes [instances]:
  "l_new_document_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs"
  using new_document_is_l_new_document get_child_nodes_is_l_get_child_nodes
  apply(simp add: l_new_document_get_child_nodes_def l_new_document_get_child_nodes_axioms_def)
  using get_child_nodes_new_document new_document_no_child_nodes
  by fast

paragraph ‹new\_shadow\_root›

locale l_new_shadow_root_get_child_nodes_S_h_a_d_o_w_{_}_D_O_M =
  l_get_child_nodes_S_h_a_d_o_w_{_}_D_O_M type_wf known_ptr type_wf_C_o_r_e_{_}_D_O_M known_ptr_C_o_r_e_{_}_D_O_M get_child_nodes
  get_child_nodes_locs get_child_nodes_C_o_r_e_{_}_D_O_M get_child_nodes_locs_C_o_r_e_{_}_D_O_M
  for type_wf :: "(_) heap ==> bool"
    and known_ptr :: "(_) object_ptr ==> bool"
    and get_child_nodes :: "(_) object_ptr ==> ((_) heap, exception, (_) node_ptr list) prog"
    and get_child_nodes_locs :: "(_) object_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
    and type_wf_C_o_r_e_{_}_D_O_M :: "(_) heap ==> bool"
    and known_ptr_C_o_r_e_{_}_D_O_M :: "(_) object_ptr ==> bool"
    and get_child_nodes_C_o_r_e_{_}_D_O_M :: "(_) object_ptr ==> ((_) heap, exception, (_) node_ptr list) prog"
    and get_child_nodes_locs_C_o_r_e_{_}_D_O_M :: "(_) object_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
begin
lemma get_child_nodes_new_shadow_root:
  "ptr' ≠ cast new_shadow_root_ptr ==> h ⊨ new_S_h_a_d_o_w_R_o_o_{t_M} →_r new_shadow_root_ptr
     ==> h ⊨ new_S_h_a_d_o_w_R_o_o_{t_M} →_h h' ==> r ∈ get_child_nodes_locs ptr' ==> r h h'"
  apply(auto simp add: get_child_nodes_locs_def)[1]
   apply (metis document_ptr_casts_commute3 insert_absorb insert_not_empty is_document_ptr_kind_none
      new_shadow_root_get_M_S_h_a_d_o_w_R_o_o_t option.case_eq_if shadow_root_ptr_casts_commute3 singletonD)
  apply(auto simp add: CD.get_child_nodes_locs_def)[1]
  using new_shadow_root_get_M_O_b_j_e_c_t apply blast
  apply (metis empty_iff insertE new_shadow_root_get_M_E_l_e_m_e_n_t)
  apply (metis empty_iff new_shadow_root_get_M_D_o_c_u_m_e_n_t singletonD)
  done

lemma new_shadow_root_no_child_nodes:
  "h ⊨ new_S_h_a_d_o_w_R_o_o_{t_M} →_r new_shadow_root_ptr ==> h ⊨ new_S_h_a_d_o_w_R_o_o_{t_M} →_h h'
     ==> h' ⊨ get_child_nodes (cast new_shadow_root_ptr) →_r []"
  apply(auto simp add: get_child_nodes_def )[1]
  apply(split CD.get_child_nodes_splits, rule conjI)+
   apply(auto simp add: CD.get_child_nodes_def CD.a_get_child_nodes_tups_def)[1]
   apply(split invoke_splits, rule conjI)+
  using NodeClass.a_known_ptr_def known_ptr_not_character_data_ptr known_ptr_not_document_ptr
    known_ptr_not_element_ptr local.CD.known_ptr_impl apply blast
     apply(auto simp add: is_document_ptr_def  split: option.splits document_ptr.splits)[1]
    apply(auto simp add: is_character_data_ptr_def  split: option.splits document_ptr.splits)[1]
   apply(auto simp add: is_element_ptr_def  split: option.splits document_ptr.splits)[1]
  apply(auto simp add: a_get_child_nodes_tups_def)[1]
  apply(split invoke_splits, rule conjI)+
   apply(auto simp add: is_shadow_root_ptr_def split: shadow_root_ptr.splits
      dest!: new_S_h_a_d_o_w_R_o_o_{t_M_is_shadow_root_ptr})[1]
  apply(auto intro!: bind_pure_returns_result_I)[1]
   apply(drule(1) new_S_h_a_d_o_w_R_o_o_{t_M_ptr_in_heap})
   apply(auto simp add: shadow_root_ptr_kinds_def document_ptr_kinds_def)[1]
   apply (metis check_in_heap_ptr_in_heap is_OK_returns_result_E old.unit.exhaust)
  using  new_shadow_root_children
  by (simp add: new_shadow_root_children get_child_nodes_s_h_a_d_o_w_{_}_r_o_o_t_{_}_p_t_{r_def})
end
interpretation i_new_shadow_root_get_child_nodes?:
  l_new_shadow_root_get_child_nodes_S_h_a_d_o_w_{_}_D_O_M type_wf known_ptr get_child_nodes get_child_nodes_locs
  DocumentClass.type_wf DocumentClass.known_ptr Core_DOM_Functions.get_child_nodes
  Core_DOM_Functions.get_child_nodes_locs
  by(unfold_locales)
declare l_new_shadow_root_get_child_nodes_S_h_a_d_o_w_{_}_D_O_{M_def}[instances]

locale l_new_shadow_root_get_child_nodes = l_get_child_nodes +
  assumes get_child_nodes_new_shadow_root:
    "ptr' ≠ cast new_shadow_root_ptr ==> h ⊨ new_S_h_a_d_o_w_R_o_o_{t_M} →_r new_shadow_root_ptr
     ==> h ⊨ new_S_h_a_d_o_w_R_o_o_{t_M} →_h h' ==> r ∈ get_child_nodes_locs ptr' ==> r h h'"
  assumes new_shadow_root_no_child_nodes:
    "h ⊨ new_S_h_a_d_o_w_R_o_o_{t_M} →_r new_shadow_root_ptr ==> h ⊨ new_S_h_a_d_o_w_R_o_o_{t_M} →_h h'
     ==> h' ⊨ get_child_nodes (cast new_shadow_root_ptr) →_r []"

lemma new_shadow_root_get_child_nodes_is_l_new_shadow_root_get_child_nodes [instances]:
  "l_new_shadow_root_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs"
  apply(simp add: l_new_shadow_root_get_child_nodes_def l_new_shadow_root_get_child_nodes_axioms_def
      instances)
  using get_child_nodes_new_shadow_root new_shadow_root_no_child_nodes
  by fast

paragraph ‹new\_element›

locale l_new_element_get_child_nodes_S_h_a_d_o_w_{_}_D_O_M =
  l_get_child_nodes_S_h_a_d_o_w_{_}_D_O_M +
  l_new_element_get_child_nodes_C_o_r_e_{_}_D_O_M type_wf_C_o_r_e_{_}_D_O_M known_ptr_C_o_r_e_{_}_D_O_M get_child_nodes_C_o_r_e_{_}_D_O_M
  get_child_nodes_locs_C_o_r_e_{_}_D_O_M
begin
lemma get_child_nodes_new_element:
  "ptr' ≠ cast new_element_ptr ==> h ⊨ new_element →_r new_element_ptr ==> h ⊨ new_element →_h h'
     ==> r ∈ get_child_nodes_locs ptr' ==> r h h'"
  by (auto simp add: get_child_nodes_locs_def CD.get_child_nodes_locs_def new_element_get_M_O_b_j_e_c_t
      new_element_get_M_E_l_e_m_e_n_t new_element_get_M_D_o_c_u_m_e_n_t new_element_get_M_S_h_a_d_o_w_R_o_o_t
      split: prod.splits if_splits option.splits
      elim!: bind_returns_result_E bind_returns_heap_E
      intro: is_element_ptr_kind_obtains)

lemma new_element_no_child_nodes:
  "h ⊨ new_element →_r new_element_ptr ==> h ⊨ new_element →_h h'
   ==> h' ⊨ get_child_nodes (cast new_element_ptr) →_r []"
  apply(auto simp add: get_child_nodes_def a_get_child_nodes_tups_def
      split: prod.splits elim!: bind_returns_result_E bind_returns_heap_E)[1]
  apply(split CD.get_child_nodes_splits, rule conjI)+
  using local.new_element_no_child_nodes apply auto[1]
  apply(auto simp add: invoke_def)[1]
  apply(auto simp add: new_element_ptr_in_heap get_child_nodes_e_l_e_m_e_n_t_{_}_p_t_{r_def} check_in_heap_def
      new_element_child_nodes intro!: bind_pure_returns_result_I
      intro: new_element_is_element_ptr elim!: new_element_ptr_in_heap)[1]
proof -
  assume " h ⊨ new_element →_r new_element_ptr"
  assume "h ⊨ new_element →_h h'"
  assume "¬ known_ptr_C_o_r_e_{_}_D_O_M (cast_e_l_e_m_e_n_t_{_}_p_t_r₂_o_b_j_e_c_t_{_}_p_t_r new_element_ptr)"
  moreover
  have "known_ptr (cast new_element_ptr)"
    using new_element_is_element_ptr ‹h ⊨ new_element →_r new_element_ptr›
    by(auto simp add: known_ptr_impl ShadowRootClass.a_known_ptr_def DocumentClass.a_known_ptr_def
        CharacterDataClass.a_known_ptr_def ElementClass.a_known_ptr_def)
  ultimately show "False"
    by(simp add: known_ptr_impl CD.known_ptr_impl ShadowRootClass.a_known_ptr_def
        is_document_ptr_kind_none)
qed
end

interpretation i_new_element_get_child_nodes?:
  l_new_element_get_child_nodes_S_h_a_d_o_w_{_}_D_O_M type_wf known_ptr DocumentClass.type_wf
  DocumentClass.known_ptr get_child_nodes get_child_nodes_locs Core_DOM_Functions.get_child_nodes
  Core_DOM_Functions.get_child_nodes_locs
  by(unfold_locales)
declare l_new_element_get_child_nodes_S_h_a_d_o_w_{_}_D_O_{M_axioms}[instances]

lemma new_element_get_child_nodes_is_l_new_element_get_child_nodes [instances]:
  "l_new_element_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs"
  using new_element_is_l_new_element get_child_nodes_is_l_get_child_nodes
  apply(auto simp add: l_new_element_get_child_nodes_def l_new_element_get_child_nodes_axioms_def)[1]
  using get_child_nodes_new_element new_element_no_child_nodes
  by fast+

subsubsection ‹delete\_shadow\_root›

locale l_delete_shadow_root_get_child_nodes_S_h_a_d_o_w_{_}_D_O_M =
  l_get_child_nodes_S_h_a_d_o_w_{_}_D_O_M
begin
lemma get_child_nodes_delete_shadow_root:
  "ptr' ≠ cast shadow_root_ptr ==> h ⊨ delete_S_h_a_d_o_w_R_o_o_{t_M} shadow_root_ptr →_h h' ==>
    r ∈ get_child_nodes_locs ptr' ==> r h h'"
  by(auto simp add: get_child_nodes_locs_def CD.get_child_nodes_locs_def
      delete_shadow_root_get_M_O_b_j_e_c_t delete_shadow_root_get_M_S_h_a_d_o_w_R_o_o_t
      delete_shadow_root_get_M_D_o_c_u_m_e_n_t delete_shadow_root_get_M_E_l_e_m_e_n_t
      split: if_splits option.splits
      intro: is_shadow_root_ptr_kind_obtains)
end

locale l_delete_shadow_root_get_child_nodes = l_get_child_nodes_defs +
  assumes get_child_nodes_delete_shadow_root:
    "ptr' ≠ cast shadow_root_ptr ==> h ⊨ delete_S_h_a_d_o_w_R_o_o_{t_M} shadow_root_ptr →_h h' ==>
      r ∈ get_child_nodes_locs ptr' ==> r h h'"

interpretation l_delete_shadow_root_get_child_nodes_S_h_a_d_o_w_{_}_D_O_M type_wf known_ptr DocumentClass.type_wf
  DocumentClass.known_ptr get_child_nodes get_child_nodes_locs Core_DOM_Functions.get_child_nodes
  Core_DOM_Functions.get_child_nodes_locs
  by(auto simp add: l_delete_shadow_root_get_child_nodes_S_h_a_d_o_w_{_}_D_O_{M_def} instances)

lemma l_delete_shadow_root_get_child_nodes_get_child_nodes_locs [instances]: "l_delete_shadow_root_get_child_nodes get_child_nodes_locs"
  apply(auto simp add: l_delete_shadow_root_get_child_nodes_def)[1]
  using get_child_nodes_delete_shadow_root apply fast
  done

subsubsection ‹set\_child\_nodes›

locale l_set_child_nodes_S_h_a_d_o_w_{_}_D_O_{M_defs} =
  CD: l_set_child_nodes_C_o_r_e_{_}_D_O_{M_defs}
begin
definition set_child_nodes_s_h_a_d_o_w_{_}_r_o_o_t_{_}_p_t_r :: "(_) shadow_root_ptr ==> (_) node_ptr list
    ==> (_, unit) dom_prog" where
  "set_child_nodes_s_h_a_d_o_w_{_}_r_o_o_t_{_}_p_t_r shadow_root_ptr = put_M shadow_root_ptr RShadowRoot.child_nodes_update"

definition a_set_child_nodes_tups :: "(((_) object_ptr ==> bool) × ((_) object_ptr ==> (_) node_ptr list
    ==> (_, unit) dom_prog)) list" where
  "a_set_child_nodes_tups ≡ [(is_shadow_root_ptr_o_b_j_e_c_t_{_}_p_t_r, set_child_nodes_s_h_a_d_o_w_{_}_r_o_o_t_{_}_p_t_r ∘ the ∘ cast)]"

definition a_set_child_nodes :: "(_) object_ptr ==> (_) node_ptr list ==> (_, unit) dom_prog" where
  "a_set_child_nodes ptr children = invoke (CD.a_set_child_nodes_tups @ a_set_child_nodes_tups)
    ptr children"

definition a_set_child_nodes_locs :: "(_) object_ptr ==> (_, unit) dom_prog set"
  where
    "a_set_child_nodes_locs ptr ≡
      (if is_shadow_root_ptr_kind ptr
      then all_args (put_M (the (cast ptr)) RShadowRoot.child_nodes_update)
      else {}) ∪ CD.a_set_child_nodes_locs ptr"
end

global_interpretation l_set_child_nodes_S_h_a_d_o_w_{_}_D_O_{M_defs} defines
  set_child_nodes = l_set_child_nodes_S_h_a_d_o_w_{_}_D_O_{M_defs}.a_set_child_nodes and
  set_child_nodes_locs = l_set_child_nodes_S_h_a_d_o_w_{_}_D_O_{M_defs}.a_set_child_nodes_locs
  .

locale l_set_child_nodes_S_h_a_d_o_w_{_}_D_O_M =
  l_type_wf type_wf +
  l_known_ptr known_ptr +
  l_set_child_nodes_S_h_a_d_o_w_{_}_D_O_{M_defs} +
  l_set_child_nodes_defs set_child_nodes set_child_nodes_locs +
  CD: l_set_child_nodes_C_o_r_e_{_}_D_O_M type_wf_C_o_r_e_{_}_D_O_M known_ptr_C_o_r_e_{_}_D_O_M set_child_nodes_C_o_r_e_{_}_D_O_M
  set_child_nodes_locs_C_o_r_e_{_}_D_O_M
  for type_wf :: "(_) heap ==> bool"
    and known_ptr :: "(_) object_ptr ==> bool"
    and type_wf_C_o_r_e_{_}_D_O_M :: "(_) heap ==> bool"
    and known_ptr_C_o_r_e_{_}_D_O_M :: "(_) object_ptr ==> bool"
    and set_child_nodes :: "(_) object_ptr ==> (_) node_ptr list ==> (_, unit) dom_prog"
    and set_child_nodes_locs :: "(_) object_ptr ==> (_, unit) dom_prog set"
    and set_child_nodes_C_o_r_e_{_}_D_O_M :: "(_) object_ptr ==> (_) node_ptr list ==> (_, unit) dom_prog"
    and set_child_nodes_locs_C_o_r_e_{_}_D_O_M :: "(_) object_ptr ==> (_, unit) dom_prog set" +
  assumes known_ptr_impl: "known_ptr = ShadowRootClass.known_ptr"
  assumes type_wf_impl: "type_wf = ShadowRootClass.type_wf"
  assumes set_child_nodes_impl: "set_child_nodes = a_set_child_nodes"
  assumes set_child_nodes_locs_impl: "set_child_nodes_locs = a_set_child_nodes_locs"
begin
lemmas set_child_nodes_def = set_child_nodes_impl[unfolded a_set_child_nodes_def set_child_nodes_def]
lemmas set_child_nodes_locs_def = set_child_nodes_locs_impl[unfolded a_set_child_nodes_locs_def
    set_child_nodes_locs_def, folded CD.set_child_nodes_locs_impl]

lemma set_child_nodes_writes: "writes (set_child_nodes_locs ptr) (set_child_nodes ptr children) h h'"
  apply (simp add: set_child_nodes_def a_set_child_nodes_tups_def set_child_nodes_locs_def)
  apply(split CD.set_child_nodes_splits, rule conjI)+
   apply (simp add: CD.set_child_nodes_writes writes_union_right_I)
  apply(split invoke_splits, rule conjI)+
   apply(auto simp add: a_set_child_nodes_def)[1]
  apply(auto simp add: set_child_nodes_s_h_a_d_o_w_{_}_r_o_o_t_{_}_p_t_{r_def}
      intro!: writes_bind_pure
      intro: writes_union_right_I writes_union_left_I
      split: list.splits)[1]
  by (simp add: is_shadow_root_ptr_kind_none)

lemma set_child_nodes_pointers_preserved:
  assumes "w ∈ set_child_nodes_locs object_ptr"
  assumes "h ⊨ w →_h h'"
  shows "object_ptr_kinds h = object_ptr_kinds h'"
  using assms(1) object_ptr_kinds_preserved[OF writes_singleton2 assms(2)]
  by(auto simp add: all_args_def set_child_nodes_locs_def CD.set_child_nodes_locs_def
      split: if_splits)

lemma set_child_nodes_types_preserved:
  assumes "w ∈ set_child_nodes_locs object_ptr"
  assumes "h ⊨ w →_h h'"
  shows "type_wf h = type_wf h'"
  using assms(1) type_wf_preserved[OF writes_singleton2 assms(2)]
  by(auto simp add: all_args_def type_wf_impl a_set_child_nodes_tups_def set_child_nodes_locs_def
      CD.set_child_nodes_locs_def
      split: if_splits option.splits)
end

interpretation
  i_set_child_nodes?: l_set_child_nodes_S_h_a_d_o_w_{_}_D_O_M type_wf known_ptr DocumentClass.type_wf
  DocumentClass.known_ptr set_child_nodes set_child_nodes_locs Core_DOM_Functions.set_child_nodes
  Core_DOM_Functions.set_child_nodes_locs
  apply(unfold_locales)
  by (auto simp add: set_child_nodes_def set_child_nodes_locs_def)
declare l_set_child_nodes_S_h_a_d_o_w_{_}_D_O_{M_axioms}[instances]

lemma set_child_nodes_is_l_set_child_nodes [instances]: "l_set_child_nodes type_wf set_child_nodes
    set_child_nodes_locs"
  using instances Shadow_DOM.i_set_child_nodes.set_child_nodes_pointers_preserved Shadow_DOM.i_set_child_nodes.set_child_nodes_writes i_set_child_nodes.set_child_nodes_types_preserved
  unfolding l_set_child_nodes_def
  by blast

paragraph ‹get\_child\_nodes›

locale l_set_child_nodes_get_child_nodes_S_h_a_d_o_w_{_}_D_O_M =
  l_get_child_nodes_S_h_a_d_o_w_{_}_D_O_M
  type_wf known_ptr type_wf_C_o_r_e_{_}_D_O_M known_ptr_C_o_r_e_{_}_D_O_M get_child_nodes get_child_nodes_locs
  get_child_nodes_C_o_r_e_{_}_D_O_M get_child_nodes_locs_C_o_r_e_{_}_D_O_M
  + l_set_child_nodes_S_h_a_d_o_w_{_}_D_O_M
  type_wf known_ptr type_wf_C_o_r_e_{_}_D_O_M known_ptr_C_o_r_e_{_}_D_O_M set_child_nodes set_child_nodes_locs
  set_child_nodes_C_o_r_e_{_}_D_O_M set_child_nodes_locs_C_o_r_e_{_}_D_O_M
  + CD: l_set_child_nodes_get_child_nodes_C_o_r_e_{_}_D_O_M
  type_wf_C_o_r_e_{_}_D_O_M known_ptr_C_o_r_e_{_}_D_O_M get_child_nodes_C_o_r_e_{_}_D_O_M get_child_nodes_locs_C_o_r_e_{_}_D_O_M
  set_child_nodes_C_o_r_e_{_}_D_O_M set_child_nodes_locs_C_o_r_e_{_}_D_O_M
  for type_wf :: "(_) heap ==> bool"
    and known_ptr :: "(_) object_ptr ==> bool"
    and type_wf_C_o_r_e_{_}_D_O_M :: "(_) heap ==> bool"
    and known_ptr_C_o_r_e_{_}_D_O_M :: "(_) object_ptr ==> bool"
    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_child_nodes_C_o_r_e_{_}_D_O_M :: "(_) object_ptr ==> ((_) heap, exception, (_) node_ptr list) prog"
    and get_child_nodes_locs_C_o_r_e_{_}_D_O_M :: "(_) object_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
    and set_child_nodes :: "(_) object_ptr ==> (_) node_ptr list ==> ((_) heap, exception, unit) prog"
    and set_child_nodes_locs :: "(_) object_ptr ==> ((_) heap, exception, unit) prog set"
    and set_child_nodes_C_o_r_e_{_}_D_O_M :: "(_) object_ptr ==> (_) node_ptr list ==> ((_) heap, exception, unit) prog"
    and set_child_nodes_locs_C_o_r_e_{_}_D_O_M :: "(_) object_ptr ==> ((_) heap, exception, unit) prog set"
begin

lemma set_child_nodes_get_child_nodes:
  assumes "known_ptr ptr"
  assumes "type_wf h"
  assumes "h ⊨ set_child_nodes ptr children →_h h'"
  shows "h' ⊨ get_child_nodes ptr →_r children"
proof -
  have "h ⊨ check_in_heap ptr →_r ()"
    using assms set_child_nodes_def invoke_ptr_in_heap
    by (metis (full_types) check_in_heap_ptr_in_heap is_OK_returns_heap_I is_OK_returns_result_E
        old.unit.exhaust)
  then have ptr_in_h: "ptr |∈| object_ptr_kinds h"
    by (simp add: check_in_heap_ptr_in_heap is_OK_returns_result_I)

  have "type_wf h'"
    apply(unfold type_wf_impl)
    apply(rule subst[where P=id, OF type_wf_preserved[OF set_child_nodes_writes assms(3),
            unfolded all_args_def], simplified])
    by(auto simp add: all_args_def assms(2)[unfolded type_wf_impl] set_child_nodes_locs_def
        CD.set_child_nodes_locs_def
        split: if_splits)
  have "h' ⊨ check_in_heap ptr →_r ()"
    using check_in_heap_reads set_child_nodes_writes assms(3) ‹h ⊨ check_in_heap ptr →_r ()›
    apply(rule reads_writes_separate_forwards)
    apply(auto simp add: all_args_def set_child_nodes_locs_def CD.set_child_nodes_locs_def)[1]
    done
  then have "ptr |∈| object_ptr_kinds h'"
    using check_in_heap_ptr_in_heap by blast
  with assms ptr_in_h ‹type_wf h'› show ?thesis
    apply(auto simp add: type_wf_impl known_ptr_impl  get_child_nodes_def a_get_child_nodes_tups_def
        set_child_nodes_def a_set_child_nodes_tups_def
        del: bind_pure_returns_result_I2
        intro!: bind_pure_returns_result_I2)[1]
    apply(split CD.get_child_nodes_splits, (rule conjI impI)+)+
     apply(split CD.set_child_nodes_splits)+
      apply(auto simp add: CD.set_child_nodes_get_child_nodes type_wf_impl CD.type_wf_impl
        dest: ShadowRootClass.type_wf_D_o_c_u_m_e_n_t)[1]
     apply(auto simp add: CD.set_child_nodes_get_child_nodes type_wf_impl CD.type_wf_impl
        dest: ShadowRootClass.type_wf_D_o_c_u_m_e_n_t)[1]
    apply(split CD.set_child_nodes_splits)+
    by(auto simp add: known_ptr_impl CD.known_ptr_impl set_child_nodes_s_h_a_d_o_w_{_}_r_o_o_t_{_}_p_t_{r_def}
        get_child_nodes_s_h_a_d_o_w_{_}_r_o_o_t_{_}_p_t_{r_def} CD.type_wf_impl ShadowRootClass.type_wf_D_o_c_u_m_e_n_t
        dest: known_ptr_new_shadow_root_ptr)[2]
qed

lemma set_child_nodes_get_child_nodes_different_pointers:
  assumes "ptr ≠ ptr'"
  assumes "w ∈ set_child_nodes_locs ptr"
  assumes "h ⊨ w →_h h'"
  assumes "r ∈ get_child_nodes_locs ptr'"
  shows "r h h'"
  using assms
  apply(auto simp add: set_child_nodes_locs_def CD.set_child_nodes_locs_def
      get_child_nodes_locs_def CD.get_child_nodes_locs_def)[1]
  by(auto simp add: all_args_def
      elim!: is_document_ptr_kind_obtains is_shadow_root_ptr_kind_obtains
      is_element_ptr_kind_obtains
      split: if_splits option.splits)

end

interpretation
  i_set_child_nodes_get_child_nodes?: l_set_child_nodes_get_child_nodes_S_h_a_d_o_w_{_}_D_O_M type_wf known_ptr
  DocumentClass.type_wf DocumentClass.known_ptr get_child_nodes get_child_nodes_locs
  Core_DOM_Functions.get_child_nodes Core_DOM_Functions.get_child_nodes_locs set_child_nodes
  set_child_nodes_locs Core_DOM_Functions.set_child_nodes Core_DOM_Functions.set_child_nodes_locs
  using instances
  by(auto simp add: l_set_child_nodes_get_child_nodes_S_h_a_d_o_w_{_}_D_O_{M_def} )
declare l_set_child_nodes_get_child_nodes_S_h_a_d_o_w_{_}_D_O_{M_axioms}[instances]

lemma set_child_nodes_get_child_nodes_is_l_set_child_nodes_get_child_nodes [instances]:
  "l_set_child_nodes_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs
    set_child_nodes set_child_nodes_locs"
  apply(auto simp add: instances l_set_child_nodes_get_child_nodes_def
      l_set_child_nodes_get_child_nodes_axioms_def)[1]
  using set_child_nodes_get_child_nodes apply fast
  using set_child_nodes_get_child_nodes_different_pointers apply fast
  done

subsubsection ‹set\_tag\_type›

locale l_set_tag_name_S_h_a_d_o_w_{_}_D_O_M =
  CD: l_set_tag_name_C_o_r_e_{_}_D_O_M type_wf_C_o_r_e_{_}_D_O_M set_tag_name set_tag_name_locs +
  l_type_wf type_wf
  for type_wf :: "(_) heap ==> bool"
    and type_wf_C_o_r_e_{_}_D_O_M :: "(_) heap ==> bool"
    and set_tag_name :: "(_) element_ptr ==> tag_name ==> (_, unit) dom_prog"
    and set_tag_name_locs :: "(_) element_ptr ==> (_, unit) dom_prog set" +
  assumes type_wf_impl: "type_wf = ShadowRootClass.type_wf"
begin
lemmas set_tag_name_def = CD.set_tag_name_impl[unfolded CD.a_set_tag_name_def set_tag_name_def]
lemmas set_tag_name_locs_def = CD.set_tag_name_locs_impl[unfolded CD.a_set_tag_name_locs_def
    set_tag_name_locs_def]

lemma set_tag_name_ok:
  "type_wf h ==> element_ptr |∈| element_ptr_kinds h ==> h ⊨ ok (set_tag_name element_ptr tag)"
  apply(unfold type_wf_impl)
  unfolding set_tag_name_impl[unfolded a_set_tag_name_def] using get_M_E_l_e_m_e_n_{t_ok} put_M_E_l_e_m_e_n_{t_ok}
  using CD.set_tag_name_ok CD.type_wf_impl ShadowRootClass.type_wf_D_o_c_u_m_e_n_t by blast

lemma set_tag_name_writes:
  "writes (set_tag_name_locs element_ptr) (set_tag_name element_ptr tag) h h'"
  using CD.set_tag_name_writes .

lemma set_tag_name_pointers_preserved:
  assumes "w ∈ set_tag_name_locs element_ptr"
  assumes "h ⊨ w →_h h'"
  shows "object_ptr_kinds h = object_ptr_kinds h'"
  using assms
  by(simp add: CD.set_tag_name_pointers_preserved)

lemma set_tag_name_typess_preserved:
  assumes "w ∈ set_tag_name_locs element_ptr"
  assumes "h ⊨ w →_h h'"
  shows "type_wf h = type_wf h'"
  apply(unfold type_wf_impl)
  apply(rule type_wf_preserved[OF writes_singleton2 assms(2)])
  using assms(1) set_tag_name_locs_def
  by(auto simp add: all_args_def set_tag_name_locs_def
      split: if_splits)
end

interpretation i_set_tag_name?: l_set_tag_name_S_h_a_d_o_w_{_}_D_O_M type_wf DocumentClass.type_wf set_tag_name
  set_tag_name_locs
  by(auto simp add: l_set_tag_name_S_h_a_d_o_w_{_}_D_O_{M_def} l_set_tag_name_S_h_a_d_o_w_{_}_D_O_{M_axioms_def} instances)
declare l_set_tag_name_S_h_a_d_o_w_{_}_D_O_{M_axioms} [instances]

lemma set_tag_name_is_l_set_tag_name [instances]:
  "l_set_tag_name type_wf set_tag_name set_tag_name_locs"
  apply(auto simp add: l_set_tag_name_def)[1]
  using set_tag_name_writes apply fast
  using set_tag_name_ok apply fast
  using set_tag_name_pointers_preserved apply (fast, fast)
  using set_tag_name_typess_preserved apply (fast, fast)
  done

paragraph ‹get\_child\_nodes›

locale l_set_tag_name_get_child_nodes_S_h_a_d_o_w_{_}_D_O_M =
  l_set_tag_name_S_h_a_d_o_w_{_}_D_O_M +
  l_get_child_nodes_S_h_a_d_o_w_{_}_D_O_M +
  CD: l_set_tag_name_get_child_nodes_C_o_r_e_{_}_D_O_M type_wf_C_o_r_e_{_}_D_O_M set_tag_name set_tag_name_locs
  known_ptr_C_o_r_e_{_}_D_O_M get_child_nodes_C_o_r_e_{_}_D_O_M get_child_nodes_locs_C_o_r_e_{_}_D_O_M
begin
lemma set_tag_name_get_child_nodes:
  "∀w ∈ set_tag_name_locs ptr. (h ⊨ w →_h h' ⟶ (∀r ∈ get_child_nodes_locs ptr'. r h h'))"
  apply(auto simp add: get_child_nodes_locs_def)[1]
    apply(auto simp add: set_tag_name_locs_def all_args_def)[1]
  using CD.set_tag_name_get_child_nodes apply(blast)
  using CD.set_tag_name_get_child_nodes apply(blast)
  done
end

interpretation
  i_set_tag_name_get_child_nodes?: l_set_tag_name_get_child_nodes_S_h_a_d_o_w_{_}_D_O_M type_wf
  DocumentClass.type_wf set_tag_name set_tag_name_locs known_ptr DocumentClass.known_ptr
  get_child_nodes get_child_nodes_locs Core_DOM_Functions.get_child_nodes
  Core_DOM_Functions.get_child_nodes_locs
  by unfold_locales
declare l_set_tag_name_get_child_nodes_S_h_a_d_o_w_{_}_D_O_{M_axioms}[instances]

lemma set_tag_name_get_child_nodes_is_l_set_tag_name_get_child_nodes [instances]:
  "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"
  using set_tag_name_is_l_set_tag_name get_child_nodes_is_l_get_child_nodes
  apply(simp add: l_set_tag_name_get_child_nodes_def
      l_set_tag_name_get_child_nodes_axioms_def)
  using set_tag_name_get_child_nodes
  by fast

subsubsection ‹get\_shadow\_root›

locale l_get_shadow_root_S_h_a_d_o_w_{_}_D_O_{M_defs}
begin
definition a_get_shadow_root :: "(_) element_ptr ==> (_, (_) shadow_root_ptr option) dom_prog"
  where
    "a_get_shadow_root element_ptr = get_M element_ptr shadow_root_opt"

definition a_get_shadow_root_locs :: "(_) element_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
  where
    "a_get_shadow_root_locs element_ptr ≡ {preserved (get_M element_ptr shadow_root_opt)}"
end

global_interpretation l_get_shadow_root_S_h_a_d_o_w_{_}_D_O_{M_defs}
  defines get_shadow_root = a_get_shadow_root
    and get_shadow_root_locs = a_get_shadow_root_locs
  .

locale l_get_shadow_root_defs =
  fixes get_shadow_root :: "(_) element_ptr ==> (_, (_) shadow_root_ptr option) dom_prog"
  fixes get_shadow_root_locs :: "(_) element_ptr ==> ((_) heap ==> (_) heap ==> bool) set"

locale l_get_shadow_root_S_h_a_d_o_w_{_}_D_O_M =
  l_get_shadow_root_S_h_a_d_o_w_{_}_D_O_{M_defs} +
  l_get_shadow_root_defs get_shadow_root get_shadow_root_locs +
  l_type_wf type_wf
  for type_wf :: "(_) heap ==> bool"
    and get_shadow_root :: "(_) element_ptr ==> ((_) heap, exception, (_) shadow_root_ptr option) prog"
    and get_shadow_root_locs :: "(_) element_ptr ==> ((_) heap ==> (_) heap ==> bool) set" +
  assumes type_wf_impl: "type_wf = ShadowRootClass.type_wf"
  assumes get_shadow_root_impl: "get_shadow_root = a_get_shadow_root"
  assumes get_shadow_root_locs_impl: "get_shadow_root_locs = a_get_shadow_root_locs"
begin
lemmas get_shadow_root_def = get_shadow_root_impl[unfolded get_shadow_root_def a_get_shadow_root_def]
lemmas get_shadow_root_locs_def = get_shadow_root_locs_impl[unfolded get_shadow_root_locs_def
    a_get_shadow_root_locs_def]

lemma get_shadow_root_ok:
  "type_wf h ==> element_ptr |∈| element_ptr_kinds h ==> h ⊨ ok (get_shadow_root element_ptr)"
  unfolding get_shadow_root_def type_wf_impl
  using ShadowRootMonad.get_M_E_l_e_m_e_n_{t_ok} by blast

lemma get_shadow_root_pure [simp]: "pure (get_shadow_root element_ptr) h"
  unfolding get_shadow_root_def by simp

lemma get_shadow_root_ptr_in_heap:
  assumes "h ⊨ get_shadow_root element_ptr →_r children"
  shows "element_ptr |∈| element_ptr_kinds h"
  using assms
  by(auto simp add: get_shadow_root_def get_M_E_l_e_m_e_n_{t_ptr_in_heap} dest: is_OK_returns_result_I)

lemma get_shadow_root_reads:
  "reads (get_shadow_root_locs element_ptr) (get_shadow_root element_ptr) h h'"
  by(simp add: get_shadow_root_def get_shadow_root_locs_def reads_bind_pure
      reads_insert_writes_set_right)
end

interpretation i_get_shadow_root?: l_get_shadow_root_S_h_a_d_o_w_{_}_D_O_M type_wf get_shadow_root
  get_shadow_root_locs
  using instances
  by (auto simp add: l_get_shadow_root_S_h_a_d_o_w_{_}_D_O_{M_def})
declare l_get_shadow_root_S_h_a_d_o_w_{_}_D_O_{M_axioms} [instances]

locale l_get_shadow_root = l_type_wf + l_get_shadow_root_defs +
  assumes get_shadow_root_reads:
    "reads (get_shadow_root_locs element_ptr) (get_shadow_root element_ptr) h h'"
  assumes get_shadow_root_ok:
    "type_wf h ==> element_ptr |∈| element_ptr_kinds h ==> h ⊨ ok (get_shadow_root element_ptr)"
  assumes get_shadow_root_ptr_in_heap:
    "h ⊨ ok (get_shadow_root element_ptr) ==> element_ptr |∈| element_ptr_kinds h"
  assumes get_shadow_root_pure [simp]:
    "pure (get_shadow_root element_ptr) h"

lemma get_shadow_root_is_l_get_shadow_root [instances]:
  "l_get_shadow_root type_wf get_shadow_root get_shadow_root_locs"
  using instances
  unfolding l_get_shadow_root_def
  by (metis (no_types, lifting) ElementClass.l_type_wf_E_l_e_m_e_n_{t_axioms} get_shadow_root_ok get_shadow_root_pure get_shadow_root_reads l_get_M_E_l_e_m_e_n_{t_lemmas}.get_M_E_l_e_m_e_n_{t_ptr_in_heap} l_get_M_E_l_e_m_e_n_{t_lemmas}.intro l_get_shadow_root_S_h_a_d_o_w_{_}_D_O_M.get_shadow_root_def)

paragraph ‹set\_disconnected\_nodes›

locale l_set_disconnected_nodes_get_shadow_root_S_h_a_d_o_w_{_}_D_O_M =
  l_set_disconnected_nodes_C_o_r_e_{_}_D_O_M type_wf_C_o_r_e_{_}_D_O_M set_disconnected_nodes set_disconnected_nodes_locs +
  l_get_shadow_root_S_h_a_d_o_w_{_}_D_O_M type_wf get_shadow_root get_shadow_root_locs
  for type_wf :: "(_) heap ==> bool"
    and type_wf_C_o_r_e_{_}_D_O_M :: "(_) heap ==> bool"
    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 get_shadow_root :: "(_) element_ptr ==> ((_) heap, exception, (_) shadow_root_ptr option) prog"
    and get_shadow_root_locs :: "(_) element_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
begin
lemma set_disconnected_nodes_get_shadow_root:
  "∀w ∈ set_disconnected_nodes_locs ptr. (h ⊨ w →_h h' ⟶ (∀r ∈ get_shadow_root_locs ptr'. r h h'))"
  by(auto simp add: set_disconnected_nodes_locs_def get_shadow_root_locs_def all_args_def)
end

locale l_set_disconnected_nodes_get_shadow_root =
  l_set_disconnected_nodes_defs +
  l_get_shadow_root_defs +
  assumes set_disconnected_nodes_get_shadow_root:
    "∀w ∈ set_disconnected_nodes_locs ptr. (h ⊨ w →_h h' ⟶ (∀r ∈ get_shadow_root_locs ptr'. r h h'))"

interpretation
  i_set_disconnected_nodes_get_shadow_root?: l_set_disconnected_nodes_get_shadow_root_S_h_a_d_o_w_{_}_D_O_M
  type_wf DocumentClass.type_wf set_disconnected_nodes set_disconnected_nodes_locs get_shadow_root
  get_shadow_root_locs
  by(auto simp add: l_set_disconnected_nodes_get_shadow_root_S_h_a_d_o_w_{_}_D_O_{M_def} instances)
declare l_set_disconnected_nodes_get_shadow_root_S_h_a_d_o_w_{_}_D_O_{M_axioms}[instances]

lemma set_disconnected_nodes_get_shadow_root_is_l_set_disconnected_nodes_get_shadow_root [instances]:
  "l_set_disconnected_nodes_get_shadow_root set_disconnected_nodes_locs get_shadow_root_locs"
  apply(auto simp add: l_set_disconnected_nodes_get_shadow_root_def)[1]
  using set_disconnected_nodes_get_shadow_root apply fast
  done

paragraph ‹set\_tag\_type›

locale l_set_tag_name_get_shadow_root_C_o_r_e_{_}_D_O_M =
  l_set_tag_name_S_h_a_d_o_w_{_}_D_O_M +
  l_get_shadow_root_S_h_a_d_o_w_{_}_D_O_M
begin
lemma set_tag_name_get_shadow_root:
  "∀w ∈ set_tag_name_locs ptr. (h ⊨ w →_h h' ⟶ (∀r ∈ get_shadow_root_locs ptr'. r h h'))"
  by(auto simp add: set_tag_name_locs_def
      get_shadow_root_locs_def all_args_def
      intro: element_put_get_preserved[where setter=tag_name_update and getter=shadow_root_opt])
end

locale l_set_tag_name_get_shadow_root = l_set_tag_name + l_get_shadow_root +
  assumes set_tag_name_get_shadow_root:
    "∀w ∈ set_tag_name_locs ptr. (h ⊨ w →_h h' ⟶ (∀r ∈ get_shadow_root_locs ptr'. r h h'))"

interpretation
  i_set_tag_name_get_shadow_root?: l_set_tag_name_get_shadow_root_C_o_r_e_{_}_D_O_M type_wf DocumentClass.type_wf
  set_tag_name set_tag_name_locs
  get_shadow_root get_shadow_root_locs
  apply(auto simp add: l_set_tag_name_get_shadow_root_C_o_r_e_{_}_D_O_{M_def} instances)[1]
  using l_set_tag_name_S_h_a_d_o_w_{_}_D_O_{M_axioms}
  by unfold_locales
declare l_set_tag_name_get_shadow_root_C_o_r_e_{_}_D_O_{M_axioms}[instances]

lemma set_tag_name_get_shadow_root_is_l_set_tag_name_get_shadow_root [instances]:
  "l_set_tag_name_get_shadow_root type_wf set_tag_name set_tag_name_locs get_shadow_root
                                  get_shadow_root_locs"
  using set_tag_name_is_l_set_tag_name get_shadow_root_is_l_get_shadow_root
  apply(simp add: l_set_tag_name_get_shadow_root_def l_set_tag_name_get_shadow_root_axioms_def)
  using set_tag_name_get_shadow_root
  by fast

paragraph ‹set\_child\_nodes›

locale l_set_child_nodes_get_shadow_root_S_h_a_d_o_w_{_}_D_O_M =
  l_set_child_nodes_S_h_a_d_o_w_{_}_D_O_M type_wf known_ptr type_wf_C_o_r_e_{_}_D_O_M known_ptr_C_o_r_e_{_}_D_O_M set_child_nodes
  set_child_nodes_locs set_child_nodes_C_o_r_e_{_}_D_O_M set_child_nodes_locs_C_o_r_e_{_}_D_O_M +
  l_get_shadow_root_S_h_a_d_o_w_{_}_D_O_M type_wf get_shadow_root get_shadow_root_locs
  for type_wf :: "(_) heap ==> bool"
    and known_ptr :: "(_) object_ptr ==> bool"
    and type_wf_C_o_r_e_{_}_D_O_M :: "(_) heap ==> bool"
    and known_ptr_C_o_r_e_{_}_D_O_M :: "(_) object_ptr ==> bool"
    and set_child_nodes :: "(_) object_ptr ==> (_) node_ptr list ==> ((_) heap, exception, unit) prog"
    and set_child_nodes_locs :: "(_) object_ptr ==> ((_) heap, exception, unit) prog set"
    and set_child_nodes_C_o_r_e_{_}_D_O_M :: "(_) object_ptr ==> (_) node_ptr list ==> ((_) heap, exception, unit) prog"
    and set_child_nodes_locs_C_o_r_e_{_}_D_O_M :: "(_) object_ptr ==> ((_) heap, exception, unit) prog set"
    and get_shadow_root :: "(_) element_ptr ==> ((_) heap, exception, (_) shadow_root_ptr option) prog"
    and get_shadow_root_locs :: "(_) element_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
begin
lemma set_child_nodes_get_shadow_root: "∀w ∈ set_child_nodes_locs ptr. (h ⊨ w →_h h' ⟶
  (∀r ∈ get_shadow_root_locs ptr'. r h h'))"
  apply(auto simp add: set_child_nodes_locs_def get_shadow_root_locs_def CD.set_child_nodes_locs_def
      all_args_def)[1]
  by(auto intro!: element_put_get_preserved[where getter=shadow_root_opt and
        setter=RElement.child_nodes_update])
end

locale l_set_child_nodes_get_shadow_root = l_set_child_nodes_defs + l_get_shadow_root_defs +
  assumes set_child_nodes_get_shadow_root:
    "∀w ∈ set_child_nodes_locs ptr. (h ⊨ w →_h h' ⟶ (∀r ∈ get_shadow_root_locs ptr'. r h h'))"

interpretation
  i_set_child_nodes_get_shadow_root?: l_set_child_nodes_get_shadow_root_S_h_a_d_o_w_{_}_D_O_M type_wf known_ptr
  DocumentClass.type_wf DocumentClass.known_ptr set_child_nodes set_child_nodes_locs
  Core_DOM_Functions.set_child_nodes Core_DOM_Functions.set_child_nodes_locs get_shadow_root
  get_shadow_root_locs
  by(auto simp add: l_set_child_nodes_get_shadow_root_S_h_a_d_o_w_{_}_D_O_{M_def} instances)
declare l_set_child_nodes_get_shadow_root_S_h_a_d_o_w_{_}_D_O_{M_axioms}[instances]

lemma set_child_nodes_get_shadow_root_is_l_set_child_nodes_get_shadow_root [instances]:
  "l_set_child_nodes_get_shadow_root set_child_nodes_locs get_shadow_root_locs"
  apply(auto simp add: l_set_child_nodes_get_shadow_root_def)[1]
  using set_child_nodes_get_shadow_root apply fast
  done

paragraph ‹delete\_shadow\_root›

locale l_delete_shadow_root_get_shadow_root_S_h_a_d_o_w_{_}_D_O_M =
  l_get_shadow_root_S_h_a_d_o_w_{_}_D_O_M
begin
lemma get_shadow_root_delete_shadow_root:  "h ⊨ delete_S_h_a_d_o_w_R_o_o_{t_M} shadow_root_ptr →_h h'
    ==> r ∈ get_shadow_root_locs ptr' ==> r h h'"
  by(auto simp add: get_shadow_root_locs_def delete_shadow_root_get_M_E_l_e_m_e_n_t)
end

locale l_delete_shadow_root_get_shadow_root = l_get_shadow_root_defs +
  assumes get_shadow_root_delete_shadow_root:  "h ⊨ delete_S_h_a_d_o_w_R_o_o_{t_M} shadow_root_ptr →_h h'
    ==> r ∈ get_shadow_root_locs ptr' ==> r h h'"
interpretation l_delete_shadow_root_get_shadow_root_S_h_a_d_o_w_{_}_D_O_M type_wf get_shadow_root
  get_shadow_root_locs
  by(auto simp add: l_delete_shadow_root_get_shadow_root_S_h_a_d_o_w_{_}_D_O_{M_def} instances)

lemma l_delete_shadow_root_get_shadow_root_get_shadow_root_locs [instances]: "l_delete_shadow_root_get_shadow_root get_shadow_root_locs"
  apply(auto simp add: l_delete_shadow_root_get_shadow_root_def)[1]
  using get_shadow_root_delete_shadow_root apply fast
  done

paragraph ‹new\_character\_data›

locale l_new_character_data_get_shadow_root_S_h_a_d_o_w_{_}_D_O_M =
  l_get_shadow_root_S_h_a_d_o_w_{_}_D_O_M type_wf get_shadow_root get_shadow_root_locs
  for type_wf :: "(_) heap ==> bool"
    and get_shadow_root :: "(_) element_ptr ==> ((_) heap, exception, (_) shadow_root_ptr option) prog"
    and get_shadow_root_locs :: "(_) element_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
begin
lemma get_shadow_root_new_character_data:
  "h ⊨ new_character_data →_r new_character_data_ptr ==> h ⊨ new_character_data →_h h'
     ==> r ∈ get_shadow_root_locs ptr' ==> r h h'"
  by (auto simp add: get_shadow_root_locs_def new_character_data_get_M_O_b_j_e_c_t
      new_character_data_get_M_E_l_e_m_e_n_t
      split: prod.splits if_splits option.splits
      elim!: bind_returns_result_E bind_returns_heap_E
      intro: is_element_ptr_kind_obtains)
end

locale l_new_character_data_get_shadow_root = l_new_character_data + l_get_shadow_root +
  assumes get_shadow_root_new_character_data:
    "h ⊨ new_character_data →_r new_character_data_ptr
            ==> h ⊨ new_character_data →_h h' ==> r ∈ get_shadow_root_locs ptr' ==> r h h'"

interpretation i_new_character_data_get_shadow_root?:
  l_new_character_data_get_shadow_root_S_h_a_d_o_w_{_}_D_O_M type_wf get_shadow_root get_shadow_root_locs
  by(unfold_locales)
declare l_new_character_data_get_shadow_root_S_h_a_d_o_w_{_}_D_O_{M_axioms} [instances]

lemma new_character_data_get_shadow_root_is_l_new_character_data_get_shadow_root [instances]:
  "l_new_character_data_get_shadow_root type_wf get_shadow_root get_shadow_root_locs"
  using new_character_data_is_l_new_character_data get_shadow_root_is_l_get_shadow_root
  apply(auto simp add: l_new_character_data_get_shadow_root_def
      l_new_character_data_get_shadow_root_axioms_def instances)[1]
  using get_shadow_root_new_character_data
  by fast

paragraph ‹new\_document›

locale l_new_document_get_shadow_root_S_h_a_d_o_w_{_}_D_O_M =
  l_get_shadow_root_S_h_a_d_o_w_{_}_D_O_M type_wf get_shadow_root get_shadow_root_locs
  for type_wf :: "(_) heap ==> bool"
    and get_shadow_root :: "(_) element_ptr ==> ((_) heap, exception, (_) shadow_root_ptr option) prog"
    and get_shadow_root_locs :: "(_) element_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
begin
lemma get_shadow_root_new_document:
  "h ⊨ new_document →_r new_document_ptr ==> h ⊨ new_document →_h h'
     ==> r ∈ get_shadow_root_locs ptr' ==> r h h'"
  by (auto simp add: get_shadow_root_locs_def new_document_get_M_O_b_j_e_c_t new_document_get_M_E_l_e_m_e_n_t
      split: prod.splits if_splits option.splits
      elim!: bind_returns_result_E bind_returns_heap_E intro: is_element_ptr_kind_obtains)
end

locale l_new_document_get_shadow_root = l_new_document + l_get_shadow_root +
  assumes get_shadow_root_new_document:
    "h ⊨ new_document →_r new_document_ptr
            ==> h ⊨ new_document →_h h' ==> r ∈ get_shadow_root_locs ptr' ==> r h h'"

interpretation i_new_document_get_shadow_root?:
  l_new_document_get_shadow_root_S_h_a_d_o_w_{_}_D_O_M type_wf get_shadow_root get_shadow_root_locs
  by(unfold_locales)
declare l_new_document_get_shadow_root_S_h_a_d_o_w_{_}_D_O_{M_axioms} [instances]

lemma new_document_get_shadow_root_is_l_new_document_get_shadow_root [instances]:
  "l_new_document_get_shadow_root type_wf get_shadow_root get_shadow_root_locs"
  using new_document_is_l_new_document get_shadow_root_is_l_get_shadow_root
  apply(auto simp add: l_new_document_get_shadow_root_def l_new_document_get_shadow_root_axioms_def
      instances)[1]
  using get_shadow_root_new_document
  by fast

paragraph ‹new\_element›

locale l_new_element_get_shadow_root_S_h_a_d_o_w_{_}_D_O_M =
  l_get_shadow_root_S_h_a_d_o_w_{_}_D_O_M type_wf get_shadow_root get_shadow_root_locs
  for type_wf :: "(_) heap ==> bool"
    and get_shadow_root :: "(_) element_ptr ==> ((_) heap, exception, (_) shadow_root_ptr option) prog"
    and get_shadow_root_locs :: "(_) element_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
begin
lemma get_shadow_root_new_element:
  "ptr' ≠ new_element_ptr ==> h ⊨ new_element →_r new_element_ptr ==> h ⊨ new_element →_h h'
     ==> r ∈ get_shadow_root_locs ptr' ==> r h h'"
  by (auto simp add: get_shadow_root_locs_def new_element_get_M_O_b_j_e_c_t new_element_get_M_E_l_e_m_e_n_t
      new_element_get_M_D_o_c_u_m_e_n_t split: prod.splits if_splits option.splits
      elim!: bind_returns_result_E bind_returns_heap_E intro: is_element_ptr_kind_obtains)

lemma new_element_no_shadow_root:
  "h ⊨ new_element →_r new_element_ptr ==> h ⊨ new_element →_h h'
   ==> h' ⊨ get_shadow_root new_element_ptr →_r None"
  by(simp add: get_shadow_root_def new_element_shadow_root_opt)
end

locale l_new_element_get_shadow_root = l_new_element + l_get_shadow_root +
  assumes get_shadow_root_new_element:
    "ptr' ≠ new_element_ptr ==> h ⊨ new_element →_r new_element_ptr
            ==> h ⊨ new_element →_h h' ==> r ∈ get_shadow_root_locs ptr' ==> r h h'"
  assumes new_element_no_shadow_root:
    "h ⊨ new_element →_r new_element_ptr ==> h ⊨ new_element →_h h'
            ==> h' ⊨ get_shadow_root new_element_ptr →_r None"

interpretation i_new_element_get_shadow_root?:
  l_new_element_get_shadow_root_S_h_a_d_o_w_{_}_D_O_M type_wf get_shadow_root get_shadow_root_locs
  by(unfold_locales)
declare l_new_element_get_shadow_root_S_h_a_d_o_w_{_}_D_O_{M_axioms} [instances]

lemma new_element_get_shadow_root_is_l_new_element_get_shadow_root [instances]:
  "l_new_element_get_shadow_root type_wf get_shadow_root get_shadow_root_locs"
  using new_element_is_l_new_element get_shadow_root_is_l_get_shadow_root
  apply(auto simp add: l_new_element_get_shadow_root_def l_new_element_get_shadow_root_axioms_def
      instances)[1]
  using get_shadow_root_new_element new_element_no_shadow_root
  by fast+

paragraph ‹new\_shadow\_root›

locale l_new_shadow_root_get_shadow_root_S_h_a_d_o_w_{_}_D_O_M =
  l_get_shadow_root_S_h_a_d_o_w_{_}_D_O_M type_wf get_shadow_root get_shadow_root_locs
  for type_wf :: "(_) heap ==> bool"
    and get_shadow_root :: "(_) element_ptr ==> ((_) heap, exception, (_) shadow_root_ptr option) prog"
    and get_shadow_root_locs :: "(_) element_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
begin
lemma get_shadow_root_new_shadow_root:
  "h ⊨ new_S_h_a_d_o_w_R_o_o_{t_M} →_r new_shadow_root_ptr ==> h ⊨ new_S_h_a_d_o_w_R_o_o_{t_M} →_h h'
     ==> r ∈ get_shadow_root_locs ptr' ==> r h h'"
  by (auto simp add: get_shadow_root_locs_def new_shadow_root_get_M_O_b_j_e_c_t new_shadow_root_get_M_E_l_e_m_e_n_t
      split: prod.splits if_splits option.splits
      elim!: bind_returns_result_E bind_returns_heap_E intro: is_element_ptr_kind_obtains)
end

locale l_new_shadow_root_get_shadow_root = l_get_shadow_root +
  assumes get_shadow_root_new_shadow_root:
    "h ⊨ new_S_h_a_d_o_w_R_o_o_{t_M} →_r new_shadow_root_ptr
            ==> h ⊨ new_S_h_a_d_o_w_R_o_o_{t_M} →_h h' ==> r ∈ get_shadow_root_locs ptr' ==> r h h'"

interpretation i_new_shadow_root_get_shadow_root?:
  l_new_shadow_root_get_shadow_root_S_h_a_d_o_w_{_}_D_O_M type_wf get_shadow_root get_shadow_root_locs
  by(unfold_locales)
declare l_new_shadow_root_get_shadow_root_S_h_a_d_o_w_{_}_D_O_{M_axioms} [instances]

lemma new_shadow_root_get_shadow_root_is_l_new_shadow_root_get_shadow_root [instances]:
  "l_new_shadow_root_get_shadow_root type_wf get_shadow_root get_shadow_root_locs"
  using get_shadow_root_is_l_get_shadow_root
  apply(auto simp add: l_new_shadow_root_get_shadow_root_def
      l_new_shadow_root_get_shadow_root_axioms_def instances)[1]
  using get_shadow_root_new_shadow_root
  by fast

subsubsection ‹set\_shadow\_root›

locale l_set_shadow_root_S_h_a_d_o_w_{_}_D_O_{M_defs}
begin
definition a_set_shadow_root :: "(_) element_ptr ==> (_) shadow_root_ptr option ==> (_, unit) dom_prog"
  where
    "a_set_shadow_root element_ptr = put_M element_ptr shadow_root_opt_update"

definition a_set_shadow_root_locs :: "(_) element_ptr ==> ((_, unit) dom_prog) set"
  where
    "a_set_shadow_root_locs element_ptr ≡ all_args (put_M element_ptr shadow_root_opt_update)"
end

global_interpretation l_set_shadow_root_S_h_a_d_o_w_{_}_D_O_{M_defs}
  defines set_shadow_root = a_set_shadow_root
    and set_shadow_root_locs = a_set_shadow_root_locs
  .

locale l_set_shadow_root_defs =
  fixes set_shadow_root :: "(_) element_ptr ==> (_) shadow_root_ptr option ==> (_, unit) dom_prog"
  fixes set_shadow_root_locs :: "(_) element_ptr ==> (_, unit) dom_prog set"

locale l_set_shadow_root_S_h_a_d_o_w_{_}_D_O_M =
  l_type_wf type_wf +
  l_set_shadow_root_defs set_shadow_root set_shadow_root_locs +
  l_set_shadow_root_S_h_a_d_o_w_{_}_D_O_{M_defs}
  for type_wf :: "(_) heap ==> bool"
    and set_shadow_root :: "(_) element_ptr ==> (_) shadow_root_ptr option ==> (_, unit) dom_prog"
    and set_shadow_root_locs :: "(_) element_ptr ==> (_, unit) dom_prog set" +
  assumes type_wf_impl: "type_wf = ShadowRootClass.type_wf"
  assumes set_shadow_root_impl: "set_shadow_root = a_set_shadow_root"
  assumes set_shadow_root_locs_impl: "set_shadow_root_locs = a_set_shadow_root_locs"
begin
lemmas set_shadow_root_def = set_shadow_root_impl[unfolded set_shadow_root_def
    a_set_shadow_root_def]
lemmas set_shadow_root_locs_def = set_shadow_root_locs_impl[unfolded set_shadow_root_locs_def
    a_set_shadow_root_locs_def]

lemma set_shadow_root_ok: "type_wf h ==> element_ptr |∈| element_ptr_kinds h ==>
  h ⊨ ok (set_shadow_root element_ptr tag)"
  apply(unfold type_wf_impl)
  unfolding set_shadow_root_def using get_M_E_l_e_m_e_n_{t_ok} put_M_E_l_e_m_e_n_{t_ok}
  by (simp add: ShadowRootMonad.put_M_E_l_e_m_e_n_{t_ok})

lemma set_shadow_root_ptr_in_heap:
  "h ⊨ ok (set_shadow_root element_ptr shadow_root) ==> element_ptr |∈| element_ptr_kinds h"
  by(simp add: set_shadow_root_def ElementMonad.put_M_ptr_in_heap)

lemma set_shadow_root_writes:
  "writes (set_shadow_root_locs element_ptr) (set_shadow_root element_ptr tag) h h'"
  by(auto simp add: set_shadow_root_def set_shadow_root_locs_def intro: writes_bind_pure)

lemma set_shadow_root_pointers_preserved:
  assumes "w ∈ set_shadow_root_locs element_ptr"
  assumes "h ⊨ w →_h h'"
  shows "object_ptr_kinds h = object_ptr_kinds h'"
  using assms(1) object_ptr_kinds_preserved[OF writes_singleton2 assms(2)]
  by(auto simp add: all_args_def set_shadow_root_locs_def split: if_splits)

lemma set_shadow_root_types_preserved:
  assumes "w ∈ set_shadow_root_locs element_ptr"
  assumes "h ⊨ w →_h h'"
  shows "type_wf h = type_wf h'"
  apply(unfold type_wf_impl)
  using assms(1) type_wf_preserved[OF writes_singleton2 assms(2)]
  by(auto simp add: all_args_def set_shadow_root_locs_def split: if_splits)
end

interpretation i_set_shadow_root?: l_set_shadow_root_S_h_a_d_o_w_{_}_D_O_M type_wf set_shadow_root
  set_shadow_root_locs
  by (auto simp add: l_set_shadow_root_S_h_a_d_o_w_{_}_D_O_{M_def} instances)
declare l_set_shadow_root_S_h_a_d_o_w_{_}_D_O_{M_axioms} [instances]

locale l_set_shadow_root = l_type_wf +l_set_shadow_root_defs +
  assumes set_shadow_root_writes:
    "writes (set_shadow_root_locs element_ptr) (set_shadow_root element_ptr disc_nodes) h h'"
  assumes set_shadow_root_ok:
    "type_wf h ==> element_ptr |∈| element_ptr_kinds h ==>
      h ⊨ ok (set_shadow_root element_ptr shadow_root)"
  assumes set_shadow_root_ptr_in_heap:
    "h ⊨ ok (set_shadow_root element_ptr shadow_root) ==> element_ptr |∈| element_ptr_kinds h"
  assumes set_shadow_root_pointers_preserved:
    "w ∈ set_shadow_root_locs element_ptr ==> h ⊨ w →_h h' ==>
      object_ptr_kinds h = object_ptr_kinds h'"
  assumes set_shadow_root_types_preserved:
    "w ∈ set_shadow_root_locs element_ptr ==> h ⊨ w →_h h' ==> type_wf h = type_wf h'"

lemma set_shadow_root_is_l_set_shadow_root [instances]:
  "l_set_shadow_root type_wf set_shadow_root set_shadow_root_locs"
  apply(auto simp add: l_set_shadow_root_def instances)[1]
  using set_shadow_root_writes apply blast
  using set_shadow_root_ok apply (blast)
  using set_shadow_root_ptr_in_heap apply blast
  using set_shadow_root_pointers_preserved apply(blast, blast)
  using set_shadow_root_types_preserved apply(blast, blast)
  done

paragraph ‹get\_shadow\_root›

locale l_set_shadow_root_get_shadow_root_S_h_a_d_o_w_{_}_D_O_M =
  l_set_shadow_root_S_h_a_d_o_w_{_}_D_O_M +
  l_get_shadow_root_S_h_a_d_o_w_{_}_D_O_M
begin
lemma set_shadow_root_get_shadow_root:
  "type_wf h ==> h ⊨ set_shadow_root ptr shadow_root_ptr_opt →_h h' ==>
  h' ⊨ get_shadow_root ptr →_r shadow_root_ptr_opt"
  by(auto simp add: set_shadow_root_def get_shadow_root_def)

lemma set_shadow_root_get_shadow_root_different_pointers: "ptr ≠ ptr' ==>
  ∀w ∈ set_shadow_root_locs ptr. (h ⊨ w →_h h' ⟶ (∀r ∈ get_shadow_root_locs ptr'. r h h'))"
  by(auto simp add: set_shadow_root_locs_def get_shadow_root_locs_def all_args_def)
end

interpretation i_set_shadow_root_get_shadow_root?: l_set_shadow_root_get_shadow_root_S_h_a_d_o_w_{_}_D_O_M type_wf
  set_shadow_root set_shadow_root_locs get_shadow_root get_shadow_root_locs
  apply(auto simp add: l_set_shadow_root_get_shadow_root_S_h_a_d_o_w_{_}_D_O_{M_def} instances)[1]
  by(unfold_locales)
declare l_set_shadow_root_get_shadow_root_S_h_a_d_o_w_{_}_D_O_{M_axioms}[instances]

locale l_set_shadow_root_get_shadow_root =
  l_type_wf +
  l_set_shadow_root_defs +
  l_get_shadow_root_defs +
  assumes set_shadow_root_get_shadow_root:
    "type_wf h ==> h ⊨ set_shadow_root ptr shadow_root_ptr_opt →_h h' ==>
      h' ⊨ get_shadow_root ptr →_r shadow_root_ptr_opt"
  assumes set_shadow_root_get_shadow_root_different_pointers:
    "ptr ≠ ptr' ==> w ∈ set_shadow_root_locs ptr ==> h ⊨ w →_h h' ==>
      r ∈ get_shadow_root_locs ptr' ==> r h h'"

lemma set_shadow_root_get_shadow_root_is_l_set_shadow_root_get_shadow_root [instances]:
  "l_set_shadow_root_get_shadow_root type_wf set_shadow_root set_shadow_root_locs get_shadow_root
  get_shadow_root_locs"
  apply(auto simp add: l_set_shadow_root_get_shadow_root_def instances)[1]
  using set_shadow_root_get_shadow_root apply fast
  using set_shadow_root_get_shadow_root_different_pointers apply fast
  done

subsubsection ‹set\_mode›

locale l_set_mode_S_h_a_d_o_w_{_}_D_O_{M_defs}
begin
definition a_set_mode :: "(_) shadow_root_ptr ==> shadow_root_mode ==> (_, unit) dom_prog"
  where
    "a_set_mode shadow_root_ptr = put_M shadow_root_ptr mode_update"

definition a_set_mode_locs :: "(_) shadow_root_ptr ==> ((_, unit) dom_prog) set"
  where
    "a_set_mode_locs shadow_root_ptr ≡ all_args (put_M shadow_root_ptr mode_update)"
end

global_interpretation l_set_mode_S_h_a_d_o_w_{_}_D_O_{M_defs}
  defines set_mode = a_set_mode
    and set_mode_locs = a_set_mode_locs
  .

locale l_set_mode_defs =
  fixes set_mode :: "(_) shadow_root_ptr ==> shadow_root_mode ==> (_, unit) dom_prog"
  fixes set_mode_locs :: "(_) shadow_root_ptr ==> (_, unit) dom_prog set"

locale l_set_mode_S_h_a_d_o_w_{_}_D_O_M =
  l_type_wf type_wf +
  l_set_mode_defs set_mode set_mode_locs +
  l_set_mode_S_h_a_d_o_w_{_}_D_O_{M_defs}
  for type_wf :: "(_) heap ==> bool"
    and set_mode :: "(_) shadow_root_ptr ==> shadow_root_mode ==> (_, unit) dom_prog"
    and set_mode_locs :: "(_) shadow_root_ptr ==> (_, unit) dom_prog set" +
  assumes type_wf_impl: "type_wf = ShadowRootClass.type_wf"
  assumes set_mode_impl: "set_mode = a_set_mode"
  assumes set_mode_locs_impl: "set_mode_locs = a_set_mode_locs"
begin
lemmas set_mode_def = set_mode_impl[unfolded set_mode_def a_set_mode_def]
lemmas set_mode_locs_def = set_mode_locs_impl[unfolded set_mode_locs_def a_set_mode_locs_def]

lemma set_mode_ok:
  "type_wf h ==> shadow_root_ptr |∈| shadow_root_ptr_kinds h ==>
    h ⊨ ok (set_mode shadow_root_ptr shadow_root_mode)"
  apply(unfold type_wf_impl)
  unfolding set_mode_def using get_M_S_h_a_d_o_w_R_o_o_{t_ok} put_M_S_h_a_d_o_w_R_o_o_{t_ok}
  by (simp add: ShadowRootMonad.put_M_S_h_a_d_o_w_R_o_o_{t_ok})

lemma set_mode_ptr_in_heap:
  "h ⊨ ok (set_mode shadow_root_ptr shadow_root_mode) ==>
    shadow_root_ptr |∈| shadow_root_ptr_kinds h"
  by(simp add: set_mode_def put_M_ptr_in_heap)

lemma set_mode_writes:
  "writes (set_mode_locs shadow_root_ptr) (set_mode shadow_root_ptr shadow_root_mode) h h'"
  by(auto simp add: set_mode_def set_mode_locs_def intro: writes_bind_pure)

lemma set_mode_pointers_preserved:
  assumes "w ∈ set_mode_locs element_ptr"
  assumes "h ⊨ w →_h h'"
  shows "object_ptr_kinds h = object_ptr_kinds h'"
  using assms(1) object_ptr_kinds_preserved[OF writes_singleton2 assms(2)]
  by(auto simp add: all_args_def set_mode_locs_def split: if_splits)

lemma set_mode_types_preserved:
  assumes "w ∈ set_mode_locs element_ptr"
  assumes "h ⊨ w →_h h'"
  shows "type_wf h = type_wf h'"
  apply(unfold type_wf_impl)
  using assms(1) type_wf_preserved[OF writes_singleton2 assms(2)]
  by(auto simp add: all_args_def set_mode_locs_def split: if_splits)
end

interpretation i_set_mode?: l_set_mode_S_h_a_d_o_w_{_}_D_O_M type_wf set_mode set_mode_locs
  by (auto simp add: l_set_mode_S_h_a_d_o_w_{_}_D_O_{M_def} instances)
declare l_set_mode_S_h_a_d_o_w_{_}_D_O_{M_axioms} [instances]

locale l_set_mode = l_type_wf +l_set_mode_defs +
  assumes set_mode_writes:
    "writes (set_mode_locs shadow_root_ptr) (set_mode shadow_root_ptr shadow_root_mode) h h'"
  assumes set_mode_ok:
    "type_wf h ==> shadow_root_ptr |∈| shadow_root_ptr_kinds h ==>
      h ⊨ ok (set_mode shadow_root_ptr shadow_root_mode)"
  assumes set_mode_ptr_in_heap:
    "h ⊨ ok (set_mode shadow_root_ptr shadow_root_mode) ==>
      shadow_root_ptr |∈| shadow_root_ptr_kinds h"
  assumes set_mode_pointers_preserved:
    "w ∈ set_mode_locs shadow_root_ptr ==>
      h ⊨ w →_h h' ==> object_ptr_kinds h = object_ptr_kinds h'"
  assumes set_mode_types_preserved:
    "w ∈ set_mode_locs shadow_root_ptr ==> h ⊨ w →_h h' ==> type_wf h = type_wf h'"

lemma set_mode_is_l_set_mode [instances]: "l_set_mode type_wf set_mode set_mode_locs"
  apply(auto simp add: l_set_mode_def instances)[1]
  using set_mode_writes apply blast
  using set_mode_ok apply (blast)
  using set_mode_ptr_in_heap apply blast
  using set_mode_pointers_preserved apply(blast, blast)
  using set_mode_types_preserved apply(blast, blast)
  done

paragraph ‹get\_child\_nodes›

locale l_set_shadow_root_get_child_nodes_S_h_a_d_o_w_{_}_D_O_M =
  l_get_child_nodes_S_h_a_d_o_w_{_}_D_O_M +
  l_set_shadow_root_S_h_a_d_o_w_{_}_D_O_M
begin
lemma set_shadow_root_get_child_nodes:
  "∀w ∈ set_shadow_root_locs ptr. (h ⊨ w →_h h' ⟶ (∀r ∈ get_child_nodes_locs ptr'. r h h'))"
  by(auto simp add: get_child_nodes_locs_def set_shadow_root_locs_def CD.get_child_nodes_locs_def
      all_args_def
      intro: element_put_get_preserved[where setter=shadow_root_opt_update])
end

interpretation i_set_shadow_root_get_child_nodes?: l_set_shadow_root_get_child_nodes_S_h_a_d_o_w_{_}_D_O_M type_wf
  known_ptr DocumentClass.type_wf DocumentClass.known_ptr get_child_nodes get_child_nodes_locs
  Core_DOM_Functions.get_child_nodes Core_DOM_Functions.get_child_nodes_locs set_shadow_root
  set_shadow_root_locs
  by(unfold_locales)
declare l_set_shadow_root_get_child_nodes_S_h_a_d_o_w_{_}_D_O_{M_axioms}[instances]

locale l_set_shadow_root_get_child_nodes = l_set_shadow_root + l_get_child_nodes +
  assumes set_shadow_root_get_child_nodes:
    "∀w ∈ set_shadow_root_locs ptr. (h ⊨ w →_h h' ⟶ (∀r ∈ get_child_nodes_locs ptr'. r h h'))"

lemma set_shadow_root_get_child_nodes_is_l_set_shadow_root_get_child_nodes [instances]:
  "l_set_shadow_root_get_child_nodes type_wf set_shadow_root set_shadow_root_locs known_ptr
    get_child_nodes get_child_nodes_locs"
  apply(auto simp add: l_set_shadow_root_get_child_nodes_def
      l_set_shadow_root_get_child_nodes_axioms_def instances)[1]
  using set_shadow_root_get_child_nodes apply blast
  done

paragraph ‹get\_shadow\_root›

locale l_set_mode_get_shadow_root_S_h_a_d_o_w_{_}_D_O_M =
  l_set_mode_S_h_a_d_o_w_{_}_D_O_M +
  l_get_shadow_root_S_h_a_d_o_w_{_}_D_O_M
begin
lemma set_mode_get_shadow_root:
  "∀w ∈ set_mode_locs ptr. (h ⊨ w →_h h' ⟶ (∀r ∈ get_shadow_root_locs ptr'. r h h'))"
  by(auto simp add: set_mode_locs_def get_shadow_root_locs_def all_args_def)
end

interpretation
  i_set_mode_get_shadow_root?: l_set_mode_get_shadow_root_S_h_a_d_o_w_{_}_D_O_M type_wf
  set_mode set_mode_locs get_shadow_root
  get_shadow_root_locs
  by unfold_locales
declare l_set_mode_get_shadow_root_S_h_a_d_o_w_{_}_D_O_{M_axioms}[instances]

locale l_set_mode_get_shadow_root = l_set_mode + l_get_shadow_root +
  assumes set_mode_get_shadow_root:
    "∀w ∈ set_mode_locs ptr. (h ⊨ w →_h h' ⟶ (∀r ∈ get_shadow_root_locs ptr'. r h h'))"

lemma set_mode_get_shadow_root_is_l_set_mode_get_shadow_root [instances]:
  "l_set_mode_get_shadow_root type_wf set_mode set_mode_locs get_shadow_root
                                         get_shadow_root_locs"
  using set_mode_is_l_set_mode get_shadow_root_is_l_get_shadow_root
  apply(simp add: l_set_mode_get_shadow_root_def
      l_set_mode_get_shadow_root_axioms_def)
  using set_mode_get_shadow_root
  by fast

paragraph ‹get\_child\_nodes›

locale l_set_mode_get_child_nodes_S_h_a_d_o_w_{_}_D_O_M =
  l_set_mode_S_h_a_d_o_w_{_}_D_O_M +
  l_get_child_nodes_S_h_a_d_o_w_{_}_D_O_M
begin
lemma set_mode_get_child_nodes:
  "∀w ∈ set_mode_locs ptr. (h ⊨ w →_h h' ⟶ (∀r ∈ get_child_nodes_locs ptr'. r h h'))"
  by(auto simp add: get_child_nodes_locs_def CD.get_child_nodes_locs_def set_mode_locs_def
      all_args_def)[1]
end

interpretation i_set_mode_get_child_nodes?: l_set_mode_get_child_nodes_S_h_a_d_o_w_{_}_D_O_M type_wf set_mode
  set_mode_locs known_ptr DocumentClass.type_wf DocumentClass.known_ptr get_child_nodes
  get_child_nodes_locs Core_DOM_Functions.get_child_nodes Core_DOM_Functions.get_child_nodes_locs
  by unfold_locales
declare l_set_mode_get_child_nodes_S_h_a_d_o_w_{_}_D_O_{M_axioms}[instances]

locale l_set_mode_get_child_nodes = l_set_mode + l_get_child_nodes +
  assumes set_mode_get_child_nodes:
    "∀w ∈ set_mode_locs ptr. (h ⊨ w →_h h' ⟶ (∀r ∈ get_child_nodes_locs ptr'. r h h'))"

lemma set_mode_get_child_nodes_is_l_set_mode_get_child_nodes [instances]:
  "l_set_mode_get_child_nodes type_wf set_mode set_mode_locs known_ptr get_child_nodes
                                         get_child_nodes_locs"
  using set_mode_is_l_set_mode get_child_nodes_is_l_get_child_nodes
  apply(simp add: l_set_mode_get_child_nodes_def
      l_set_mode_get_child_nodes_axioms_def)
  using set_mode_get_child_nodes
  by fast

subsubsection ‹get\_host›

locale l_get_host_S_h_a_d_o_w_{_}_D_O_{M_defs} =
  l_get_shadow_root_defs get_shadow_root get_shadow_root_locs
  for get_shadow_root
    :: "(_::linorder) element_ptr ==> ((_) heap, exception, (_) shadow_root_ptr option) prog"
    and get_shadow_root_locs :: "(_) element_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
begin
definition a_get_host :: "(_) shadow_root_ptr ==> (_, (_) element_ptr) dom_prog"  where
  "a_get_host shadow_root_ptr = do {
      host_ptrs ← element_ptr_kinds_M 🍋 filter_M (λelement_ptr. do {
        shadow_root_opt ← get_shadow_root element_ptr;
        return (shadow_root_opt = Some shadow_root_ptr)
      });
      (case host_ptrs of host_ptr#[] ==> return host_ptr | _ ==> error HierarchyRequestError)
    }"
definition "a_get_host_locs ≡ (∪element_ptr. (get_shadow_root_locs element_ptr)) ∪
  (∪ptr. {preserved (get_M_O_b_j_e_c_t ptr RObject.nothing)})"

end

global_interpretation l_get_host_S_h_a_d_o_w_{_}_D_O_{M_defs} get_shadow_root get_shadow_root_locs
  defines get_host = "a_get_host"
    and get_host_locs = "a_get_host_locs"
  .

locale l_get_host_defs =
  fixes get_host :: "(_) shadow_root_ptr ==> (_, (_) element_ptr) dom_prog"
  fixes get_host_locs :: "((_) heap ==> (_) heap ==> bool) set"

locale l_get_host_S_h_a_d_o_w_{_}_D_O_M =
  l_get_host_S_h_a_d_o_w_{_}_D_O_{M_defs} +
  l_get_host_defs +
  l_get_shadow_root +
  assumes get_host_impl: "get_host = a_get_host"
  assumes get_host_locs_impl: "get_host_locs = a_get_host_locs"
begin
lemmas get_host_def = get_host_impl[unfolded a_get_host_def]
lemmas get_host_locs_def = get_host_locs_impl[unfolded a_get_host_locs_def]

lemma get_host_pure [simp]: "pure (get_host element_ptr) h"
  by(auto simp add: get_host_def intro!: bind_pure_I filter_M_pure_I split: list.splits)

lemma get_host_reads: "reads get_host_locs (get_host element_ptr) h h'"
  using get_shadow_root_reads[unfolded reads_def]
  by(auto simp add: get_host_def get_host_locs_def
      intro!: reads_bind_pure reads_subset[OF check_in_heap_reads] reads_subset[OF error_reads]
      reads_subset[OF get_shadow_root_reads] reads_subset[OF return_reads]
      reads_subset[OF element_ptr_kinds_M_reads] filter_M_reads filter_M_pure_I
      bind_pure_I
      split: list.splits)
end

locale l_get_host = l_get_host_defs +
  assumes get_host_pure [simp]: "pure (get_host element_ptr) h"
  assumes get_host_reads: "reads get_host_locs (get_host node_ptr) h h'"

interpretation i_get_host?: l_get_host_S_h_a_d_o_w_{_}_D_O_M get_shadow_root get_shadow_root_locs get_host
  get_host_locs type_wf
  using instances
  by (simp add: l_get_host_S_h_a_d_o_w_{_}_D_O_{M_def} l_get_host_S_h_a_d_o_w_{_}_D_O_{M_axioms_def} get_host_def get_host_locs_def)
declare l_get_host_S_h_a_d_o_w_{_}_D_O_{M_axioms} [instances]

lemma get_host_is_l_get_host [instances]: "l_get_host get_host get_host_locs"
  apply(auto simp add: l_get_host_def)[1]
  using get_host_reads apply fast
  done

subsubsection ‹get\_mode›

locale l_get_mode_S_h_a_d_o_w_{_}_D_O_{M_defs}
begin
definition a_get_mode :: "(_) shadow_root_ptr ==> (_, shadow_root_mode) dom_prog"
  where
    "a_get_mode shadow_root_ptr = get_M shadow_root_ptr mode"

definition a_get_mode_locs :: "(_) shadow_root_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
  where
    "a_get_mode_locs shadow_root_ptr ≡ {preserved (get_M shadow_root_ptr mode)}"
end

global_interpretation l_get_mode_S_h_a_d_o_w_{_}_D_O_{M_defs}
  defines get_mode = a_get_mode
    and get_mode_locs = a_get_mode_locs
  .

locale l_get_mode_defs =
  fixes get_mode :: "(_) shadow_root_ptr ==> (_, shadow_root_mode) dom_prog"
  fixes get_mode_locs :: "(_) shadow_root_ptr ==> ((_) heap ==> (_) heap ==> bool) set"

locale l_get_mode_S_h_a_d_o_w_{_}_D_O_M =
  l_get_mode_S_h_a_d_o_w_{_}_D_O_{M_defs} +
  l_get_mode_defs get_mode get_mode_locs +
  l_type_wf type_wf
  for get_mode :: "(_) shadow_root_ptr ==> ((_) heap, exception, shadow_root_mode) prog"
    and get_mode_locs :: "(_) shadow_root_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
    and type_wf :: "(_) heap ==> bool" +
  assumes type_wf_impl: "type_wf = ShadowRootClass.type_wf"
  assumes get_mode_impl: "get_mode = a_get_mode"
  assumes get_mode_locs_impl: "get_mode_locs = a_get_mode_locs"
begin
lemmas get_mode_def = get_mode_impl[unfolded get_mode_def a_get_mode_def]
lemmas get_mode_locs_def = get_mode_locs_impl[unfolded get_mode_locs_def a_get_mode_locs_def]

lemma get_mode_ok: "type_wf h ==> shadow_root_ptr |∈| shadow_root_ptr_kinds h ==>
  h ⊨ ok (get_mode shadow_root_ptr)"
  unfolding get_mode_def type_wf_impl
  using ShadowRootMonad.get_M_S_h_a_d_o_w_R_o_o_{t_ok} by blast

lemma get_mode_pure [simp]: "pure (get_mode element_ptr) h"
  unfolding get_mode_def by simp

lemma get_mode_ptr_in_heap:
  assumes "h ⊨ get_mode shadow_root_ptr →_r children"
  shows "shadow_root_ptr |∈| shadow_root_ptr_kinds h"
  using assms
  by(auto simp add: get_mode_def get_M_S_h_a_d_o_w_R_o_o_{t_ptr_in_heap} dest: is_OK_returns_result_I)

lemma get_mode_reads: "reads (get_mode_locs element_ptr) (get_mode element_ptr) h h'"
  by(simp add: get_mode_def get_mode_locs_def reads_bind_pure reads_insert_writes_set_right)
end

interpretation i_get_mode?: l_get_mode_S_h_a_d_o_w_{_}_D_O_M get_mode get_mode_locs type_wf
  using instances
  by (auto simp add: l_get_mode_S_h_a_d_o_w_{_}_D_O_{M_def})
declare l_get_mode_S_h_a_d_o_w_{_}_D_O_{M_axioms} [instances]

locale l_get_mode = l_type_wf + l_get_mode_defs +
  assumes get_mode_reads: "reads (get_mode_locs shadow_root_ptr) (get_mode shadow_root_ptr) h h'"
  assumes get_mode_ok:
    "type_wf h ==> shadow_root_ptr |∈| shadow_root_ptr_kinds h ==> h ⊨ ok (get_mode shadow_root_ptr)"
  assumes get_mode_ptr_in_heap:
    "h ⊨ ok (get_mode shadow_root_ptr) ==> shadow_root_ptr |∈| shadow_root_ptr_kinds h"
  assumes get_mode_pure [simp]: "pure (get_mode shadow_root_ptr) h"

lemma get_mode_is_l_get_mode [instances]: "l_get_mode type_wf get_mode get_mode_locs"
  apply(auto simp add: l_get_mode_def instances)[1]
  using get_mode_reads apply blast
  using get_mode_ok apply blast
  using get_mode_ptr_in_heap apply blast
  done

subsubsection ‹get\_shadow\_root\_safe›

locale l_get_shadow_root_safe_S_h_a_d_o_w_{_}_D_O_{M_defs} =
  l_get_shadow_root_defs get_shadow_root get_shadow_root_locs +
  l_get_mode_defs get_mode get_mode_locs
  for get_shadow_root :: "(_) element_ptr ==> (_, (_) shadow_root_ptr option) dom_prog"
    and get_shadow_root_locs :: "(_) element_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
    and get_mode :: "(_) shadow_root_ptr ==> (_, shadow_root_mode) dom_prog"
    and get_mode_locs :: "(_) shadow_root_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
begin
definition a_get_shadow_root_safe :: "(_) element_ptr ==> (_, (_) shadow_root_ptr option) dom_prog"
  where
    "a_get_shadow_root_safe element_ptr = do {
      shadow_root_ptr_opt ← get_shadow_root element_ptr;
      (case shadow_root_ptr_opt of
        Some shadow_root_ptr ==> do {
          mode ← get_mode shadow_root_ptr;
          (if mode = Open then
            return (Some shadow_root_ptr)
          else
            return None
          )
        } | None ==> return None)
    }"

definition a_get_shadow_root_safe_locs
  :: "(_) element_ptr ==> (_) shadow_root_ptr ==> ((_) heap ==> (_) heap ==> bool) set" where
  "a_get_shadow_root_safe_locs element_ptr shadow_root_ptr ≡
    (get_shadow_root_locs element_ptr) ∪ (get_mode_locs shadow_root_ptr)"
end

global_interpretation l_get_shadow_root_safe_S_h_a_d_o_w_{_}_D_O_{M_defs} get_shadow_root get_shadow_root_locs
  get_mode get_mode_locs
  defines get_shadow_root_safe = a_get_shadow_root_safe
    and get_shadow_root_safe_locs = a_get_shadow_root_safe_locs
  .

locale l_get_shadow_root_safe_defs =
  fixes get_shadow_root_safe :: "(_) element_ptr ==> (_, (_) shadow_root_ptr option) dom_prog"
  fixes get_shadow_root_safe_locs ::
    "(_) element_ptr ==> (_) shadow_root_ptr ==> ((_) heap ==> (_) heap ==> bool) set"

locale l_get_shadow_root_safe_S_h_a_d_o_w_{_}_D_O_M =
  l_get_shadow_root_safe_S_h_a_d_o_w_{_}_D_O_{M_defs} get_shadow_root get_shadow_root_locs get_mode get_mode_locs +
  l_get_shadow_root_safe_defs get_shadow_root_safe get_shadow_root_safe_locs +
  l_get_shadow_root  type_wf get_shadow_root get_shadow_root_locs +
  l_get_mode  type_wf get_mode get_mode_locs +
  l_type_wf type_wf
  for type_wf :: "(_) heap ==> bool"
    and get_shadow_root_safe ::
    "(_) element_ptr ==> ((_) heap, exception, (_) shadow_root_ptr option) prog"
    and get_shadow_root_safe_locs ::
    "(_) element_ptr ==> (_) shadow_root_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
    and get_shadow_root :: "(_) element_ptr ==> (_, (_) shadow_root_ptr option) dom_prog"
    and get_shadow_root_locs :: "(_) element_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
    and get_mode :: "(_) shadow_root_ptr ==> (_, shadow_root_mode) dom_prog"
    and get_mode_locs :: "(_) shadow_root_ptr ==> ((_) heap ==> (_) heap ==> bool) set" +
  assumes type_wf_impl: "type_wf = ShadowRootClass.type_wf"
  assumes get_shadow_root_safe_impl: "get_shadow_root_safe = a_get_shadow_root_safe"
  assumes get_shadow_root_safe_locs_impl: "get_shadow_root_safe_locs = a_get_shadow_root_safe_locs"
begin
lemmas get_shadow_root_safe_def =
  get_shadow_root_safe_impl[unfolded get_shadow_root_safe_def a_get_shadow_root_safe_def]
lemmas get_shadow_root_safe_locs_def =
  get_shadow_root_safe_locs_impl[unfolded get_shadow_root_safe_locs_def a_get_shadow_root_safe_locs_def]

lemma get_shadow_root_safe_pure [simp]: "pure (get_shadow_root_safe element_ptr) h"
  by (auto simp add: get_shadow_root_safe_def bind_pure_I option.case_eq_if)
end

interpretation i_get_shadow_root_safe?: l_get_shadow_root_safe_S_h_a_d_o_w_{_}_D_O_M type_wf get_shadow_root_safe
  get_shadow_root_safe_locs get_shadow_root get_shadow_root_locs get_mode get_mode_locs
  using instances
  by (auto simp add: l_get_shadow_root_safe_S_h_a_d_o_w_{_}_D_O_{M_def} l_get_shadow_root_safe_S_h_a_d_o_w_{_}_D_O_{M_axioms_def}
      get_shadow_root_safe_def get_shadow_root_safe_locs_def)
declare l_get_shadow_root_safe_S_h_a_d_o_w_{_}_D_O_{M_axioms} [instances]

locale l_get_shadow_root_safe = l_get_shadow_root_safe_defs +
  assumes get_shadow_root_safe_pure [simp]: "pure (get_shadow_root_safe element_ptr) h"

lemma get_shadow_root_safe_is_l_get_shadow_root_safe [instances]:
  "l_get_shadow_root_safe get_shadow_root_safe"
  using instances
  apply(auto simp add: l_get_shadow_root_safe_def)[1]
  done

subsubsection ‹set\_disconnected\_nodes›

locale l_set_disconnected_nodes_S_h_a_d_o_w_{_}_D_O_M =
  CD: l_set_disconnected_nodes_C_o_r_e_{_}_D_O_M type_wf_C_o_r_e_{_}_D_O_M set_disconnected_nodes set_disconnected_nodes_locs +
  l_type_wf type_wf
  for type_wf :: "(_) heap ==> bool"
    and type_wf_C_o_r_e_{_}_D_O_M :: "(_) heap ==> bool"
    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" +
  assumes type_wf_impl: "type_wf = ShadowRootClass.type_wf"
begin
lemma set_disconnected_nodes_ok:
  "type_wf h ==> document_ptr |∈| document_ptr_kinds h ==>
    h ⊨ ok (set_disconnected_nodes document_ptr node_ptrs)"
  using CD.set_disconnected_nodes_ok CD.type_wf_impl ShadowRootClass.type_wf_defs local.type_wf_impl
  by blast

lemma set_disconnected_nodes_typess_preserved:
  assumes "w ∈ set_disconnected_nodes_locs object_ptr"
  assumes "h ⊨ w →_h h'"
  shows "type_wf h = type_wf h'"
  using assms(1) type_wf_preserved[OF writes_singleton2 assms(2)]
  apply(unfold type_wf_impl)
  by(auto simp add: all_args_def CD.set_disconnected_nodes_locs_def split: if_splits)
end

interpretation i_set_disconnected_nodes?: l_set_disconnected_nodes_S_h_a_d_o_w_{_}_D_O_M type_wf
  DocumentClass.type_wf set_disconnected_nodes set_disconnected_nodes_locs
  by(auto simp add: l_set_disconnected_nodes_S_h_a_d_o_w_{_}_D_O_{M_def}
      l_set_disconnected_nodes_S_h_a_d_o_w_{_}_D_O_{M_axioms_def} instances)
declare l_set_disconnected_nodes_S_h_a_d_o_w_{_}_D_O_{M_axioms} [instances]

lemma set_disconnected_nodes_is_l_set_disconnected_nodes [instances]:
  "l_set_disconnected_nodes type_wf set_disconnected_nodes set_disconnected_nodes_locs"
  apply(auto simp add: l_set_disconnected_nodes_def)[1]
        apply (simp add: i_set_disconnected_nodes.set_disconnected_nodes_writes)
  using set_disconnected_nodes_ok apply blast
      apply (simp add: i_set_disconnected_nodes.set_disconnected_nodes_ptr_in_heap)
  using i_set_disconnected_nodes.set_disconnected_nodes_pointers_preserved apply (blast, blast)
  using set_disconnected_nodes_typess_preserved apply(blast, blast)
  done

paragraph ‹get\_child\_nodes›

locale l_set_disconnected_nodes_get_child_nodes_S_h_a_d_o_w_{_}_D_O_M =
  l_set_disconnected_nodes_C_o_r_e_{_}_D_O_M type_wf_C_o_r_e_{_}_D_O_M set_disconnected_nodes set_disconnected_nodes_locs +
  l_get_child_nodes_S_h_a_d_o_w_{_}_D_O_M type_wf known_ptr type_wf_C_o_r_e_{_}_D_O_M known_ptr_C_o_r_e_{_}_D_O_M get_child_nodes
  get_child_nodes_locs get_child_nodes_C_o_r_e_{_}_D_O_M get_child_nodes_locs_C_o_r_e_{_}_D_O_M
  for type_wf :: "(_) heap ==> bool"
    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 known_ptr :: "(_) object_ptr ==> bool"
    and type_wf_C_o_r_e_{_}_D_O_M :: "(_) heap ==> bool"
    and known_ptr_C_o_r_e_{_}_D_O_M :: "(_) object_ptr ==> bool"
    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_child_nodes_C_o_r_e_{_}_D_O_M :: "(_) object_ptr ==> ((_) heap, exception, (_) node_ptr list) prog"
    and get_child_nodes_locs_C_o_r_e_{_}_D_O_M :: "(_) object_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
begin

lemma set_disconnected_nodes_get_child_nodes:
  "∀w ∈ set_disconnected_nodes_locs ptr. (h ⊨ w →_h h' ⟶ (∀r ∈ get_child_nodes_locs ptr'. r h h'))"
  by(auto simp add: set_disconnected_nodes_locs_def get_child_nodes_locs_def
      CD.get_child_nodes_locs_def all_args_def)
end

interpretation i_set_disconnected_nodes_get_child_nodes?:
  l_set_disconnected_nodes_get_child_nodes_S_h_a_d_o_w_{_}_D_O_M type_wf set_disconnected_nodes
  set_disconnected_nodes_locs known_ptr DocumentClass.type_wf DocumentClass.known_ptr get_child_nodes
  get_child_nodes_locs Core_DOM_Functions.get_child_nodes Core_DOM_Functions.get_child_nodes_locs
  by(auto simp add: l_set_disconnected_nodes_get_child_nodes_S_h_a_d_o_w_{_}_D_O_{M_def} instances)
declare l_set_disconnected_nodes_get_child_nodes_S_h_a_d_o_w_{_}_D_O_{M_axioms}[instances]

lemma set_disconnected_nodes_get_child_nodes_is_l_set_disconnected_nodes_get_child_nodes [instances]:
  "l_set_disconnected_nodes_get_child_nodes set_disconnected_nodes_locs get_child_nodes_locs"
  apply(auto simp add: l_set_disconnected_nodes_get_child_nodes_def)[1]
  using set_disconnected_nodes_get_child_nodes apply fast
  done

paragraph ‹get\_host›

locale l_set_disconnected_nodes_get_host_S_h_a_d_o_w_{_}_D_O_M =
  l_set_disconnected_nodes_S_h_a_d_o_w_{_}_D_O_M +
  l_get_shadow_root_S_h_a_d_o_w_{_}_D_O_M +
  l_get_host_S_h_a_d_o_w_{_}_D_O_M
begin
lemma set_disconnected_nodes_get_host:
  "∀w ∈ set_disconnected_nodes_locs ptr. (h ⊨ w →_h h' ⟶ (∀r ∈ get_host_locs. r h h'))"
  by(auto simp add: CD.set_disconnected_nodes_locs_def get_shadow_root_locs_def get_host_locs_def all_args_def)
end

interpretation i_set_disconnected_nodes_get_host?: l_set_disconnected_nodes_get_host_S_h_a_d_o_w_{_}_D_O_M type_wf
  DocumentClass.type_wf set_disconnected_nodes set_disconnected_nodes_locs get_shadow_root
  get_shadow_root_locs get_host get_host_locs
  by(auto simp add: l_set_disconnected_nodes_get_host_S_h_a_d_o_w_{_}_D_O_{M_def} instances)
declare l_set_disconnected_nodes_get_host_S_h_a_d_o_w_{_}_D_O_{M_axioms} [instances]

locale l_set_disconnected_nodes_get_host = l_set_disconnected_nodes_defs + l_get_host_defs +
  assumes set_disconnected_nodes_get_host:
    "∀w ∈ set_disconnected_nodes_locs ptr. (h ⊨ w →_h h' ⟶ (∀r ∈ get_host_locs. r h h'))"

lemma set_disconnected_nodes_get_host_is_l_set_disconnected_nodes_get_host [instances]:
  "l_set_disconnected_nodes_get_host set_disconnected_nodes_locs get_host_locs"
  apply(auto simp add: l_set_disconnected_nodes_get_host_def instances)[1]
  using set_disconnected_nodes_get_host
  by fast

subsubsection ‹get\_tag\_name›

locale l_get_tag_name_S_h_a_d_o_w_{_}_D_O_M =
  CD: l_get_tag_name_C_o_r_e_{_}_D_O_M type_wf_C_o_r_e_{_}_D_O_M get_tag_name get_tag_name_locs +
  l_type_wf type_wf
  for type_wf :: "(_) heap ==> bool"
    and type_wf_C_o_r_e_{_}_D_O_M :: "(_) heap ==> bool"
    and get_tag_name :: "(_) element_ptr ==> (_, tag_name) dom_prog"
    and get_tag_name_locs :: "(_) element_ptr ==> ((_) heap ==> (_) heap ==> bool) set" +
  assumes type_wf_impl: "type_wf = ShadowRootClass.type_wf"
begin

lemma get_tag_name_ok:
  "type_wf h ==> element_ptr |∈| element_ptr_kinds h ==> h ⊨ ok (get_tag_name element_ptr)"
  apply(unfold type_wf_impl get_tag_name_impl[unfolded a_get_tag_name_def])
  using CD.get_tag_name_ok CD.type_wf_impl ShadowRootClass.type_wf_D_o_c_u_m_e_n_t
  by blast
end

interpretation i_get_tag_name?: l_get_tag_name_S_h_a_d_o_w_{_}_D_O_M type_wf DocumentClass.type_wf get_tag_name
  get_tag_name_locs
  by(auto simp add: l_get_tag_name_S_h_a_d_o_w_{_}_D_O_{M_def} l_get_tag_name_S_h_a_d_o_w_{_}_D_O_{M_axioms_def} instances)
declare l_get_tag_name_S_h_a_d_o_w_{_}_D_O_{M_axioms} [instances]

lemma get_tag_name_is_l_get_tag_name [instances]: "l_get_tag_name type_wf get_tag_name
  get_tag_name_locs"
  apply(auto simp add: l_get_tag_name_def)[1]
  using get_tag_name_reads apply fast
  using get_tag_name_ok apply fast
  done

paragraph ‹set\_disconnected\_nodes›

locale l_set_disconnected_nodes_get_tag_name_S_h_a_d_o_w_{_}_D_O_M =
  l_set_disconnected_nodes_S_h_a_d_o_w_{_}_D_O_M +
  l_get_tag_name_S_h_a_d_o_w_{_}_D_O_M
begin
lemma set_disconnected_nodes_get_tag_name:
  "∀w ∈ set_disconnected_nodes_locs ptr. (h ⊨ w →_h h' ⟶ (∀r ∈ get_tag_name_locs ptr'. r h h'))"
  by(auto simp add: CD.set_disconnected_nodes_locs_def CD.get_tag_name_locs_def all_args_def)
end

interpretation i_set_disconnected_nodes_get_tag_name?: l_set_disconnected_nodes_get_tag_name_S_h_a_d_o_w_{_}_D_O_M
  type_wf DocumentClass.type_wf set_disconnected_nodes set_disconnected_nodes_locs get_tag_name
  get_tag_name_locs
  by(auto simp add: l_set_disconnected_nodes_get_tag_name_S_h_a_d_o_w_{_}_D_O_{M_def} instances)
declare l_set_disconnected_nodes_get_tag_name_S_h_a_d_o_w_{_}_D_O_{M_axioms} [instances]

lemma set_disconnected_nodes_get_tag_name_is_l_set_disconnected_nodes_get_tag_name [instances]:
  "l_set_disconnected_nodes_get_tag_name type_wf set_disconnected_nodes set_disconnected_nodes_locs
  get_tag_name get_tag_name_locs"
  apply(auto simp add: l_set_disconnected_nodes_get_tag_name_def
      l_set_disconnected_nodes_get_tag_name_axioms_def instances)[1]
  using set_disconnected_nodes_get_tag_name
  by fast

paragraph ‹set\_child\_nodes›

locale l_set_child_nodes_get_tag_name_S_h_a_d_o_w_{_}_D_O_M =
  l_set_child_nodes_S_h_a_d_o_w_{_}_D_O_M +
  l_get_tag_name_S_h_a_d_o_w_{_}_D_O_M
begin
lemma set_child_nodes_get_tag_name:
  "∀w ∈ set_child_nodes_locs ptr. (h ⊨ w →_h h' ⟶ (∀r ∈ get_tag_name_locs ptr'. r h h'))"
  by(auto simp add: CD.set_child_nodes_locs_def set_child_nodes_locs_def CD.get_tag_name_locs_def
      all_args_def
      intro: element_put_get_preserved[where getter=tag_name and
        setter=RElement.child_nodes_update])
end

interpretation i_set_child_nodes_get_tag_name?: l_set_child_nodes_get_tag_name_S_h_a_d_o_w_{_}_D_O_M type_wf
  known_ptr DocumentClass.type_wf DocumentClass.known_ptr set_child_nodes set_child_nodes_locs
  Core_DOM_Functions.set_child_nodes Core_DOM_Functions.set_child_nodes_locs get_tag_name
  get_tag_name_locs
  by(auto simp add: l_set_child_nodes_get_tag_name_S_h_a_d_o_w_{_}_D_O_{M_def} instances)
declare l_set_child_nodes_get_tag_name_S_h_a_d_o_w_{_}_D_O_{M_axioms} [instances]

lemma set_child_nodes_get_tag_name_is_l_set_child_nodes_get_tag_name [instances]:
  "l_set_child_nodes_get_tag_name type_wf set_child_nodes set_child_nodes_locs get_tag_name
  get_tag_name_locs"
  apply(auto simp add: l_set_child_nodes_get_tag_name_def l_set_child_nodes_get_tag_name_axioms_def
      instances)[1]
  using set_child_nodes_get_tag_name
  by fast

paragraph ‹delete\_shadow\_root›

locale l_delete_shadow_root_get_tag_name_S_h_a_d_o_w_{_}_D_O_M =
  l_get_tag_name_S_h_a_d_o_w_{_}_D_O_M
begin
lemma get_tag_name_delete_shadow_root:  "h ⊨ delete_S_h_a_d_o_w_R_o_o_{t_M} shadow_root_ptr →_h h'
    ==> r ∈ get_tag_name_locs ptr' ==> r h h'"
  by(auto simp add: CD.get_tag_name_locs_def delete_shadow_root_get_M_E_l_e_m_e_n_t)
end

locale l_delete_shadow_root_get_tag_name = l_get_tag_name_defs +
  assumes get_tag_name_delete_shadow_root:  "h ⊨ delete_S_h_a_d_o_w_R_o_o_{t_M} shadow_root_ptr →_h h'
    ==> r ∈ get_tag_name_locs ptr' ==> r h h'"
interpretation l_delete_shadow_root_get_tag_name_S_h_a_d_o_w_{_}_D_O_M type_wf DocumentClass.type_wf get_tag_name
  get_tag_name_locs
  by(auto simp add: l_delete_shadow_root_get_tag_name_S_h_a_d_o_w_{_}_D_O_{M_def} instances)

lemma l_delete_shadow_root_get_tag_name_get_tag_name_locs [instances]: "l_delete_shadow_root_get_tag_name get_tag_name_locs"
  apply(auto simp add: l_delete_shadow_root_get_tag_name_def)[1]
  using get_tag_name_delete_shadow_root apply fast
  done

paragraph ‹set\_shadow\_root›

locale l_set_shadow_root_get_tag_name_S_h_a_d_o_w_{_}_D_O_M =
  l_set_shadow_root_S_h_a_d_o_w_{_}_D_O_M +
  l_get_tag_name_S_h_a_d_o_w_{_}_D_O_M
begin
lemma set_shadow_root_get_tag_name:
  "∀w ∈ set_shadow_root_locs ptr. (h ⊨ w →_h h' ⟶ (∀r ∈ get_tag_name_locs ptr'. r h h'))"
  by(auto simp add: set_shadow_root_locs_def CD.get_tag_name_locs_def all_args_def
      element_put_get_preserved[where setter=shadow_root_opt_update])
end

interpretation i_set_shadow_root_get_tag_name?: l_set_shadow_root_get_tag_name_S_h_a_d_o_w_{_}_D_O_M type_wf
  set_shadow_root set_shadow_root_locs DocumentClass.type_wf get_tag_name get_tag_name_locs
  apply(auto simp add: l_set_shadow_root_get_tag_name_S_h_a_d_o_w_{_}_D_O_{M_def} instances)[1]
  by(unfold_locales)
declare l_set_shadow_root_get_tag_name_S_h_a_d_o_w_{_}_D_O_{M_axioms}[instances]

locale l_set_shadow_root_get_tag_name = l_set_shadow_root_defs + l_get_tag_name_defs +
  assumes set_shadow_root_get_tag_name:
    "∀w ∈ set_shadow_root_locs ptr. (h ⊨ w →_h h' ⟶ (∀r ∈ get_tag_name_locs ptr'. r h h'))"

lemma set_shadow_root_get_tag_name_is_l_set_shadow_root_get_tag_name [instances]:
  "l_set_shadow_root_get_tag_name set_shadow_root_locs get_tag_name_locs"
  using set_shadow_root_is_l_set_shadow_root get_tag_name_is_l_get_tag_name
  apply(simp add: l_set_shadow_root_get_tag_name_def )
  using set_shadow_root_get_tag_name
  by fast

paragraph ‹new\_element›

locale l_new_element_get_tag_name_S_h_a_d_o_w_{_}_D_O_M =
  l_get_tag_name_S_h_a_d_o_w_{_}_D_O_M type_wf type_wf_C_o_r_e_{_}_D_O_M get_tag_name get_tag_name_locs
  for type_wf :: "(_) heap ==> bool"
    and type_wf_C_o_r_e_{_}_D_O_M :: "(_) heap ==> bool"
    and get_tag_name :: "(_) element_ptr ==> (_, tag_name) dom_prog"
    and get_tag_name_locs :: "(_) element_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
begin
lemma get_tag_name_new_element:
  "ptr' ≠ new_element_ptr ==> h ⊨ new_element →_r new_element_ptr ==> h ⊨ new_element →_h h'
     ==> r ∈ get_tag_name_locs ptr' ==> r h h'"
  by (auto simp add: CD.get_tag_name_locs_def new_element_get_M_O_b_j_e_c_t new_element_get_M_E_l_e_m_e_n_t
      new_element_get_M_D_o_c_u_m_e_n_t split: prod.splits if_splits option.splits
      elim!: bind_returns_result_E bind_returns_heap_E intro: is_element_ptr_kind_obtains)

lemma new_element_empty_tag_name:
  "h ⊨ new_element →_r new_element_ptr ==> h ⊨ new_element →_h h'
   ==> h' ⊨ get_tag_name new_element_ptr →_r ''''"
  by(simp add: CD.get_tag_name_def new_element_tag_name)
end

locale l_new_element_get_tag_name = l_new_element + l_get_tag_name +
  assumes get_tag_name_new_element:
    "ptr' ≠ new_element_ptr ==> h ⊨ new_element →_r new_element_ptr
            ==> h ⊨ new_element →_h h' ==> r ∈ get_tag_name_locs ptr' ==> r h h'"
  assumes new_element_empty_tag_name:
    "h ⊨ new_element →_r new_element_ptr ==> h ⊨ new_element →_h h'
            ==> h' ⊨ get_tag_name new_element_ptr →_r ''''"

interpretation i_new_element_get_tag_name?:
  l_new_element_get_tag_name_S_h_a_d_o_w_{_}_D_O_M type_wf DocumentClass.type_wf get_tag_name get_tag_name_locs
  by(auto simp add: l_new_element_get_tag_name_S_h_a_d_o_w_{_}_D_O_{M_def} instances)
declare l_new_element_get_tag_name_S_h_a_d_o_w_{_}_D_O_{M_axioms} [instances]

lemma new_element_get_tag_name_is_l_new_element_get_tag_name [instances]:
  "l_new_element_get_tag_name type_wf get_tag_name get_tag_name_locs"
  using new_element_is_l_new_element get_tag_name_is_l_get_tag_name
  apply(auto simp add: l_new_element_get_tag_name_def l_new_element_get_tag_name_axioms_def
      instances)[1]
  using get_tag_name_new_element new_element_empty_tag_name
  by fast+

paragraph ‹get\_shadow\_root›

locale l_set_mode_get_tag_name_S_h_a_d_o_w_{_}_D_O_M =
  l_set_mode_S_h_a_d_o_w_{_}_D_O_M +
  l_get_tag_name_S_h_a_d_o_w_{_}_D_O_M
begin
lemma set_mode_get_tag_name:
  "∀w ∈ set_mode_locs ptr. (h ⊨ w →_h h' ⟶ (∀r ∈ get_tag_name_locs ptr'. r h h'))"
  by(auto simp add: set_mode_locs_def CD.get_tag_name_locs_def all_args_def)
end

interpretation
  i_set_mode_get_tag_name?: l_set_mode_get_tag_name_S_h_a_d_o_w_{_}_D_O_M type_wf
  set_mode set_mode_locs  DocumentClass.type_wf get_tag_name
  get_tag_name_locs
  by unfold_locales
declare l_set_mode_get_tag_name_S_h_a_d_o_w_{_}_D_O_{M_axioms}[instances]

locale l_set_mode_get_tag_name = l_set_mode + l_get_tag_name +
  assumes set_mode_get_tag_name:
    "∀w ∈ set_mode_locs ptr. (h ⊨ w →_h h' ⟶ (∀r ∈ get_tag_name_locs ptr'. r h h'))"

lemma set_mode_get_tag_name_is_l_set_mode_get_tag_name [instances]:
  "l_set_mode_get_tag_name type_wf set_mode set_mode_locs get_tag_name
                                         get_tag_name_locs"
  using set_mode_is_l_set_mode get_tag_name_is_l_get_tag_name
  apply(simp add: l_set_mode_get_tag_name_def
      l_set_mode_get_tag_name_axioms_def)
  using set_mode_get_tag_name
  by fast

paragraph ‹new\_document›

locale l_new_document_get_tag_name_S_h_a_d_o_w_{_}_D_O_M =
  l_get_tag_name_S_h_a_d_o_w_{_}_D_O_M type_wf type_wf_C_o_r_e_{_}_D_O_M get_tag_name get_tag_name_locs
  for type_wf :: "(_) heap ==> bool"
    and type_wf_C_o_r_e_{_}_D_O_M :: "(_) heap ==> bool"
    and get_tag_name :: "(_) element_ptr ==> ((_) heap, exception, tag_name) prog"
    and get_tag_name_locs :: "(_) element_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
begin
lemma get_tag_name_new_document:
  "h ⊨ new_document →_r new_document_ptr ==> h ⊨ new_document →_h h'
       ==> r ∈ get_tag_name_locs ptr' ==> r h h'"
  by(auto simp add: CD.get_tag_name_locs_def new_document_get_M_E_l_e_m_e_n_t)
end

locale l_new_document_get_tag_name = l_get_tag_name_defs +
  assumes get_tag_name_new_document:
    "h ⊨ new_document →_r new_document_ptr ==> h ⊨ new_document →_h h'
       ==> r ∈ get_tag_name_locs ptr' ==> r h h'"

interpretation i_new_document_get_tag_name?:
  l_new_document_get_tag_name_S_h_a_d_o_w_{_}_D_O_M type_wf DocumentClass.type_wf get_tag_name
  get_tag_name_locs
  by unfold_locales
declare l_new_document_get_tag_name_S_h_a_d_o_w_{_}_D_O_{M_def}[instances]

lemma new_document_get_tag_name_is_l_new_document_get_tag_name [instances]:
  "l_new_document_get_tag_name get_tag_name_locs"
  unfolding l_new_document_get_tag_name_def
  unfolding get_tag_name_locs_def
  using new_document_get_M_E_l_e_m_e_n_t by blast

paragraph ‹new\_shadow\_root›

locale l_new_shadow_root_get_tag_name_S_h_a_d_o_w_{_}_D_O_M =
  l_get_tag_name_S_h_a_d_o_w_{_}_D_O_M
begin
lemma get_tag_name_new_shadow_root:
  "h ⊨ new_S_h_a_d_o_w_R_o_o_{t_M} →_r new_shadow_root_ptr ==> h ⊨ new_S_h_a_d_o_w_R_o_o_{t_M} →_h h'
     ==> r ∈ get_tag_name_locs ptr' ==> r h h'"
  by (auto simp add: CD.get_tag_name_locs_def new_shadow_root_get_M_O_b_j_e_c_t new_shadow_root_get_M_E_l_e_m_e_n_t
      split: prod.splits if_splits option.splits
      elim!: bind_returns_result_E bind_returns_heap_E intro: is_element_ptr_kind_obtains)
end

locale l_new_shadow_root_get_tag_name = l_get_tag_name +
  assumes get_tag_name_new_shadow_root:
    "h ⊨ new_S_h_a_d_o_w_R_o_o_{t_M} →_r new_shadow_root_ptr
            ==> h ⊨ new_S_h_a_d_o_w_R_o_o_{t_M} →_h h' ==> r ∈ get_tag_name_locs ptr' ==> r h h'"

interpretation i_new_shadow_root_get_tag_name?:
  l_new_shadow_root_get_tag_name_S_h_a_d_o_w_{_}_D_O_M type_wf DocumentClass.type_wf get_tag_name get_tag_name_locs
  by(unfold_locales)
declare l_new_shadow_root_get_tag_name_S_h_a_d_o_w_{_}_D_O_{M_axioms} [instances]

lemma new_shadow_root_get_tag_name_is_l_new_shadow_root_get_tag_name [instances]:
  "l_new_shadow_root_get_tag_name type_wf get_tag_name get_tag_name_locs"
  using get_tag_name_is_l_get_tag_name
  apply(auto simp add: l_new_shadow_root_get_tag_name_def l_new_shadow_root_get_tag_name_axioms_def
      instances)[1]
  using get_tag_name_new_shadow_root
  by fast

paragraph ‹new\_character\_data›

locale l_new_character_data_get_tag_name_S_h_a_d_o_w_{_}_D_O_M =
  l_get_tag_name_S_h_a_d_o_w_{_}_D_O_M type_wf type_wf_C_o_r_e_{_}_D_O_M get_tag_name get_tag_name_locs
  for type_wf :: "(_) heap ==> bool"
    and type_wf_C_o_r_e_{_}_D_O_M :: "(_) heap ==> bool"
    and get_tag_name :: "(_) element_ptr ==> ((_) heap, exception, tag_name) prog"
    and get_tag_name_locs :: "(_) element_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
begin
lemma get_tag_name_new_character_data:
  "h ⊨ new_character_data →_r new_character_data_ptr ==> h ⊨ new_character_data →_h h'
       ==> r ∈ get_tag_name_locs ptr' ==> r h h'"
  by(auto simp add: CD.get_tag_name_locs_def new_character_data_get_M_E_l_e_m_e_n_t)
end

locale l_new_character_data_get_tag_name = l_get_tag_name_defs +
  assumes get_tag_name_new_character_data:
    "h ⊨ new_character_data →_r new_character_data_ptr ==> h ⊨ new_character_data →_h h'
       ==> r ∈ get_tag_name_locs ptr' ==> r h h'"

interpretation i_new_character_data_get_tag_name?:
  l_new_character_data_get_tag_name_S_h_a_d_o_w_{_}_D_O_M type_wf DocumentClass.type_wf get_tag_name
  get_tag_name_locs
  by unfold_locales
declare l_new_character_data_get_tag_name_S_h_a_d_o_w_{_}_D_O_{M_def}[instances]

lemma new_character_data_get_tag_name_is_l_new_character_data_get_tag_name [instances]:
  "l_new_character_data_get_tag_name get_tag_name_locs"
  unfolding l_new_character_data_get_tag_name_def
  unfolding get_tag_name_locs_def
  using new_character_data_get_M_E_l_e_m_e_n_t by blast

paragraph ‹get\_tag\_type›

locale l_set_tag_name_get_tag_name_S_h_a_d_o_w_{_}_D_O_M = l_get_tag_name_S_h_a_d_o_w_{_}_D_O_M
  + l_set_tag_name_S_h_a_d_o_w_{_}_D_O_M
begin
lemma set_tag_name_get_tag_name:
  assumes "h ⊨ CD.a_set_tag_name element_ptr tag →_h h'"
  shows "h' ⊨ CD.a_get_tag_name element_ptr →_r tag"
  using assms
  by(auto simp add: CD.a_get_tag_name_def CD.a_set_tag_name_def)

lemma set_tag_name_get_tag_name_different_pointers:
  assumes "ptr ≠ ptr'"
  assumes "w ∈ CD.a_set_tag_name_locs ptr"
  assumes "h ⊨ w →_h h'"
  assumes "r ∈ CD.a_get_tag_name_locs ptr'"
  shows "r h h'"
  using assms
  by(auto simp add: all_args_def CD.a_set_tag_name_locs_def CD.a_get_tag_name_locs_def
      split: if_splits option.splits )
end

interpretation i_set_tag_name_get_tag_name?:
  l_set_tag_name_get_tag_name_S_h_a_d_o_w_{_}_D_O_M type_wf DocumentClass.type_wf get_tag_name
  get_tag_name_locs set_tag_name set_tag_name_locs
  by unfold_locales
declare l_set_tag_name_get_tag_name_S_h_a_d_o_w_{_}_D_O_{M_axioms}[instances]

lemma set_tag_name_get_tag_name_is_l_set_tag_name_get_tag_name [instances]:
  "l_set_tag_name_get_tag_name type_wf get_tag_name get_tag_name_locs
                                                    set_tag_name set_tag_name_locs"
  using set_tag_name_is_l_set_tag_name get_tag_name_is_l_get_tag_name
  apply(simp add: l_set_tag_name_get_tag_name_def
      l_set_tag_name_get_tag_name_axioms_def)
  using set_tag_name_get_tag_name
    set_tag_name_get_tag_name_different_pointers
  by fast+

subsubsection ‹attach\_shadow\_root›

locale l_attach_shadow_root_S_h_a_d_o_w_{_}_D_O_{M_defs} =
  l_set_shadow_root_defs set_shadow_root set_shadow_root_locs +
  l_set_mode_defs set_mode set_mode_locs +
  l_get_tag_name_defs get_tag_name get_tag_name_locs +
  l_get_shadow_root_defs get_shadow_root get_shadow_root_locs
  for set_shadow_root ::
    "(_) element_ptr ==> (_) shadow_root_ptr option ==> ((_) heap, exception, unit) prog"
    and set_shadow_root_locs :: "(_) element_ptr ==> ((_) heap, exception, unit) prog set"
    and set_mode :: "(_) shadow_root_ptr ==> shadow_root_mode ==> ((_) heap, exception, unit) prog"
    and set_mode_locs :: "(_) shadow_root_ptr ==> ((_) heap, exception, unit) prog set"
    and get_tag_name :: "(_) element_ptr ==> (_, char list) dom_prog"
    and get_tag_name_locs :: "(_) element_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
    and get_shadow_root :: "(_) element_ptr ==> ((_) heap, exception, (_) shadow_root_ptr option) prog"
    and get_shadow_root_locs :: "(_) element_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
begin
definition a_attach_shadow_root ::
  "(_) element_ptr ==> shadow_root_mode ==> (_, (_) shadow_root_ptr) dom_prog"
  where
    "a_attach_shadow_root element_ptr shadow_root_mode = do {
      tag ← get_tag_name element_ptr;
      (if tag ∉ safe_shadow_root_element_types then error NotSupportedError else return ());
      prev_shadow_root ← get_shadow_root element_ptr;
      (if prev_shadow_root ≠ None then error NotSupportedError else return ());
      new_shadow_root_ptr ← new_S_h_a_d_o_w_R_o_o_{t_M};
      set_mode new_shadow_root_ptr shadow_root_mode;
      set_shadow_root element_ptr (Some new_shadow_root_ptr);
      return new_shadow_root_ptr
    }"
end

locale l_attach_shadow_root_defs =
  fixes attach_shadow_root ::
    "(_) element_ptr ==> shadow_root_mode ==> (_, (_) shadow_root_ptr) dom_prog"

global_interpretation l_attach_shadow_root_S_h_a_d_o_w_{_}_D_O_{M_defs} set_shadow_root set_shadow_root_locs
  set_mode set_mode_locs get_tag_name get_tag_name_locs get_shadow_root get_shadow_root_locs
  defines attach_shadow_root = a_attach_shadow_root
  .

locale l_attach_shadow_root_S_h_a_d_o_w_{_}_D_O_M =
  l_attach_shadow_root_S_h_a_d_o_w_{_}_D_O_{M_defs} set_shadow_root set_shadow_root_locs set_mode set_mode_locs
  get_tag_name get_tag_name_locs get_shadow_root get_shadow_root_locs +
  l_attach_shadow_root_defs attach_shadow_root +
  l_set_shadow_root_S_h_a_d_o_w_{_}_D_O_M type_wf set_shadow_root set_shadow_root_locs +
  l_set_mode type_wf set_mode set_mode_locs +
  l_get_tag_name type_wf get_tag_name get_tag_name_locs +
  l_get_shadow_root type_wf get_shadow_root get_shadow_root_locs +
  l_known_ptr known_ptr
  for known_ptr :: "(_) object_ptr ==> bool"
    and set_shadow_root ::
    "(_) element_ptr ==> (_) shadow_root_ptr option ==> ((_) heap, exception, unit) prog"
    and set_shadow_root_locs :: "(_) element_ptr ==> ((_) heap, exception, unit) prog set"
    and set_mode :: "(_) shadow_root_ptr ==> shadow_root_mode ==> ((_) heap, exception, unit) prog"
    and set_mode_locs :: "(_) shadow_root_ptr ==> ((_) heap, exception, unit) prog set"
    and attach_shadow_root ::
    "(_) element_ptr ==> shadow_root_mode ==> ((_) heap, exception, (_) shadow_root_ptr) prog"
    and type_wf :: "(_) heap ==> bool"
    and get_tag_name :: "(_) element_ptr ==> (_, char list) dom_prog"
    and get_tag_name_locs :: "(_) element_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
    and get_shadow_root :: "(_) element_ptr ==> ((_) heap, exception, (_) shadow_root_ptr option) prog"
    and get_shadow_root_locs :: "(_) element_ptr ==> ((_) heap ==> (_) heap ==> bool) set" +
  assumes known_ptr_impl: "known_ptr = a_known_ptr"
  assumes attach_shadow_root_impl: "attach_shadow_root = a_attach_shadow_root"
begin
lemmas attach_shadow_root_def = a_attach_shadow_root_def[folded attach_shadow_root_impl]

lemma attach_shadow_root_element_ptr_in_heap:
  assumes "h ⊨ ok (attach_shadow_root element_ptr shadow_root_mode)"
  shows "element_ptr |∈| element_ptr_kinds h"
proof -
  obtain h' where "h ⊨ attach_shadow_root element_ptr shadow_root_mode →_h h'"
    using assms by auto
  then
  obtain h2 h3 new_shadow_root_ptr where
    h2: "h ⊨ new_S_h_a_d_o_w_R_o_o_{t_M} →_h h2" and
    new_shadow_root_ptr: "h ⊨ new_S_h_a_d_o_w_R_o_o_{t_M} →_r new_shadow_root_ptr" and
    h3: "h2 ⊨ set_mode new_shadow_root_ptr shadow_root_mode →_h h3" and
    "h3 ⊨ set_shadow_root element_ptr (Some new_shadow_root_ptr) →_h h'"
    by(auto simp add: attach_shadow_root_def
        elim!: bind_returns_heap_E bind_returns_heap_E2[rotated, OF get_tag_name_pure, rotated]
        bind_returns_heap_E2[rotated, OF get_shadow_root_pure, rotated] split: if_splits)

  then have "element_ptr |∈| element_ptr_kinds h3"
    using set_shadow_root_ptr_in_heap by blast

  moreover
  have "object_ptr_kinds h2 = object_ptr_kinds h |∪| {|cast new_shadow_root_ptr|}"
    using h2 new_S_h_a_d_o_w_R_o_o_{t_M_new_ptr} new_shadow_root_ptr by auto

  moreover
  have "object_ptr_kinds h2 = object_ptr_kinds h3"
    apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h = object_ptr_kinds h'",
          OF set_mode_writes h3])
    using set_mode_pointers_preserved
      apply blast
    by (auto simp add: reflp_def transp_def)
  ultimately
  show ?thesis
    by(auto simp add: element_ptr_kinds_def node_ptr_kinds_def)
qed

lemma create_shadow_root_known_ptr:
  assumes "h ⊨ attach_shadow_root element_ptr shadow_root_mode →_r new_shadow_root_ptr"
  shows "known_ptr (cast_s_h_a_d_o_w_{_}_r_o_o_t_{_}_p_t_r₂_o_b_j_e_c_t_{_}_p_t_r new_shadow_root_ptr)"
  using assms
  by(auto simp add: attach_shadow_root_def known_ptr_impl ShadowRootClass.a_known_ptr_def
      new_S_h_a_d_o_w_R_o_o_{t_M_def} new_S_h_a_d_o_w_R_o_o_{t_def} Let_def
      elim!: bind_returns_result_E)
end

locale l_attach_shadow_root = l_attach_shadow_root_defs

interpretation
  i_attach_shadow_root?: l_attach_shadow_root_S_h_a_d_o_w_{_}_D_O_M known_ptr set_shadow_root set_shadow_root_locs
  set_mode set_mode_locs attach_shadow_root type_wf get_tag_name get_tag_name_locs get_shadow_root
  get_shadow_root_locs
  by(auto simp add: l_attach_shadow_root_S_h_a_d_o_w_{_}_D_O_{M_def} l_attach_shadow_root_S_h_a_d_o_w_{_}_D_O_{M_axioms_def}
      attach_shadow_root_def instances)
declare l_attach_shadow_root_S_h_a_d_o_w_{_}_D_O_{M_axioms} [instances]

subsubsection ‹get\_parent›

global_interpretation l_get_parent_C_o_r_e_{_}_D_O_{M_defs} get_child_nodes get_child_nodes_locs
  defines get_parent = "l_get_parent_C_o_r_e_{_}_D_O_{M_defs}.a_get_parent get_child_nodes"
    and get_parent_locs = "l_get_parent_C_o_r_e_{_}_D_O_{M_defs}.a_get_parent_locs get_child_nodes_locs"
  .

interpretation i_get_parent?: l_get_parent_C_o_r_e_{_}_D_O_M known_ptr type_wf get_child_nodes
  get_child_nodes_locs known_ptrs get_parent get_parent_locs
  by(simp add: l_get_parent_C_o_r_e_{_}_D_O_{M_def} l_get_parent_C_o_r_e_{_}_D_O_{M_axioms_def} get_parent_def
      get_parent_locs_def instances)
declare l_get_parent_C_o_r_e_{_}_D_O_{M_axioms} [instances]

lemma get_parent_is_l_get_parent [instances]: "l_get_parent type_wf known_ptr known_ptrs get_parent
    get_parent_locs get_child_nodes get_child_nodes_locs"
  apply(simp add: l_get_parent_def l_get_parent_axioms_def instances)
  using get_parent_reads get_parent_ok get_parent_ptr_in_heap get_parent_pure
    get_parent_parent_in_heap get_parent_child_dual get_parent_reads_pointers
  by blast

paragraph ‹set\_disconnected\_nodes›

locale l_set_disconnected_nodes_get_parent_S_h_a_d_o_w_{_}_D_O_M =
  l_set_disconnected_nodes_get_child_nodes
  + l_set_disconnected_nodes_S_h_a_d_o_w_{_}_D_O_M
  + l_get_parent_C_o_r_e_{_}_D_O_M
begin
lemma set_disconnected_nodes_get_parent [simp]:
  "∀w ∈ set_disconnected_nodes_locs ptr. (h ⊨ w →_h h' ⟶ (∀r ∈ get_parent_locs. r h h'))"
  by(auto simp add: get_parent_locs_def CD.set_disconnected_nodes_locs_def all_args_def)
end
interpretation i_set_disconnected_nodes_get_parent?: l_set_disconnected_nodes_get_parent_S_h_a_d_o_w_{_}_D_O_M
  set_disconnected_nodes set_disconnected_nodes_locs get_child_nodes get_child_nodes_locs type_wf
  DocumentClass.type_wf known_ptr known_ptrs get_parent get_parent_locs
  by (simp add: l_set_disconnected_nodes_get_parent_S_h_a_d_o_w_{_}_D_O_{M_def} instances)
declare l_set_disconnected_nodes_get_parent_C_o_r_e_{_}_D_O_{M_axioms}[instances]

lemma set_disconnected_nodes_get_parent_is_l_set_disconnected_nodes_get_parent [instances]:
  "l_set_disconnected_nodes_get_parent set_disconnected_nodes_locs get_parent_locs"
  by(simp add: l_set_disconnected_nodes_get_parent_def)

subsubsection ‹get\_root\_node›

global_interpretation l_get_root_node_C_o_r_e_{_}_D_O_{M_defs} get_parent get_parent_locs
  defines get_root_node = "l_get_root_node_C_o_r_e_{_}_D_O_{M_defs}.a_get_root_node get_parent"
    and get_root_node_locs = "l_get_root_node_C_o_r_e_{_}_D_O_{M_defs}.a_get_root_node_locs get_parent_locs"
    and get_ancestors = "l_get_root_node_C_o_r_e_{_}_D_O_{M_defs}.a_get_ancestors get_parent"
    and get_ancestors_locs = "l_get_root_node_C_o_r_e_{_}_D_O_{M_defs}.a_get_ancestors_locs get_parent_locs"
  .
declare a_get_ancestors.simps [code]

interpretation i_get_root_node?: l_get_root_node_C_o_r_e_{_}_D_O_M type_wf known_ptr known_ptrs get_parent
  get_parent_locs get_child_nodes get_child_nodes_locs get_ancestors get_ancestors_locs
  get_root_node get_root_node_locs
  by(simp add: l_get_root_node_C_o_r_e_{_}_D_O_{M_def} l_get_root_node_C_o_r_e_{_}_D_O_{M_axioms_def} get_root_node_def
      get_root_node_locs_def get_ancestors_def get_ancestors_locs_def instances)
declare l_get_root_node_C_o_r_e_{_}_D_O_{M_axioms} [instances]

lemma get_ancestors_is_l_get_ancestors [instances]: "l_get_ancestors get_ancestors"
  apply(auto simp add: l_get_ancestors_def)[1]
  using get_ancestors_ptr_in_heap apply fast
  using get_ancestors_ptr apply fast
  done

lemma get_root_node_is_l_get_root_node [instances]: "l_get_root_node get_root_node get_parent"
  by (simp add: l_get_root_node_def Shadow_DOM.i_get_root_node.get_root_node_no_parent)

subsubsection ‹get\_root\_node\_si›

locale l_get_root_node_si_S_h_a_d_o_w_{_}_D_O_{M_defs} =
  l_get_parent_defs get_parent get_parent_locs +
  l_get_host_defs get_host get_host_locs
  for get_parent :: "(_) node_ptr ==> ((_) heap, exception, (_::linorder) object_ptr option) prog"
    and get_parent_locs :: "((_) heap ==> (_) heap ==> bool) set"
    and get_host :: "(_) shadow_root_ptr ==> ((_) heap, exception, (_) element_ptr) prog"
    and get_host_locs :: "((_) heap ==> (_) heap ==> bool) set"
begin
partial_function (dom_prog) a_get_ancestors_si ::
  "(_::linorder) object_ptr ==> (_, (_) object_ptr list) dom_prog"
  where
    "a_get_ancestors_si ptr = do {
      check_in_heap ptr;
      ancestors ← (case cast_o_b_j_e_c_t_{_}_p_t_r₂_n_o_d_e_{_}_p_t_r ptr of
        Some node_ptr ==> do {
          parent_ptr_opt ← get_parent node_ptr;
          (case parent_ptr_opt of
            Some parent_ptr ==> a_get_ancestors_si parent_ptr
          | None ==> return [])
        }
      | None ==> (case cast ptr of
          Some shadow_root_ptr ==> do {
            host ← get_host shadow_root_ptr;
            a_get_ancestors_si (cast host)
          } |
          None ==> return []));
      return (ptr # ancestors)
    }"

definition "a_get_ancestors_si_locs = get_parent_locs ∪ get_host_locs"

definition a_get_root_node_si :: "(_) object_ptr ==> (_, (_) object_ptr) dom_prog"
  where
    "a_get_root_node_si ptr = do {
      ancestors ← a_get_ancestors_si ptr;
      return (last ancestors)
    }"
definition "a_get_root_node_si_locs = a_get_ancestors_si_locs"
end

locale l_get_ancestors_si_defs =
  fixes get_ancestors_si :: "(_::linorder) object_ptr ==> (_, (_) object_ptr list) dom_prog"
  fixes get_ancestors_si_locs :: "((_) heap ==> (_) heap ==> bool) set"

locale l_get_root_node_si_defs =
  fixes get_root_node_si :: "(_) object_ptr ==> (_, (_) object_ptr) dom_prog"
  fixes get_root_node_si_locs :: "((_) heap ==> (_) heap ==> bool) set"

locale l_get_root_node_si_S_h_a_d_o_w_{_}_D_O_M =
  l_get_parent +
  l_get_host +
  l_get_root_node_si_S_h_a_d_o_w_{_}_D_O_{M_defs} +
  l_get_ancestors_si_defs +
  l_get_root_node_si_defs +
  assumes get_ancestors_si_impl: "get_ancestors_si = a_get_ancestors_si"
  assumes get_ancestors_si_locs_impl: "get_ancestors_si_locs = a_get_ancestors_si_locs"
  assumes get_root_node_si_impl: "get_root_node_si = a_get_root_node_si"
  assumes get_root_node_si_locs_impl: "get_root_node_si_locs = a_get_root_node_si_locs"
begin
lemmas get_ancestors_si_def = a_get_ancestors_si.simps[folded get_ancestors_si_impl]
lemmas get_ancestors_si_locs_def = a_get_ancestors_si_locs_def[folded get_ancestors_si_locs_impl]
lemmas get_root_node_si_def =
  a_get_root_node_si_def[folded get_root_node_si_impl get_ancestors_si_impl]
lemmas get_root_node_si_locs_def =
  a_get_root_node_si_locs_def[folded get_root_node_si_locs_impl get_ancestors_si_locs_impl]

lemma get_ancestors_si_pure [simp]:
  "pure (get_ancestors_si ptr) h"
proof -
  have "∀ptr h h' x. h ⊨ get_ancestors_si ptr →_r x ⟶ h ⊨ get_ancestors_si ptr →_h h' ⟶ h = h'"
  proof (induct rule: a_get_ancestors_si.fixp_induct[folded get_ancestors_si_impl])
    case 1
    then show ?case
      by(rule admissible_dom_prog)
  next
    case 2
    then show ?case
      by simp
  next
    case (3 f)
    then show ?case
      using get_parent_pure get_host_pure
      apply(auto simp add: pure_returns_heap_eq pure_def
          split: option.splits
          elim!: bind_returns_heap_E bind_returns_result_E
          dest!: pure_returns_heap_eq[rotated, OF check_in_heap_pure])[1]
        apply (meson option.simps(3) returns_result_eq)
       apply(metis get_parent_pure pure_returns_heap_eq)
      apply(metis get_host_pure pure_returns_heap_eq)
      done
  qed
  then show ?thesis
    by (meson pure_eq_iff)
qed

lemma get_root_node_si_pure [simp]: "pure (get_root_node_si ptr) h"
  by(auto simp add: get_root_node_si_def bind_pure_I)

lemma get_ancestors_si_ptr_in_heap:
  assumes "h ⊨ ok (get_ancestors_si ptr)"
  shows "ptr |∈| object_ptr_kinds h"
  using assms
  by(auto simp add: get_ancestors_si_def check_in_heap_ptr_in_heap elim!: bind_is_OK_E
      dest: is_OK_returns_result_I)

lemma get_ancestors_si_ptr:
  assumes "h ⊨ get_ancestors_si ptr →_r ancestors"
  shows "ptr ∈ set ancestors"
  using assms
  by(simp add: get_ancestors_si_def)
    (auto elim!: bind_returns_result_E2
      split: option.splits
      intro!: bind_pure_I)

lemma get_ancestors_si_never_empty:
  assumes "h ⊨ get_ancestors_si child →_r ancestors"
  shows "ancestors ≠ []"
  using assms
  apply(simp add: get_ancestors_si_def)
  by(auto elim!: bind_returns_result_E2 split: option.splits)

lemma get_root_node_si_no_parent:
  "h ⊨ get_parent node_ptr →_r None ==> h ⊨ get_root_node_si (cast node_ptr) →_r cast node_ptr"
  apply(auto simp add: check_in_heap_def get_root_node_si_def get_ancestors_si_def
      intro!: bind_pure_returns_result_I )[1]
  using get_parent_ptr_in_heap by blast

lemma get_root_node_si_root_not_shadow_root:
  assumes "h ⊨ get_root_node_si ptr →_r root"
  shows "¬ is_shadow_root_ptr_o_b_j_e_c_t_{_}_p_t_r root"
  using assms
proof(auto simp add: get_root_node_si_def elim!: bind_returns_result_E2)
  fix y
  assume "h ⊨ get_ancestors_si ptr →_r y"
    and "is_shadow_root_ptr_o_b_j_e_c_t_{_}_p_t_r (last y)"
    and "root = last y"
  then
  show False
  proof(induct y arbitrary: ptr)
    case Nil
    then show ?case
      using assms(1) get_ancestors_si_never_empty by blast
  next
    case (Cons a x)
    then show ?case
      apply(auto simp add: get_ancestors_si_def[of ptr]
          elim!: bind_returns_result_E2
          split: option.splits if_splits)[1]
      using get_ancestors_si_never_empty apply blast
      using Cons.prems(2) apply auto[1]
      using ‹is_shadow_root_ptr_o_b_j_e_c_t_{_}_p_t_r (last y)› ‹root = last y› by auto
  qed
qed
end

global_interpretation l_get_root_node_si_S_h_a_d_o_w_{_}_D_O_{M_defs} get_parent get_parent_locs get_host get_host_locs
  defines get_root_node_si = a_get_root_node_si
    and get_root_node_si_locs = a_get_root_node_si_locs
    and get_ancestors_si = a_get_ancestors_si
    and get_ancestors_si_locs = a_get_ancestors_si_locs
  .
declare a_get_ancestors_si.simps [code]

interpretation i_get_root_node_si?: l_get_root_node_si_S_h_a_d_o_w_{_}_D_O_M type_wf known_ptr known_ptrs
  get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_host get_host_locs
  get_ancestors_si get_ancestors_si_locs get_root_node_si get_root_node_si_locs
  apply(auto simp add: l_get_root_node_si_S_h_a_d_o_w_{_}_D_O_{M_def} l_get_root_node_si_S_h_a_d_o_w_{_}_D_O_{M_axioms_def}
      instances)[1]
  by(auto simp add: get_root_node_si_def get_root_node_si_locs_def get_ancestors_si_def
      get_ancestors_si_locs_def)
declare l_get_root_node_si_S_h_a_d_o_w_{_}_D_O_{M_axioms}[instances]

lemma get_ancestors_si_is_l_get_ancestors [instances]: "l_get_ancestors get_ancestors_si"
  unfolding l_get_ancestors_def
  using get_ancestors_si_pure get_ancestors_si_ptr get_ancestors_si_ptr_in_heap
  by blast

lemma get_root_node_si_is_l_get_root_node [instances]: "l_get_root_node get_root_node_si get_parent"
  apply(simp add: l_get_root_node_def)
  using get_root_node_si_no_parent
  by fast

paragraph ‹set\_disconnected\_nodes›

locale l_set_disconnected_nodes_get_ancestors_si_C_o_r_e_{_}_D_O_M =
  l_set_disconnected_nodes_get_parent
  + l_get_root_node_si_S_h_a_d_o_w_{_}_D_O_M
  + l_set_disconnected_nodes_S_h_a_d_o_w_{_}_D_O_M
  + l_set_disconnected_nodes_get_host
begin
lemma set_disconnected_nodes_get_ancestors_si:
  "∀w ∈ set_disconnected_nodes_locs ptr. (h ⊨ w →_h h' ⟶ (∀r ∈ get_ancestors_si_locs. r h h'))"
  by(auto simp add: get_parent_locs_def set_disconnected_nodes_locs_def
      set_disconnected_nodes_get_host get_ancestors_si_locs_def all_args_def)
end

locale l_set_disconnected_nodes_get_ancestors_si =
  l_set_disconnected_nodes_defs +
  l_get_ancestors_si_defs +
  assumes set_disconnected_nodes_get_ancestors_si:
    "∀w ∈ set_disconnected_nodes_locs ptr. (h ⊨ w →_h h' ⟶ (∀r ∈ get_ancestors_si_locs. r h h'))"

interpretation i_set_disconnected_nodes_get_ancestors_si?:
  l_set_disconnected_nodes_get_ancestors_si_C_o_r_e_{_}_D_O_M set_disconnected_nodes set_disconnected_nodes_locs
  get_parent get_parent_locs type_wf known_ptr known_ptrs get_child_nodes get_child_nodes_locs
  get_host get_host_locs get_ancestors_si get_ancestors_si_locs get_root_node_si
  get_root_node_si_locs DocumentClass.type_wf

  by (auto simp add: l_set_disconnected_nodes_get_ancestors_si_C_o_r_e_{_}_D_O_{M_def} instances)
declare l_set_disconnected_nodes_get_ancestors_si_C_o_r_e_{_}_D_O_{M_axioms}[instances]

lemma set_disconnected_nodes_get_ancestors_si_is_l_set_disconnected_nodes_get_ancestors_si [instances]:
  "l_set_disconnected_nodes_get_ancestors_si set_disconnected_nodes_locs get_ancestors_si_locs"
  using instances
  apply(simp add: l_set_disconnected_nodes_get_ancestors_si_def)
  using set_disconnected_nodes_get_ancestors_si
  by fast

subsubsection ‹get\_attribute›

lemma get_attribute_is_l_get_attribute [instances]:
  "l_get_attribute type_wf get_attribute get_attribute_locs"
  apply(auto simp add: l_get_attribute_def)[1]
  using i_get_attribute.get_attribute_reads apply fast
  using type_wf_D_o_c_u_m_e_n_t i_get_attribute.get_attribute_ok apply blast
  using i_get_attribute.get_attribute_ptr_in_heap apply fast
  done

subsubsection ‹to\_tree\_order›

global_interpretation l_to_tree_order_C_o_r_e_{_}_D_O_{M_defs} get_child_nodes get_child_nodes_locs defines
  to_tree_order = "l_to_tree_order_C_o_r_e_{_}_D_O_{M_defs}.a_to_tree_order get_child_nodes" .
declare a_to_tree_order.simps [code]

interpretation i_to_tree_order?: l_to_tree_order_C_o_r_e_{_}_D_O_M ShadowRootClass.known_ptr
  ShadowRootClass.type_wf Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs to_tree_order
  by(auto simp add: l_to_tree_order_C_o_r_e_{_}_D_O_{M_def} l_to_tree_order_C_o_r_e_{_}_D_O_{M_axioms_def} to_tree_order_def
      instances)
declare l_to_tree_order_C_o_r_e_{_}_D_O_{M_axioms} [instances]

subsubsection ‹to\_tree\_order\_si›

locale l_to_tree_order_si_C_o_r_e_{_}_D_O_{M_defs} =
  l_get_child_nodes_defs get_child_nodes get_child_nodes_locs +
  l_get_shadow_root_defs get_shadow_root get_shadow_root_locs
  for get_child_nodes :: "(_::linorder) object_ptr ==> ((_) heap, exception, (_) node_ptr list) prog"
    and get_child_nodes_locs :: "(_) object_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
    and get_shadow_root :: "(_) element_ptr ==> ((_) heap, exception, (_) shadow_root_ptr option) prog"
    and get_shadow_root_locs :: "(_) element_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
begin
partial_function (dom_prog) a_to_tree_order_si :: "(_) object_ptr ==> (_, (_) object_ptr list) dom_prog"
  where
    "a_to_tree_order_si ptr = (do {
      children ← get_child_nodes ptr;
      shadow_root_part ← (case cast ptr of
        Some element_ptr ==> do {
          shadow_root_opt ← get_shadow_root element_ptr;
          (case shadow_root_opt of
            Some shadow_root_ptr ==> return [cast shadow_root_ptr]
            | None ==> return [])
        } |
        None ==> return []);
      treeorders ← map_M a_to_tree_order_si ((map (cast) children) @ shadow_root_part);
      return (ptr # concat treeorders)
    })"
end

locale l_to_tree_order_si_defs =
  fixes to_tree_order_si :: "(_) object_ptr ==> (_, (_) object_ptr list) dom_prog"

global_interpretation l_to_tree_order_si_C_o_r_e_{_}_D_O_{M_defs} get_child_nodes get_child_nodes_locs
  get_shadow_root get_shadow_root_locs
  defines to_tree_order_si = "a_to_tree_order_si" .
declare a_to_tree_order_si.simps [code]

locale l_to_tree_order_si_S_h_a_d_o_w_{_}_D_O_M =
  l_to_tree_order_si_defs +
  l_to_tree_order_si_C_o_r_e_{_}_D_O_{M_defs} +
  l_get_child_nodes +
  l_get_shadow_root +
  assumes to_tree_order_si_impl: "to_tree_order_si = a_to_tree_order_si"
begin
lemmas to_tree_order_si_def = a_to_tree_order_si.simps[folded to_tree_order_si_impl]

lemma to_tree_order_si_pure [simp]: "pure (to_tree_order_si ptr) h"
proof -
  have "\<forall>ptr h h' x. h \<turnstile> to_tree_order_si ptr \<rightarrow>\<^sub>r x \<longrightarrow> h \<turnstile> to_tree_order_si ptr \<rightarrow>\<^sub>h h' \<longrightarrow> h = h'"
  proof (induct rule: a_to_tree_order_si.fixp_induct[folded to_tree_order_si_impl])
    case 1
    then show ?case
      by (rule admissible_dom_prog)
  next
    case 2
    then show ?case
      by simp
  next
    case (3 f)
    then have "\<And>x h. pure (f x) h"
      by (metis is_OK_returns_heap_E is_OK_returns_result_E pure_def)
    then have "\<And>xs h. pure (map_M f xs) h"
      by(rule map_M_pure_I)
    then show ?case
      by(auto elim!: bind_returns_heap_E2 split: option.splits)
  qed
  then show ?thesis
    unfolding pure_def
    by (metis is_OK_returns_heap_E is_OK_returns_result_E)
qed
end

interpretation i_to_tree_order_si?: l_to_tree_order_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M to_tree_order_si get_child_nodes
  get_child_nodes_locs get_shadow_root get_shadow_root_locs type_wf known_ptr
  by(auto simp add: l_to_tree_order_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def l_to_tree_order_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def
      to_tree_order_si_def instances)
declare l_to_tree_order_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms [instances]

subsubsection \<open>first\_in\_tree\_order\<close>

global_interpretation l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs to_tree_order defines
  first_in_tree_order = "l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_first_in_tree_order to_tree_order" .

interpretation i_first_in_tree_order?: l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M to_tree_order first_in_tree_order
  by(auto simp add: l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def first_in_tree_order_def)
declare l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms [instances]

lemma to_tree_order_is_l_to_tree_order [instances]: "l_to_tree_order to_tree_order"
  by(auto simp add: l_to_tree_order_def)

subsubsection \<open>first\_in\_tree\_order\<close>

global_interpretation l_dummy defines
  first_in_tree_order_si = "l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_first_in_tree_order to_tree_order_si"
  .

subsubsection \<open>get\_element\_by\<close>

global_interpretation l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs to_tree_order first_in_tree_order get_attribute
  get_attribute_locs
  defines
    get_element_by_id =
    "l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_element_by_id first_in_tree_order get_attribute" and
    get_elements_by_class_name =
    "l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_elements_by_class_name to_tree_order get_attribute" and
    get_elements_by_tag_name =
    "l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_elements_by_tag_name to_tree_order" .

interpretation i_get_element_by?: l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M to_tree_order first_in_tree_order
  get_attribute get_attribute_locs get_element_by_id get_elements_by_class_name
  get_elements_by_tag_name type_wf
  by(auto simp add: l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def
      get_element_by_id_def get_elements_by_class_name_def get_elements_by_tag_name_def
      instances)
declare l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances]

lemma get_element_by_is_l_get_element_by [instances]:
  "l_get_element_by get_element_by_id get_elements_by_tag_name to_tree_order"
  apply(auto simp add: l_get_element_by_def)[1]
  using get_element_by_id_result_in_tree_order apply fast
  done

subsubsection \<open>get\_element\_by\_si\<close>

global_interpretation l_dummy defines
  get_element_by_id_si =
  "l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_element_by_id first_in_tree_order_si get_attribute" and
  get_elements_by_class_name_si =
  "l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_elements_by_class_name to_tree_order_si get_attribute" and
  get_elements_by_tag_name_si =
  "l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_elements_by_tag_name to_tree_order_si"
  .

subsubsection \<open>find\_slot\<close>

locale l_find_slot\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs =
  l_get_parent_defs get_parent get_parent_locs +
  l_get_shadow_root_defs get_shadow_root get_shadow_root_locs +
  l_get_mode_defs get_mode get_mode_locs +
  l_get_attribute_defs get_attribute get_attribute_locs +
  l_get_tag_name_defs get_tag_name get_tag_name_locs +
  l_first_in_tree_order_defs first_in_tree_order
  for get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_::linorder) object_ptr option) prog"
    and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"
    and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and get_mode :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, shadow_root_mode) prog"
    and get_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and get_attribute :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, char list option) prog"
    and get_attribute_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"
    and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and first_in_tree_order ::
    "(_) object_ptr \<Rightarrow> ((_) object_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog) \<Rightarrow>
    ((_) heap, exception, (_) element_ptr option) prog"
begin
definition a_find_slot :: "bool \<Rightarrow> (_) node_ptr \<Rightarrow> (_, (_) element_ptr option) dom_prog"
  where
    "a_find_slot open_flag slotable = do {
      parent_opt \<leftarrow> get_parent slotable;
      (case parent_opt of
        Some parent \<Rightarrow>
          if is_element_ptr_kind parent
          then do {
            shadow_root_ptr_opt \<leftarrow> get_shadow_root (the (cast parent));
            (case shadow_root_ptr_opt of
              Some shadow_root_ptr \<Rightarrow> do {
                shadow_root_mode \<leftarrow> get_mode shadow_root_ptr;
                if open_flag \<and> shadow_root_mode \<noteq> Open
                then return None
                else first_in_tree_order (cast shadow_root_ptr) (\<lambda>ptr. if is_element_ptr_kind ptr
                  then do {
                    tag \<leftarrow> get_tag_name (the (cast ptr));
                    name_attr \<leftarrow> get_attribute (the (cast ptr)) ''name'';
                    slotable_name_attr \<leftarrow> (if is_element_ptr_kind slotable
                                           then get_attribute (the (cast slotable)) ''slot''
                                           else return None);
                    (if (tag = ''slot'' \<and> (name_attr = slotable_name_attr \<or>
                        (name_attr = None \<and> slotable_name_attr = Some '''') \<or>
                        (name_attr = Some '''' \<and> slotable_name_attr = None)))
                    then return (Some (the (cast ptr)))
                    else return None)}
                  else return None)}
              | None \<Rightarrow> return None)}
          else return None
    | _ \<Rightarrow> return None)}"

definition a_assigned_slot :: "(_) node_ptr \<Rightarrow> (_, (_) element_ptr option) dom_prog"
  where
    "a_assigned_slot = a_find_slot True"
end

global_interpretation l_find_slot\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs get_parent get_parent_locs get_shadow_root
  get_shadow_root_locs get_mode get_mode_locs get_attribute get_attribute_locs get_tag_name
  get_tag_name_locs first_in_tree_order
  defines find_slot = a_find_slot
    and assigned_slot = a_assigned_slot
  .

locale l_find_slot_defs =
  fixes find_slot :: "bool \<Rightarrow> (_) node_ptr \<Rightarrow> (_, (_) element_ptr option) dom_prog"
    and assigned_slot :: "(_) node_ptr \<Rightarrow> (_, (_) element_ptr option) dom_prog"

locale l_find_slot\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M =
  l_find_slot\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs +
  l_find_slot_defs +
  l_get_parent +
  l_get_shadow_root +
  l_get_mode +
  l_get_attribute +
  l_get_tag_name +
  l_to_tree_order +
  l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M +
  assumes find_slot_impl: "find_slot = a_find_slot"
  assumes assigned_slot_impl: "assigned_slot = a_assigned_slot"
begin
lemmas find_slot_def = find_slot_impl[unfolded a_find_slot_def]
lemmas assigned_slot_def = assigned_slot_impl[unfolded a_assigned_slot_def]

lemma find_slot_ptr_in_heap:
  assumes "h \<turnstile> find_slot open_flag slotable \<rightarrow>\<^sub>r slot_opt"
  shows "slotable |\<in>| node_ptr_kinds h"
  using assms
  apply(auto simp add: find_slot_def elim!: bind_returns_result_E2)[1]
  using get_parent_ptr_in_heap by blast

lemma find_slot_slot_in_heap:
  assumes "h \<turnstile> find_slot open_flag slotable \<rightarrow>\<^sub>r Some slot"
  shows "slot |\<in>| element_ptr_kinds h"
  using assms
  apply(auto simp add: find_slot_def first_in_tree_order_def
      elim!: bind_returns_result_E2 map_filter_M_pure_E[where y=slot]
      split: option.splits if_splits list.splits
      intro!: map_filter_M_pure bind_pure_I)[1]
  using get_tag_name_ptr_in_heap by blast+

lemma find_slot_pure [simp]: "pure (find_slot open_flag slotable) h"
  by(auto simp add: find_slot_def first_in_tree_order_def
      intro!: bind_pure_I map_filter_M_pure
      split: option.splits list.splits)
end

interpretation i_find_slot?: l_find_slot\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_parent get_parent_locs get_shadow_root
  get_shadow_root_locs get_mode get_mode_locs get_attribute get_attribute_locs get_tag_name
  get_tag_name_locs first_in_tree_order find_slot assigned_slot type_wf known_ptr known_ptrs
  get_child_nodes get_child_nodes_locs to_tree_order
  by (auto simp add: find_slot_def assigned_slot_def l_find_slot\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def
      l_find_slot\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def instances)
declare l_find_slot\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms [instances]

locale l_find_slot = l_find_slot_defs +
  assumes find_slot_ptr_in_heap:
    "h \<turnstile> find_slot open_flag slotable \<rightarrow>\<^sub>r slot_opt \<Longrightarrow> slotable |\<in>| node_ptr_kinds h"
  assumes find_slot_slot_in_heap:
    "h \<turnstile> find_slot open_flag slotable \<rightarrow>\<^sub>r Some slot \<Longrightarrow> slot |\<in>| element_ptr_kinds h"
  assumes find_slot_pure [simp]:
    "pure (find_slot open_flag slotable) h"

lemma find_slot_is_l_find_slot [instances]: "l_find_slot find_slot"
  apply(auto simp add: l_find_slot_def)[1]
  using find_slot_ptr_in_heap apply fast
  using find_slot_slot_in_heap apply fast
  done

subsubsection \<open>get\_disconnected\_nodes\<close>

locale l_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M =
  CD: l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes
  get_disconnected_nodes_locs +
  l_type_wf type_wf
  for type_wf :: "(_) heap \<Rightarrow> bool"
    and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"
    and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_, (_) node_ptr list) dom_prog"
    and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" +
  assumes type_wf_impl: "type_wf = ShadowRootClass.type_wf"
begin

lemma get_disconnected_nodes_ok:
  "type_wf h \<Longrightarrow> document_ptr |\<in>| document_ptr_kinds h \<Longrightarrow>
    h \<turnstile> ok (get_disconnected_nodes document_ptr)"
  apply(unfold type_wf_impl get_disconnected_nodes_impl[unfolded a_get_disconnected_nodes_def])
  using CD.get_disconnected_nodes_ok CD.type_wf_impl ShadowRootClass.type_wf\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t
  by blast
end

interpretation i_get_disconnected_nodes?: l_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf
  DocumentClass.type_wf get_disconnected_nodes get_disconnected_nodes_locs
  by(auto simp add: l_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def l_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def
      instances)
declare l_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms [instances]

lemma get_disconnected_nodes_is_l_get_disconnected_nodes [instances]:
  "l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs"
  apply(auto simp add: l_get_disconnected_nodes_def)[1]
  using i_get_disconnected_nodes.get_disconnected_nodes_reads apply fast
  using get_disconnected_nodes_ok apply fast
  using i_get_disconnected_nodes.get_disconnected_nodes_ptr_in_heap apply fast
  done

paragraph \<open>set\_child\_nodes\<close>

locale l_set_child_nodes_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M =
  l_set_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M +
  l_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M
begin
lemma set_child_nodes_get_disconnected_nodes:
  "\<forall>w \<in> set_child_nodes_locs ptr. (h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r \<in> get_disconnected_nodes_locs ptr'. r h h'))"
  by(auto simp add: set_child_nodes_locs_def CD.set_child_nodes_locs_def
      CD.get_disconnected_nodes_locs_def all_args_def)
end

interpretation
  i_set_child_nodes_get_disconnected_nodes?: l_set_child_nodes_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf
  known_ptr DocumentClass.type_wf DocumentClass.known_ptr set_child_nodes set_child_nodes_locs
  Core_DOM_Functions.set_child_nodes Core_DOM_Functions.set_child_nodes_locs
  get_disconnected_nodes get_disconnected_nodes_locs
  apply(auto simp add: l_set_child_nodes_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def instances)[1]
  by(unfold_locales)
declare l_set_child_nodes_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances]

lemma set_child_nodes_get_disconnected_nodes_is_l_set_child_nodes_get_disconnected_nodes [instances]:
  "l_set_child_nodes_get_disconnected_nodes type_wf set_child_nodes set_child_nodes_locs
    get_disconnected_nodes get_disconnected_nodes_locs"
  using set_child_nodes_is_l_set_child_nodes get_disconnected_nodes_is_l_get_disconnected_nodes
  apply(simp add: l_set_child_nodes_get_disconnected_nodes_def
      l_set_child_nodes_get_disconnected_nodes_axioms_def)
  using set_child_nodes_get_disconnected_nodes
  by fast

paragraph \<open>set\_disconnected\_nodes\<close>

lemma set_disconnected_nodes_get_disconnected_nodes_l_set_disconnected_nodes_get_disconnected_nodes
  [instances]:
  "l_set_disconnected_nodes_get_disconnected_nodes ShadowRootClass.type_wf get_disconnected_nodes
    get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs"
  apply(auto simp add: l_set_disconnected_nodes_get_disconnected_nodes_def
      l_set_disconnected_nodes_get_disconnected_nodes_axioms_def instances)[1]
  using i_set_disconnected_nodes_get_disconnected_nodes.set_disconnected_nodes_get_disconnected_nodes
   apply fast
  using i_set_disconnected_nodes_get_disconnected_nodes.set_disconnected_nodes_get_disconnected_nodes_different_pointers
  apply fast
  done

paragraph \<open>delete\_shadow\_root\<close>

locale l_delete_shadow_root_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M =
  l_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M
begin
lemma get_disconnected_nodes_delete_shadow_root:  "h \<turnstile> delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr \<rightarrow>\<^sub>h h'
    \<Longrightarrow> r \<in> get_disconnected_nodes_locs ptr' \<Longrightarrow> r h h'"
  by(auto simp add: CD.get_disconnected_nodes_locs_def delete_shadow_root_get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t)
end

locale l_delete_shadow_root_get_disconnected_nodes = l_get_disconnected_nodes_defs +
  assumes get_disconnected_nodes_delete_shadow_root:  "h \<turnstile> delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr \<rightarrow>\<^sub>h h'
    \<Longrightarrow> r \<in> get_disconnected_nodes_locs ptr' \<Longrightarrow> r h h'"
interpretation l_delete_shadow_root_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf DocumentClass.type_wf
  get_disconnected_nodes get_disconnected_nodes_locs
  by(auto simp add: l_delete_shadow_root_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def instances)

lemma l_delete_shadow_root_get_disconnected_nodes_get_disconnected_nodes_locs [instances]: "l_delete_shadow_root_get_disconnected_nodes get_disconnected_nodes_locs"
  apply(auto simp add: l_delete_shadow_root_get_disconnected_nodes_def)[1]
  using get_disconnected_nodes_delete_shadow_root apply fast
  done

paragraph \<open>set\_shadow\_root\<close>

locale l_set_shadow_root_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M =
  l_set_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M +
  l_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M
begin
lemma set_shadow_root_get_disconnected_nodes:
  "\<forall>w \<in> set_shadow_root_locs ptr. (h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r \<in> get_disconnected_nodes_locs ptr'. r h h'))"
  by(auto simp add: set_shadow_root_locs_def CD.get_disconnected_nodes_locs_def all_args_def)
end

interpretation i_set_shadow_root_get_disconnected_nodes?:
  l_set_shadow_root_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_shadow_root set_shadow_root_locs
  DocumentClass.type_wf get_disconnected_nodes get_disconnected_nodes_locs
  apply(auto simp add: l_set_shadow_root_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def instances)[1]
  by(unfold_locales)
declare l_set_shadow_root_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances]

locale l_set_shadow_root_get_disconnected_nodes =
  l_set_shadow_root_defs +
  l_get_disconnected_nodes_defs +
  assumes set_shadow_root_get_disconnected_nodes:
    "\<forall>w \<in> set_shadow_root_locs ptr. (h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r \<in> get_disconnected_nodes_locs ptr'. r h h'))"

lemma set_shadow_root_get_disconnected_nodes_is_l_set_shadow_root_get_disconnected_nodes [instances]:
  "l_set_shadow_root_get_disconnected_nodes set_shadow_root_locs get_disconnected_nodes_locs"
  using set_shadow_root_is_l_set_shadow_root get_disconnected_nodes_is_l_get_disconnected_nodes
  apply(simp add: l_set_shadow_root_get_disconnected_nodes_def )
  using set_shadow_root_get_disconnected_nodes
  by fast

paragraph \<open>set\_mode\<close>

locale l_set_mode_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M =
  l_set_mode\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M +
  l_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M
begin
lemma set_mode_get_disconnected_nodes:
  "\<forall>w \<in> set_mode_locs ptr. (h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r \<in> get_disconnected_nodes_locs ptr'. r h h'))"
  by(auto simp add: set_mode_locs_def
      CD.get_disconnected_nodes_locs_impl[unfolded CD.a_get_disconnected_nodes_locs_def]
      all_args_def)
end

interpretation i_set_mode_get_disconnected_nodes?: l_set_mode_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf
  set_mode set_mode_locs DocumentClass.type_wf get_disconnected_nodes get_disconnected_nodes_locs
  by unfold_locales
declare l_set_mode_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances]

locale l_set_mode_get_disconnected_nodes = l_set_mode + l_get_disconnected_nodes +
  assumes set_mode_get_disconnected_nodes:
    "\<forall>w \<in> set_mode_locs ptr. (h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r \<in> get_disconnected_nodes_locs ptr'. r h h'))"

lemma set_mode_get_disconnected_nodes_is_l_set_mode_get_disconnected_nodes [instances]:
  "l_set_mode_get_disconnected_nodes type_wf set_mode set_mode_locs get_disconnected_nodes
                                         get_disconnected_nodes_locs"
  using set_mode_is_l_set_mode get_disconnected_nodes_is_l_get_disconnected_nodes
  apply(simp add: l_set_mode_get_disconnected_nodes_def
      l_set_mode_get_disconnected_nodes_axioms_def)
  using set_mode_get_disconnected_nodes
  by fast

paragraph \<open>new\_shadow\_root\<close>

locale l_new_shadow_root_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M =
  l_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes
  get_disconnected_nodes_locs
  for type_wf :: "(_) heap \<Rightarrow> bool"
    and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"
    and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_, (_) node_ptr list) dom_prog"
    and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
begin
lemma get_disconnected_nodes_new_shadow_root:
  "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr \<Longrightarrow> h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h'
    \<Longrightarrow> r \<in> get_disconnected_nodes_locs ptr' \<Longrightarrow> r h h'"
  by(auto simp add: CD.get_disconnected_nodes_locs_def new_shadow_root_get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t)
end

interpretation i_new_shadow_root_get_disconnected_nodes?:
  l_new_shadow_root_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf DocumentClass.type_wf
  get_disconnected_nodes get_disconnected_nodes_locs
  by unfold_locales
declare l_new_shadow_root_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances]

locale l_new_shadow_root_get_disconnected_nodes = l_get_disconnected_nodes_defs +
  assumes get_disconnected_nodes_new_shadow_root:
    "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr \<Longrightarrow> h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h'
    \<Longrightarrow> r \<in> get_disconnected_nodes_locs ptr' \<Longrightarrow> r h h'"

lemma new_shadow_root_get_disconnected_nodes_is_l_new_shadow_root_get_disconnected_nodes [instances]:
  "l_new_shadow_root_get_disconnected_nodes get_disconnected_nodes_locs"
  apply (auto simp add: l_new_shadow_root_get_disconnected_nodes_def)[1]
  using get_disconnected_nodes_new_shadow_root apply fast
  done

subsubsection \<open>remove\_shadow\_root\<close>

locale l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs =
  l_get_child_nodes_defs get_child_nodes get_child_nodes_locs +
  l_get_shadow_root_defs get_shadow_root get_shadow_root_locs +
  l_set_shadow_root_defs set_shadow_root set_shadow_root_locs +
  l_get_disconnected_nodes_defs get_disconnected_nodes get_disconnected_nodes_locs
  for get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"
    and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"
    and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and set_shadow_root ::
    "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"
    and set_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"
    and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"
    and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
begin

definition a_remove_shadow_root :: "(_) element_ptr \<Rightarrow> (_, unit) dom_prog" where
  "a_remove_shadow_root element_ptr = do {
    shadow_root_ptr_opt \<leftarrow> get_shadow_root element_ptr;
    (case shadow_root_ptr_opt of
      Some shadow_root_ptr \<Rightarrow> do {
        children \<leftarrow> get_child_nodes (cast shadow_root_ptr);
        (if children = []
        then do {
          set_shadow_root element_ptr None;
          delete_M shadow_root_ptr
        } else do {
          error HierarchyRequestError
        })
      } |
      None \<Rightarrow> error HierarchyRequestError)
    }"

definition a_remove_shadow_root_locs ::
  "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr \<Rightarrow> ((_, unit) dom_prog) set"
  where
    "a_remove_shadow_root_locs element_ptr shadow_root_ptr \<equiv>
      set_shadow_root_locs element_ptr \<union> {delete_M shadow_root_ptr}"
end

global_interpretation l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs get_child_nodes get_child_nodes_locs
  get_shadow_root get_shadow_root_locs set_shadow_root set_shadow_root_locs get_disconnected_nodes
  get_disconnected_nodes_locs
  defines remove_shadow_root = "a_remove_shadow_root"
    and remove_shadow_root_locs = a_remove_shadow_root_locs
  .

locale l_remove_shadow_root_defs =
  fixes remove_shadow_root :: "(_) element_ptr \<Rightarrow> (_, unit) dom_prog"
  fixes remove_shadow_root_locs ::
    "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr \<Rightarrow> ((_, unit) dom_prog) set"

locale l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M =
  l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs +
  l_remove_shadow_root_defs +
  l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M +
  l_set_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M +
  l_get_child_nodes +
  l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs +
  assumes remove_shadow_root_impl: "remove_shadow_root = a_remove_shadow_root"
  assumes remove_shadow_root_locs_impl: "remove_shadow_root_locs = a_remove_shadow_root_locs"
begin
lemmas remove_shadow_root_def =
  remove_shadow_root_impl[unfolded remove_shadow_root_def a_remove_shadow_root_def]
lemmas remove_shadow_root_locs_def =
  remove_shadow_root_locs_impl[unfolded remove_shadow_root_locs_def a_remove_shadow_root_locs_def]

lemma remove_shadow_root_writes:
  "writes (remove_shadow_root_locs element_ptr (the |h \<turnstile> get_shadow_root element_ptr|\<^sub>r))
    (remove_shadow_root element_ptr) h h'"
  apply(auto simp add: remove_shadow_root_locs_def remove_shadow_root_def all_args_def
      writes_union_right_I writes_union_left_I set_shadow_root_writes
      intro!: writes_bind writes_bind_pure[OF get_shadow_root_pure]
      writes_bind_pure[OF get_child_nodes_pure]
      intro: writes_subset[OF set_shadow_root_writes] writes_subset[OF writes_singleton2]
      split: option.splits)[1]
  using  writes_union_left_I[OF set_shadow_root_writes]
   apply (metis inf_sup_aci(5) insert_is_Un)
  using writes_union_right_I[OF writes_singleton[of delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M]]
  by (smt (verit, best) insert_is_Un writes_singleton2 writes_union_left_I)
end

interpretation i_remove_shadow_root?: l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes
  get_child_nodes_locs get_shadow_root get_shadow_root_locs set_shadow_root set_shadow_root_locs
  get_disconnected_nodes get_disconnected_nodes_locs remove_shadow_root remove_shadow_root_locs
  type_wf known_ptr
  by(auto simp add: l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def
      remove_shadow_root_def remove_shadow_root_locs_def instances)
declare l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms [instances]

paragraph \<open>get\_child\_nodes\<close>

locale l_remove_shadow_root_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M =
  l_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M +
  l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M
begin

lemma remove_shadow_root_get_child_nodes_different_pointers:
  assumes "ptr \<noteq> cast shadow_root_ptr"
  assumes "w \<in> remove_shadow_root_locs element_ptr shadow_root_ptr"
  assumes "h \<turnstile> w \<rightarrow>\<^sub>h h'"
  assumes "r \<in> get_child_nodes_locs ptr"
  shows "r h h'"
  using assms
  apply(auto simp add: all_args_def get_child_nodes_locs_def CD.get_child_nodes_locs_def
      remove_shadow_root_locs_def set_shadow_root_locs_def
      delete_shadow_root_get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t delete_shadow_root_get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t
      delete_shadow_root_get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t delete_shadow_root_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t
      delete_shadow_root_get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t[rotated]
      element_put_get_preserved[where setter=shadow_root_opt_update]
      intro: is_shadow_root_ptr_kind_obtains
      elim: is_document_ptr_kind_obtains is_shadow_root_ptr_kind_obtains
      split: if_splits option.splits)[1]
  done
end

locale l_remove_shadow_root_get_child_nodes = l_get_child_nodes_defs + l_remove_shadow_root_defs +
  assumes remove_shadow_root_get_child_nodes_different_pointers:
    "ptr \<noteq> cast shadow_root_ptr \<Longrightarrow> w \<in> remove_shadow_root_locs element_ptr shadow_root_ptr \<Longrightarrow>
      h \<turnstile> w \<rightarrow>\<^sub>h h' \<Longrightarrow> r \<in> get_child_nodes_locs ptr \<Longrightarrow> r h h'"

interpretation i_remove_shadow_root_get_child_nodes?: l_remove_shadow_root_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M
  type_wf known_ptr DocumentClass.type_wf DocumentClass.known_ptr get_child_nodes
  get_child_nodes_locs Core_DOM_Functions.get_child_nodes Core_DOM_Functions.get_child_nodes_locs
  get_shadow_root get_shadow_root_locs set_shadow_root set_shadow_root_locs get_disconnected_nodes
  get_disconnected_nodes_locs remove_shadow_root remove_shadow_root_locs
  by(auto simp add: l_remove_shadow_root_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def instances)
declare l_remove_shadow_root_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances]

lemma remove_shadow_root_get_child_nodes_is_l_remove_shadow_root_get_child_nodes [instances]:
  "l_remove_shadow_root_get_child_nodes get_child_nodes_locs remove_shadow_root_locs"
  apply(auto simp add: l_remove_shadow_root_get_child_nodes_def instances )[1]
  using remove_shadow_root_get_child_nodes_different_pointers apply fast
  done

paragraph \<open>get\_tag\_name\<close>

locale l_remove_shadow_root_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M =
  l_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M +
  l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M
begin

lemma remove_shadow_root_get_tag_name:
  assumes "w \<in> remove_shadow_root_locs element_ptr shadow_root_ptr"
  assumes "h \<turnstile> w \<rightarrow>\<^sub>h h'"
  assumes "r \<in> get_tag_name_locs ptr"
  shows "r h h'"
  using assms
  by(auto simp add: all_args_def remove_shadow_root_locs_def set_shadow_root_locs_def
      CD.get_tag_name_locs_def delete_shadow_root_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t
      element_put_get_preserved[where setter=shadow_root_opt_update]
      split: if_splits option.splits)
end

locale l_remove_shadow_root_get_tag_name = l_get_tag_name_defs + l_remove_shadow_root_defs +
  assumes remove_shadow_root_get_tag_name:
    "w \<in> remove_shadow_root_locs element_ptr shadow_root_ptr \<Longrightarrow> h \<turnstile> w \<rightarrow>\<^sub>h h' \<Longrightarrow>
    r \<in> get_tag_name_locs ptr \<Longrightarrow> r h h'"

interpretation i_remove_shadow_root_get_tag_name?: l_remove_shadow_root_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf
  DocumentClass.type_wf get_tag_name get_tag_name_locs get_child_nodes get_child_nodes_locs
  get_shadow_root get_shadow_root_locs set_shadow_root set_shadow_root_locs get_disconnected_nodes
  get_disconnected_nodes_locs remove_shadow_root remove_shadow_root_locs known_ptr
  by(auto simp add: l_remove_shadow_root_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def instances)
declare l_remove_shadow_root_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances]

lemma remove_shadow_root_get_tag_name_is_l_remove_shadow_root_get_tag_name [instances]:
  "l_remove_shadow_root_get_tag_name get_tag_name_locs remove_shadow_root_locs"
  apply(auto simp add: l_remove_shadow_root_get_tag_name_def instances )[1]
  using remove_shadow_root_get_tag_name apply fast
  done

subsubsection \<open>get\_owner\_document\<close>

locale l_get_owner_document\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs =
  l_get_host_defs get_host get_host_locs +
  CD: l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs get_root_node get_root_node_locs get_disconnected_nodes
  get_disconnected_nodes_locs
  for get_root_node :: "(_::linorder) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"
    and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"
    and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"
    and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
begin
definition a_get_owner_document\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ::
  "(_) shadow_root_ptr \<Rightarrow> unit \<Rightarrow> (_, (_) document_ptr) dom_prog"
  where
    "a_get_owner_document\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr _ = do {
      host \<leftarrow> get_host shadow_root_ptr;
      CD.a_get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r (cast host) ()
    }"

definition a_get_owner_document_tups :: "(((_) object_ptr \<Rightarrow> bool) \<times> ((_) object_ptr \<Rightarrow> unit
  \<Rightarrow> (_, (_) document_ptr) dom_prog)) list"
  where
    "a_get_owner_document_tups = [(is_shadow_root_ptr, a_get_owner_document\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r \<circ> the \<circ> cast)]"

definition a_get_owner_document :: "(_::linorder) object_ptr \<Rightarrow> (_, (_) document_ptr) dom_prog"
  where
    "a_get_owner_document ptr = invoke (CD.a_get_owner_document_tups @ a_get_owner_document_tups) ptr ()"
end

global_interpretation l_get_owner_document\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs get_root_node_si get_root_node_si_locs
  get_disconnected_nodes get_disconnected_nodes_locs get_host get_host_locs
  defines get_owner_document_tups = a_get_owner_document_tups
    and get_owner_document = a_get_owner_document
    and get_owner_document\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r = a_get_owner_document\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r
    and get_owner_document_tups\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M =
      "l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_owner_document_tups
        get_root_node_si get_disconnected_nodes"
    and get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r =
      "l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r
        get_root_node_si get_disconnected_nodes"
  .

locale l_get_owner_document\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M =
  l_known_ptr known_ptr +
  l_get_owner_document\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs get_root_node_si get_root_node_si_locs get_disconnected_nodes
  get_disconnected_nodes_locs get_host get_host_locs +
  l_get_owner_document_defs get_owner_document +
  l_get_host get_host get_host_locs +
  CD: l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M
  get_parent get_parent_locs known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes
  get_disconnected_nodes_locs get_root_node_si get_root_node_si_locs get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M
  for known_ptr :: "(_::linorder) object_ptr \<Rightarrow> bool"
    and known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_::linorder) object_ptr \<Rightarrow> bool"
    and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"
    and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and type_wf :: "(_) heap \<Rightarrow> bool"
    and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"
    and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and get_root_node_si :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"
    and get_root_node_si_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"
    and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"
    and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog" +
  assumes known_ptr_impl: "known_ptr = a_known_ptr"
  assumes get_owner_document_impl: "get_owner_document = a_get_owner_document"
begin
lemmas known_ptr_def = known_ptr_impl[unfolded a_known_ptr_def]
lemmas get_owner_document_def = a_get_owner_document_def[folded get_owner_document_impl]

lemma get_owner_document_pure [simp]:
  "pure (get_owner_document ptr) h"
proof -
  have "\<And>shadow_root_ptr. pure (a_get_owner_document\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr ()) h"
    apply(auto simp add: a_get_owner_document\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def
        intro!: bind_pure_I filter_M_pure_I
        split: option.splits)[1]
    by(auto simp add: CD.a_get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def
        intro!: bind_pure_I filter_M_pure_I
        split: option.splits)
  then show ?thesis
    apply(auto simp add: get_owner_document_def)[1]
    apply(split CD.get_owner_document_splits, rule conjI)+
     apply(simp)
    apply(auto simp add: a_get_owner_document_tups_def)[1]
    apply(split invoke_splits, rule conjI)+
     apply simp
    by(auto intro!: bind_pure_I)
qed

lemma get_owner_document_ptr_in_heap:
  assumes "h \<turnstile> ok (get_owner_document ptr)"
  shows "ptr |\<in>| object_ptr_kinds h"
  using assms
  by(auto simp add: get_owner_document_def invoke_ptr_in_heap dest: is_OK_returns_heap_I)

end

interpretation i_get_owner_document?: l_get_owner_document\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr DocumentClass.known_ptr
  get_parent get_parent_locs type_wf get_disconnected_nodes get_disconnected_nodes_locs
  get_root_node_si get_root_node_si_locs CD.a_get_owner_document get_host get_host_locs
  get_owner_document
  by(auto simp add: instances l_get_owner_document\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def l_get_owner_document\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def
      l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def
      get_owner_document_def Core_DOM_Functions.get_owner_document_def)
declare l_get_owner_document\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms [instances]

lemma get_owner_document_is_l_get_owner_document [instances]:
  "l_get_owner_document get_owner_document"
  apply(auto simp add: l_get_owner_document_def)[1]
  using get_owner_document_ptr_in_heap apply fast
  done

subsubsection \<open>remove\_child\<close>

global_interpretation l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs get_child_nodes get_child_nodes_locs set_child_nodes
  set_child_nodes_locs get_parent get_parent_locs get_owner_document get_disconnected_nodes
  get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs
  defines remove = "l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_remove get_child_nodes set_child_nodes get_parent
    get_owner_document get_disconnected_nodes set_disconnected_nodes"
    and remove_child = "l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_remove_child get_child_nodes set_child_nodes
    get_owner_document get_disconnected_nodes set_disconnected_nodes"
    and remove_child_locs = "l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_remove_child_locs set_child_nodes_locs
    set_disconnected_nodes_locs"
  .

interpretation i_remove_child?: l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M Shadow_DOM.get_child_nodes
  Shadow_DOM.get_child_nodes_locs Shadow_DOM.set_child_nodes Shadow_DOM.set_child_nodes_locs
  Shadow_DOM.get_parent Shadow_DOM.get_parent_locs Shadow_DOM.get_owner_document
  get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes
  set_disconnected_nodes_locs remove_child remove_child_locs remove ShadowRootClass.type_wf
  ShadowRootClass.known_ptr ShadowRootClass.known_ptrs
  by(auto simp add: l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def remove_child_def
      remove_child_locs_def remove_def instances)
declare l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms [instances]

subsubsection \<open>get\_disconnected\_document\<close>

locale l_get_disconnected_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs =
  l_get_disconnected_nodes_defs get_disconnected_nodes get_disconnected_nodes_locs
  for get_disconnected_nodes ::
    "(_::linorder) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"
    and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
begin
definition a_get_disconnected_document :: "(_) node_ptr \<Rightarrow> (_, (_) document_ptr) dom_prog"
  where
    "a_get_disconnected_document node_ptr = do {
      check_in_heap (cast node_ptr);
      ptrs \<leftarrow> document_ptr_kinds_M;
      candidates \<leftarrow> filter_M (\<lambda>document_ptr. do {
        disconnected_nodes \<leftarrow> get_disconnected_nodes document_ptr;
        return (node_ptr \<in> set disconnected_nodes)
      }) ptrs;
      (case candidates of
        Cons document_ptr [] \<Rightarrow> return document_ptr |
        _ \<Rightarrow> error HierarchyRequestError
      )
    }"
definition "a_get_disconnected_document_locs =
  (\<Union>document_ptr. get_disconnected_nodes_locs document_ptr) \<union>
  (\<Union>ptr. {preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr RObject.nothing)})"
end

locale l_get_disconnected_document_defs =
  fixes get_disconnected_document :: "(_) node_ptr \<Rightarrow> (_, (_::linorder) document_ptr) dom_prog"
  fixes get_disconnected_document_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"

locale l_get_disconnected_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M =
  l_get_disconnected_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs +
  l_get_disconnected_document_defs +
  l_get_disconnected_nodes +
  assumes get_disconnected_document_impl:
    "get_disconnected_document = a_get_disconnected_document"
  assumes get_disconnected_document_locs_impl:
    "get_disconnected_document_locs = a_get_disconnected_document_locs"
begin
lemmas get_disconnected_document_def =
  get_disconnected_document_impl[unfolded a_get_disconnected_document_def]
lemmas get_disconnected_document_locs_def =
  get_disconnected_document_locs_impl[unfolded a_get_disconnected_document_locs_def]

lemma get_disconnected_document_pure [simp]: "pure (get_disconnected_document ptr) h"
  using get_disconnected_nodes_pure
  by(auto simp add: get_disconnected_document_def
      intro!: bind_pure_I filter_M_pure_I
      split: list.splits)

lemma get_disconnected_document_ptr_in_heap [simp]:
  "h \<turnstile> ok (get_disconnected_document node_ptr) \<Longrightarrow> node_ptr |\<in>| node_ptr_kinds h"
  using get_disconnected_document_def is_OK_returns_result_I check_in_heap_ptr_in_heap
  by (metis (no_types, lifting) bind_returns_heap_E get_disconnected_document_pure
      node_ptr_kinds_commutes pure_pure)

lemma get_disconnected_document_disconnected_document_in_heap:
  assumes "h \<turnstile> get_disconnected_document child_node \<rightarrow>\<^sub>r disconnected_document"
  shows "disconnected_document |\<in>| document_ptr_kinds h"
  using assms get_disconnected_nodes_pure
  by(auto simp add: get_disconnected_document_def elim!: bind_returns_result_E2
      dest!: filter_M_not_more_elements[where x=disconnected_document]
      intro!: filter_M_pure_I bind_pure_I
      split: if_splits list.splits)

lemma get_disconnected_document_reads:
  "reads get_disconnected_document_locs (get_disconnected_document node_ptr) h h'"
  using get_disconnected_nodes_reads[unfolded reads_def]
  by(auto simp add: get_disconnected_document_def get_disconnected_document_locs_def
      intro!: reads_bind_pure reads_subset[OF check_in_heap_reads]
      reads_subset[OF error_reads]
      reads_subset[OF get_disconnected_nodes_reads] reads_subset[OF return_reads]
      reads_subset[OF document_ptr_kinds_M_reads] filter_M_reads filter_M_pure_I
      bind_pure_I
      split: list.splits)

end

locale l_get_disconnected_document = l_get_disconnected_document_defs +
  assumes get_disconnected_document_reads:
    "reads get_disconnected_document_locs (get_disconnected_document node_ptr) h h'"
  assumes get_disconnected_document_ptr_in_heap:
    "h \<turnstile> ok (get_disconnected_document node_ptr) \<Longrightarrow> node_ptr |\<in>| node_ptr_kinds h"
  assumes get_disconnected_document_pure [simp]:
    "pure (get_disconnected_document node_ptr) h"
  assumes get_disconnected_document_disconnected_document_in_heap:
    "h \<turnstile> get_disconnected_document child_node \<rightarrow>\<^sub>r disconnected_document \<Longrightarrow>
    disconnected_document |\<in>| document_ptr_kinds h"

global_interpretation l_get_disconnected_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs get_disconnected_nodes
  get_disconnected_nodes_locs defines
  get_disconnected_document = a_get_disconnected_document and
  get_disconnected_document_locs = a_get_disconnected_document_locs .

interpretation i_get_disconnected_document?: l_get_disconnected_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M
  get_disconnected_nodes get_disconnected_nodes_locs get_disconnected_document
  get_disconnected_document_locs type_wf
  by(auto simp add: l_get_disconnected_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def l_get_disconnected_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def
      get_disconnected_document_def get_disconnected_document_locs_def instances)
declare l_get_disconnected_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms [instances]

lemma get_disconnected_document_is_l_get_disconnected_document [instances]:
  "l_get_disconnected_document get_disconnected_document get_disconnected_document_locs"
  apply(auto simp add: l_get_disconnected_document_def instances)[1]
  using get_disconnected_document_ptr_in_heap get_disconnected_document_pure
    get_disconnected_document_disconnected_document_in_heap get_disconnected_document_reads
  by blast+

paragraph \<open>get\_disconnected\_nodes\<close>

locale l_set_tag_name_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M =
  l_set_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M +
  l_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M
begin
lemma set_tag_name_get_disconnected_nodes:
  "\<forall>w \<in> set_tag_name_locs ptr. (h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r \<in> get_disconnected_nodes_locs ptr'. r h h'))"
  by(auto simp add: CD.set_tag_name_locs_impl[unfolded CD.a_set_tag_name_locs_def]
      CD.get_disconnected_nodes_locs_impl[unfolded CD.a_get_disconnected_nodes_locs_def]
      all_args_def)
end

interpretation
  i_set_tag_name_get_disconnected_nodes?: l_set_tag_name_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf
  DocumentClass.type_wf set_tag_name set_tag_name_locs get_disconnected_nodes
  get_disconnected_nodes_locs
  by unfold_locales
declare l_set_tag_name_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances]

lemma set_tag_name_get_disconnected_nodes_is_l_set_tag_name_get_disconnected_nodes [instances]:
  "l_set_tag_name_get_disconnected_nodes type_wf set_tag_name set_tag_name_locs
    get_disconnected_nodes get_disconnected_nodes_locs"
  using set_tag_name_is_l_set_tag_name get_disconnected_nodes_is_l_get_disconnected_nodes
  apply(simp add: l_set_tag_name_get_disconnected_nodes_def
      l_set_tag_name_get_disconnected_nodes_axioms_def)
  using set_tag_name_get_disconnected_nodes
  by fast

subsubsection \<open>adopt\_node\<close>

global_interpretation l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs get_owner_document get_parent get_parent_locs
  remove_child remove_child_locs get_disconnected_nodes get_disconnected_nodes_locs
  set_disconnected_nodes set_disconnected_nodes_locs
  defines adopt_node = a_adopt_node
    and adopt_node_locs = a_adopt_node_locs
  .
interpretation i_adopt_node?: l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_owner_document get_parent get_parent_locs
  remove_child remove_child_locs get_disconnected_nodes get_disconnected_nodes_locs
  set_disconnected_nodes set_disconnected_nodes_locs adopt_node adopt_node_locs known_ptr type_wf
  get_child_nodes get_child_nodes_locs known_ptrs set_child_nodes set_child_nodes_locs remove
  by(auto simp add: l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def adopt_node_def
      adopt_node_locs_def instances)
declare l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances]

lemma adopt_node_is_l_adopt_node [instances]: "l_adopt_node type_wf known_ptr known_ptrs get_parent
  adopt_node adopt_node_locs get_child_nodes get_owner_document"
  apply(auto simp add: l_adopt_node_def l_adopt_node_axioms_def instances)[1]
  using adopt_node_writes apply fast
  using adopt_node_pointers_preserved apply (fast, fast)
  using adopt_node_types_preserved apply (fast, fast)
  using adopt_node_child_in_heap apply fast
  using adopt_node_children_subset apply fast
  done

paragraph \<open>get\_shadow\_root\<close>

locale l_adopt_node_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M =
  l_set_child_nodes_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M +
  l_set_disconnected_nodes_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M +
  l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M
begin
lemma adopt_node_get_shadow_root:
  "\<forall>w \<in> adopt_node_locs parent owner_document document_ptr.
    (h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r \<in> get_shadow_root_locs ptr'. r h h'))"
  by(auto simp add: adopt_node_locs_def remove_child_locs_def all_args_def
      set_disconnected_nodes_get_shadow_root set_child_nodes_get_shadow_root)
end

locale l_adopt_node_get_shadow_root = l_adopt_node_defs + l_get_shadow_root_defs +
  assumes adopt_node_get_shadow_root:
    "\<forall>w \<in> adopt_node_locs parent owner_document document_ptr.
    (h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r \<in> get_shadow_root_locs ptr'. r h h'))"

interpretation i_adopt_node_get_shadow_root?: l_adopt_node_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr
  DocumentClass.type_wf DocumentClass.known_ptr set_child_nodes set_child_nodes_locs
  Core_DOM_Functions.set_child_nodes Core_DOM_Functions.set_child_nodes_locs get_shadow_root
  get_shadow_root_locs set_disconnected_nodes set_disconnected_nodes_locs get_owner_document
  get_parent get_parent_locs remove_child remove_child_locs get_disconnected_nodes
  get_disconnected_nodes_locs adopt_node adopt_node_locs get_child_nodes get_child_nodes_locs
  known_ptrs remove
  by(auto simp add: l_adopt_node_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def instances)
declare l_adopt_node_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances]

lemma adopt_node_get_shadow_root_is_l_adopt_node_get_shadow_root [instances]:
  "l_adopt_node_get_shadow_root adopt_node_locs get_shadow_root_locs"
  apply(auto simp add: l_adopt_node_get_shadow_root_def)[1]
  using adopt_node_get_shadow_root apply fast
  done

subsubsection \<open>insert\_before\<close>

global_interpretation l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs get_parent get_parent_locs get_child_nodes
  get_child_nodes_locs set_child_nodes set_child_nodes_locs get_ancestors_si get_ancestors_si_locs
  adopt_node adopt_node_locs set_disconnected_nodes set_disconnected_nodes_locs
  get_disconnected_nodes get_disconnected_nodes_locs get_owner_document
  defines
    next_sibling = a_next_sibling and
    insert_node = a_insert_node and
    ensure_pre_insertion_validity = a_ensure_pre_insertion_validity and
    insert_before = a_insert_before and
    insert_before_locs = a_insert_before_locs
  .
global_interpretation l_append_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs insert_before
  defines append_child = "l_append_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_append_child insert_before"
  .

interpretation i_insert_before?: l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_parent get_parent_locs get_child_nodes
  get_child_nodes_locs set_child_nodes set_child_nodes_locs get_ancestors_si get_ancestors_si_locs
  adopt_node adopt_node_locs set_disconnected_nodes set_disconnected_nodes_locs get_disconnected_nodes
  get_disconnected_nodes_locs get_owner_document insert_before insert_before_locs append_child type_wf
  known_ptr known_ptrs
  by(auto simp add: l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def insert_before_def
      insert_before_locs_def instances)
declare l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms [instances]

interpretation i_append_child?: l_append_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M append_child insert_before insert_before_locs
  by(simp add: l_append_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def instances append_child_def)
declare l_append_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances]

subsubsection \<open>get\_assigned\_nodes\<close>

fun map_filter_M2 :: "('x \<Rightarrow> ('heap, 'e, 'y option) prog) \<Rightarrow> 'x list
  \<Rightarrow> ('heap, 'e, 'y list) prog"
  where
    "map_filter_M2 f [] = return []" |
    "map_filter_M2 f (x # xs) = do {
      res \<leftarrow> f x;
      remainder \<leftarrow> map_filter_M2 f xs;
      return ((case res of  Some r \<Rightarrow> [r] | None \<Rightarrow> []) @ remainder)
    }"
lemma map_filter_M2_pure [simp]:
  assumes "\<And>x. x \<in> set xs \<Longrightarrow> pure (f x) h"
  shows "pure (map_filter_M2 f xs) h"
  using assms
  apply(induct xs arbitrary: h)
  by(auto elim!: bind_returns_result_E2 intro!: bind_pure_I)

lemma map_filter_pure_no_monad:
  assumes "\<And>x. x \<in> set xs \<Longrightarrow> pure (f x) h"
  assumes "h \<turnstile> map_filter_M2 f xs \<rightarrow>\<^sub>r ys"
  shows
    "ys = map the (filter (\<lambda>x. x \<noteq> None) (map (\<lambda>x. |h \<turnstile> f x|\<^sub>r) xs))" and
    "\<And>x. x \<in> set xs \<Longrightarrow> h \<turnstile> ok (f x)"
  using assms
   apply(induct xs arbitrary: h ys)
  by(auto elim!: bind_returns_result_E2)

lemma map_filter_pure_foo:
  assumes "\<And>x. x \<in> set xs \<Longrightarrow> pure (f x) h"
  assumes "h \<turnstile> map_filter_M2 f xs \<rightarrow>\<^sub>r ys"
  assumes "y \<in> set ys"
  obtains x where "h \<turnstile> f x \<rightarrow>\<^sub>r Some y" and "x \<in> set xs"
  using assms
  apply(induct xs arbitrary: ys)
  by(auto elim!: bind_returns_result_E2)

lemma map_filter_M2_in_result:
  assumes "h \<turnstile> map_filter_M2 P xs \<rightarrow>\<^sub>r ys"
  assumes "a \<in> set xs"
  assumes "\<And>x. x \<in> set xs \<Longrightarrow> pure (P x) h"
  assumes "h \<turnstile> P a \<rightarrow>\<^sub>r Some b"
  shows "b \<in> set ys"
  using assms
  apply(induct xs arbitrary: h ys)
  by(auto elim!: bind_returns_result_E2 )

locale l_assigned_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs =
  l_get_tag_name_defs get_tag_name get_tag_name_locs +
  l_get_root_node_defs get_root_node get_root_node_locs +
  l_get_host_defs get_host get_host_locs +
  l_get_child_nodes_defs get_child_nodes get_child_nodes_locs +
  l_find_slot_defs find_slot assigned_slot +
  l_remove_defs remove +
  l_insert_before_defs insert_before insert_before_locs +
  l_append_child_defs append_child +
  l_remove_shadow_root_defs remove_shadow_root remove_shadow_root_locs
  for get_child_nodes :: "(_::linorder) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"
    and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"
    and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"
    and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"
    and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and find_slot :: "bool \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"
    and assigned_slot :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"
    and remove :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"
    and insert_before ::
    "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> (_) node_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"
    and insert_before_locs ::  "(_) object_ptr \<Rightarrow> (_) object_ptr option \<Rightarrow> (_) document_ptr \<Rightarrow>
    (_) document_ptr \<Rightarrow> (_, unit) dom_prog set"
    and append_child :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"
    and remove_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog"
    and remove_shadow_root_locs :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr \<Rightarrow>
    ((_) heap, exception, unit) prog set"
begin
definition a_assigned_nodes :: "(_) element_ptr \<Rightarrow> (_, (_) node_ptr list) dom_prog"
  where
    "a_assigned_nodes slot = do {
      tag \<leftarrow> get_tag_name slot;
      (if tag \<noteq> ''slot''
      then error HierarchyRequestError
      else return ());
      root \<leftarrow> get_root_node (cast slot);
      if is_shadow_root_ptr_kind root
      then do {
        host \<leftarrow> get_host (the (cast root));
        children \<leftarrow> get_child_nodes (cast host);
        filter_M (\<lambda>slotable. do {
          found_slot \<leftarrow> find_slot False slotable;
          return (found_slot = Some slot)}) children}
      else return []}"

partial_function (dom_prog) a_assigned_nodes_flatten ::
  "(_) element_ptr \<Rightarrow> (_, (_) node_ptr list) dom_prog"
  where
    "a_assigned_nodes_flatten slot = do {
      tag \<leftarrow> get_tag_name slot;
      (if tag \<noteq> ''slot''
      then error HierarchyRequestError
      else return ());
      root \<leftarrow> get_root_node (cast slot);
      (if is_shadow_root_ptr_kind root
      then do {
        slotables \<leftarrow> a_assigned_nodes slot;
        slotables_or_child_nodes \<leftarrow> (if slotables = []
        then do {
          get_child_nodes (cast slot)
        } else do {
          return slotables
        });
        list_of_lists \<leftarrow> map_M (\<lambda>node_ptr. do {
          (case cast node_ptr of
            Some element_ptr \<Rightarrow> do {
              tag \<leftarrow> get_tag_name element_ptr;
              (if tag = ''slot''
              then do {
                root \<leftarrow> get_root_node (cast element_ptr);
                (if is_shadow_root_ptr_kind root
                then do {
                  a_assigned_nodes_flatten element_ptr
                } else do {
                  return [node_ptr]
                })
              } else do {
                return [node_ptr]
              })
            }
            | None \<Rightarrow> return [node_ptr])
        }) slotables_or_child_nodes;
        return (concat list_of_lists)
      } else return [])
    }"

definition a_flatten_dom :: "(_, unit) dom_prog" where
  "a_flatten_dom = do {
    tups \<leftarrow> element_ptr_kinds_M \<bind> map_filter_M2 (\<lambda>element_ptr. do {
      tag \<leftarrow> get_tag_name element_ptr;
      assigned_nodes \<leftarrow> a_assigned_nodes element_ptr;
      (if tag = ''slot'' \<and> assigned_nodes \<noteq> []
       then return (Some (element_ptr, assigned_nodes))
       else return None)});
    forall_M (\<lambda>(slot, assigned_nodes). do {
      get_child_nodes (cast slot) \<bind> forall_M remove;
      forall_M (append_child (cast slot)) assigned_nodes
    }) tups;
    shadow_root_ptr_kinds_M \<bind> forall_M (\<lambda>shadow_root_ptr. do {
      host \<leftarrow> get_host shadow_root_ptr;
      get_child_nodes (cast host) \<bind> forall_M remove;
      get_child_nodes (cast shadow_root_ptr) \<bind> forall_M (append_child (cast host));
      remove_shadow_root host
    });
    return ()
  }"
end

global_interpretation l_assigned_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs get_child_nodes get_child_nodes_locs
  get_tag_name get_tag_name_locs get_root_node get_root_node_locs get_host get_host_locs find_slot
  assigned_slot remove insert_before insert_before_locs append_child remove_shadow_root
  remove_shadow_root_locs
  defines assigned_nodes = a_assigned_nodes
    and assigned_nodes_flatten = a_assigned_nodes_flatten
    and flatten_dom = a_flatten_dom
  .

declare a_assigned_nodes_flatten.simps [code]

locale l_assigned_nodes_defs =
  fixes assigned_nodes :: "(_) element_ptr \<Rightarrow> (_, (_) node_ptr list) dom_prog"
  fixes assigned_nodes_flatten :: "(_) element_ptr \<Rightarrow> (_, (_) node_ptr list) dom_prog"
  fixes flatten_dom :: "(_, unit) dom_prog"

locale l_assigned_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M =
  l_assigned_nodes_defs
  assigned_nodes assigned_nodes_flatten flatten_dom
  + l_assigned_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs
  get_child_nodes get_child_nodes_locs get_tag_name get_tag_name_locs get_root_node
  get_root_node_locs get_host get_host_locs find_slot assigned_slot remove insert_before
  insert_before_locs append_child remove_shadow_root remove_shadow_root_locs
  + l_get_shadow_root
  type_wf get_shadow_root get_shadow_root_locs
  + l_set_shadow_root
  type_wf set_shadow_root set_shadow_root_locs
  + l_remove
  + l_insert_before
  insert_before insert_before_locs
  + l_find_slot
  find_slot assigned_slot
  + l_get_tag_name
  type_wf get_tag_name get_tag_name_locs
  + l_get_root_node
  get_root_node get_root_node_locs get_parent get_parent_locs
  + l_get_host
  get_host get_host_locs
  + l_get_child_nodes
  type_wf known_ptr get_child_nodes get_child_nodes_locs
  + l_to_tree_order
  to_tree_order
  for known_ptr :: "(_::linorder) object_ptr \<Rightarrow> bool"
    and assigned_nodes :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"
    and assigned_nodes_flatten :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"
    and flatten_dom :: "((_) heap, exception, unit) prog"
    and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"
    and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"
    and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"
    and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"
    and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and find_slot :: "bool \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"
    and assigned_slot :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"
    and remove :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"
    and insert_before ::
    "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> (_) node_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"
    and insert_before_locs ::  "(_) object_ptr \<Rightarrow> (_) object_ptr option \<Rightarrow> (_) document_ptr \<Rightarrow>
    (_) document_ptr \<Rightarrow> (_, unit) dom_prog set"
    and append_child :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"
    and remove_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog"
    and remove_shadow_root_locs :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr \<Rightarrow>
    ((_) heap, exception, unit) prog set"
    and type_wf :: "(_) heap \<Rightarrow> bool"
    and get_shadow_root ::
    "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"
    and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and set_shadow_root ::
    "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"
    and set_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"
    and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"
    and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog" +
  assumes assigned_nodes_impl: "assigned_nodes = a_assigned_nodes"
  assumes flatten_dom_impl: "flatten_dom = a_flatten_dom"
begin
lemmas assigned_nodes_def = assigned_nodes_impl[unfolded a_assigned_nodes_def]
lemmas flatten_dom_def = flatten_dom_impl[unfolded a_flatten_dom_def, folded assigned_nodes_impl]

lemma assigned_nodes_pure [simp]: "pure (assigned_nodes slot) h"
  by(auto simp add: assigned_nodes_def intro!: bind_pure_I filter_M_pure_I)

lemma assigned_nodes_ptr_in_heap:
  assumes "h \<turnstile> ok (assigned_nodes slot)"
  shows "slot |\<in>| element_ptr_kinds h"
  using assms
  apply(auto simp add: assigned_nodes_def)[1]
  by (meson bind_is_OK_E is_OK_returns_result_I local.get_tag_name_ptr_in_heap)

lemma assigned_nodes_slot_is_slot:
  assumes "h \<turnstile> ok (assigned_nodes slot)"
  shows "h \<turnstile> get_tag_name slot \<rightarrow>\<^sub>r ''slot''"
  using assms
  by(auto simp add: assigned_nodes_def elim!: bind_is_OK_E split: if_splits)

lemma assigned_nodes_different_ptr:
  assumes "h \<turnstile> assigned_nodes slot \<rightarrow>\<^sub>r nodes"
  assumes "h \<turnstile> assigned_nodes slot' \<rightarrow>\<^sub>r nodes'"
  assumes "slot \<noteq> slot'"
  shows "set nodes \<inter> set nodes' = {}"
proof (rule ccontr)
  assume "set nodes \<inter> set nodes' \<noteq> {} "
  then obtain common_ptr where "common_ptr \<in> set nodes" and "common_ptr \<in> set nodes'"
    by auto

  have "h \<turnstile> find_slot False common_ptr \<rightarrow>\<^sub>r Some slot"
    using \<open>common_ptr \<in> set nodes\<close>
    using assms(1)
    by(auto simp add: assigned_nodes_def
        elim!: bind_returns_result_E2
        split: if_splits
        dest!: filter_M_holds_for_result[where x=common_ptr]
        intro!: bind_pure_I)
  moreover
  have "h \<turnstile> find_slot False common_ptr \<rightarrow>\<^sub>r Some slot'"
    using \<open>common_ptr \<in> set nodes'\<close>
    using assms(2)
    by(auto simp add: assigned_nodes_def
        elim!: bind_returns_result_E2
        split: if_splits
        dest!: filter_M_holds_for_result[where x=common_ptr]
        intro!: bind_pure_I)
  ultimately
  show False
    using assms(3)
    by (meson option.inject returns_result_eq)
qed
end

interpretation i_assigned_nodes?: l_assigned_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr assigned_nodes
  assigned_nodes_flatten flatten_dom get_child_nodes get_child_nodes_locs get_tag_name
  get_tag_name_locs get_root_node get_root_node_locs get_host get_host_locs find_slot
  assigned_slot remove insert_before insert_before_locs append_child remove_shadow_root
  remove_shadow_root_locs type_wf get_shadow_root get_shadow_root_locs set_shadow_root
  set_shadow_root_locs get_parent get_parent_locs to_tree_order
  by(auto simp add: instances l_assigned_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def l_assigned_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def
      assigned_nodes_def flatten_dom_def)
declare l_assigned_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms [instances]

locale l_assigned_nodes = l_assigned_nodes_defs +
  assumes assigned_nodes_pure [simp]: "pure (assigned_nodes slot) h"
  assumes assigned_nodes_ptr_in_heap:
    "h \<turnstile> ok (assigned_nodes slot) \<Longrightarrow> slot |\<in>| element_ptr_kinds h"
  assumes assigned_nodes_slot_is_slot:
    "h \<turnstile> ok (assigned_nodes slot) \<Longrightarrow> h \<turnstile> get_tag_name slot \<rightarrow>\<^sub>r ''slot''"
  assumes assigned_nodes_different_ptr:
    "h \<turnstile> assigned_nodes slot \<rightarrow>\<^sub>r nodes \<Longrightarrow> h \<turnstile> assigned_nodes slot' \<rightarrow>\<^sub>r nodes' \<Longrightarrow>
  slot \<noteq> slot' \<Longrightarrow> set nodes \<inter> set nodes' = {}"

lemma assigned_nodes_is_l_assigned_nodes [instances]: "l_assigned_nodes assigned_nodes"
  apply(auto simp add: l_assigned_nodes_def)[1]
  using assigned_nodes_ptr_in_heap apply fast
  using assigned_nodes_slot_is_slot apply fast
  using assigned_nodes_different_ptr apply fast
  done

subsubsection \<open>set\_val\<close>

locale l_set_val\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M =
  CD: l_set_val\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_val set_val_locs +
  l_type_wf type_wf
  for type_wf :: "(_) heap \<Rightarrow> bool"
    and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"
    and set_val :: "(_) character_data_ptr \<Rightarrow> char list \<Rightarrow> (_, unit) dom_prog"
    and set_val_locs :: "(_) character_data_ptr \<Rightarrow> (_, unit) dom_prog set"  +
  assumes type_wf_impl: "type_wf = ShadowRootClass.type_wf"
begin

lemma set_val_ok:
  "type_wf h \<Longrightarrow> character_data_ptr |\<in>| character_data_ptr_kinds h \<Longrightarrow>
  h \<turnstile> ok (set_val character_data_ptr tag)"
  using CD.set_val_ok CD.type_wf_impl ShadowRootClass.type_wf\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t local.type_wf_impl by blast

lemma set_val_writes:
  "writes (set_val_locs character_data_ptr) (set_val character_data_ptr tag) h h'"
  using CD.set_val_writes .

lemma set_val_pointers_preserved:
  assumes "w \<in> set_val_locs character_data_ptr"
  assumes "h \<turnstile> w \<rightarrow>\<^sub>h h'"
  shows "object_ptr_kinds h = object_ptr_kinds h'"
  using assms CD.set_val_pointers_preserved by simp

lemma set_val_typess_preserved:
  assumes "w \<in> set_val_locs character_data_ptr"
  assumes "h \<turnstile> w \<rightarrow>\<^sub>h h'"
  shows "type_wf h = type_wf h'"
  apply(unfold type_wf_impl)
  using assms(1) type_wf_preserved[OF writes_singleton2 assms(2)]
  by(auto simp add: all_args_def CD.set_val_locs_impl[unfolded CD.a_set_val_locs_def]
      split: if_splits)
end

interpretation
  i_set_val?: l_set_val\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf DocumentClass.type_wf set_val set_val_locs
  apply(unfold_locales)
  by (auto simp add: set_val_def set_val_locs_def)
declare l_set_val\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances]

lemma set_val_is_l_set_val [instances]: "l_set_val type_wf set_val set_val_locs"
  apply(simp add: l_set_val_def)
  using set_val_ok set_val_writes set_val_pointers_preserved set_val_typess_preserved
  by blast

paragraph \<open>get\_shadow\_root\<close>

locale l_set_val_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M =
  l_set_val\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M +
  l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M
begin
lemma set_val_get_shadow_root:
  "\<forall>w \<in> set_val_locs ptr. (h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r \<in> get_shadow_root_locs ptr'. r h h'))"
  by(auto simp add: CD.set_val_locs_impl[unfolded CD.a_set_val_locs_def]
      get_shadow_root_locs_def all_args_def)
end

locale l_set_val_get_shadow_root = l_set_val + l_get_shadow_root +
  assumes set_val_get_shadow_root:
    "\<forall>w \<in> set_val_locs ptr. (h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r \<in> get_shadow_root_locs ptr'. r h h'))"

interpretation
  i_set_val_get_shadow_root?: l_set_val_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf DocumentClass.type_wf
  set_val set_val_locs
  get_shadow_root get_shadow_root_locs
  apply(auto simp add: l_set_val_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def  instances)[1]
  using l_set_val\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms
  by unfold_locales
declare l_set_val_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances]

lemma set_val_get_shadow_root_is_l_set_val_get_shadow_root [instances]:
  "l_set_val_get_shadow_root type_wf set_val set_val_locs get_shadow_root
                                  get_shadow_root_locs"
  using set_val_is_l_set_val get_shadow_root_is_l_get_shadow_root
  apply(simp add: l_set_val_get_shadow_root_def l_set_val_get_shadow_root_axioms_def)
  using set_val_get_shadow_root
  by fast

paragraph \<open>get\_tag\_type\<close>

locale l_set_val_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M =
  l_set_val\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M +
  l_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M
begin
lemma set_val_get_tag_name:
  "\<forall>w \<in> set_val_locs ptr. (h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r \<in> get_tag_name_locs ptr'. r h h'))"
  by(auto simp add: CD.set_val_locs_impl[unfolded CD.a_set_val_locs_def]
      CD.get_tag_name_locs_impl[unfolded CD.a_get_tag_name_locs_def]
      all_args_def)
end

locale l_set_val_get_tag_name = l_set_val + l_get_tag_name +
  assumes set_val_get_tag_name:
    "\<forall>w \<in> set_val_locs ptr. (h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r \<in> get_tag_name_locs ptr'. r h h'))"

interpretation
  i_set_val_get_tag_name?: l_set_val_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf DocumentClass.type_wf set_val
  set_val_locs get_tag_name get_tag_name_locs
  by unfold_locales
declare l_set_val_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances]

lemma set_val_get_tag_name_is_l_set_val_get_tag_name [instances]:
  "l_set_val_get_tag_name type_wf set_val set_val_locs get_tag_name get_tag_name_locs"
  using set_val_is_l_set_val get_tag_name_is_l_get_tag_name
  apply(simp add: l_set_val_get_tag_name_def l_set_val_get_tag_name_axioms_def)
  using set_val_get_tag_name
  by fast

subsubsection \<open>create\_character\_data\<close>

locale l_create_character_data\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M =
  CD: l_create_character_data\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M _ _ _ _ _ _ type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M _ known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M +
  l_known_ptr known_ptr
  for known_ptr :: "(_) object_ptr \<Rightarrow> bool"
    and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"
    and known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> bool" +
  assumes known_ptr_impl: "known_ptr = a_known_ptr"
begin

lemma create_character_data_document_in_heap:
  assumes "h \<turnstile> ok (create_character_data document_ptr text)"
  shows "document_ptr |\<in>| document_ptr_kinds h"
  using assms CD.create_character_data_document_in_heap by simp

lemma create_character_data_known_ptr:
  assumes "h \<turnstile> create_character_data document_ptr text \<rightarrow>\<^sub>r new_character_data_ptr"
  shows "known_ptr (cast new_character_data_ptr)"
  using assms CD.create_character_data_known_ptr
  by(simp add: known_ptr_impl CD.known_ptr_impl ShadowRootClass.a_known_ptr_def)
end

locale l_create_character_data = l_create_character_data_defs

interpretation i_create_character_data?: l_create_character_data\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes
  get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs set_val set_val_locs
  create_character_data known_ptr DocumentClass.type_wf DocumentClass.known_ptr
  by(auto simp add: l_create_character_data\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def l_create_character_data\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def
      instances)
declare l_create_character_data\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms [instances]

subsubsection \<open>create\_element\<close>

locale l_create_element\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M =
  CD: l_create_element\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes
  set_disconnected_nodes_locs set_tag_name set_tag_name_locs type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M create_element
  known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M +
  l_known_ptr known_ptr
  for get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"
    and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and set_disconnected_nodes ::
    "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"
    and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"
    and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"
    and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"
    and type_wf :: "(_) heap \<Rightarrow> bool"
    and create_element ::
    "(_) document_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"
    and known_ptr :: "(_) object_ptr \<Rightarrow> bool"
    and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"
    and known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> bool" +
  assumes known_ptr_impl: "known_ptr = a_known_ptr"
begin
lemmas create_element_def = CD.create_element_def

lemma create_element_document_in_heap:
  assumes "h \<turnstile> ok (create_element document_ptr tag)"
  shows "document_ptr |\<in>| document_ptr_kinds h"
  using CD.create_element_document_in_heap assms .

lemma create_element_known_ptr:
  assumes "h \<turnstile> create_element document_ptr tag \<rightarrow>\<^sub>r new_element_ptr"
  shows "known_ptr (cast new_element_ptr)"
proof -
  have "is_element_ptr new_element_ptr"
    using assms
    apply(auto simp add: create_element_def elim!: bind_returns_result_E)[1]
    using new_element_is_element_ptr
    by blast
  then show ?thesis
    by(auto simp add: known_ptr_impl known_ptr_defs DocumentClass.known_ptr_defs
        CharacterDataClass.known_ptr_defs ElementClass.known_ptr_defs)
qed
end

interpretation i_create_element?: l_create_element\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>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 DocumentClass.type_wf DocumentClass.known_ptr
  by(auto simp add: l_create_element\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def l_create_element\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def instances)
declare l_create_element\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances]

subsection \<open>A wellformed heap (Core DOM)\<close>

subsubsection \<open>wellformed\_heap\<close>

locale l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs =
  CD: l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs get_child_nodes get_child_nodes_locs get_disconnected_nodes
  get_disconnected_nodes_locs +
  l_get_shadow_root_defs get_shadow_root get_shadow_root_locs +
  l_get_tag_name_defs get_tag_name get_tag_name_locs
  for get_child_nodes :: "(_::linorder) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"
    and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"
    and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"
    and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"
    and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
begin
definition a_host_shadow_root_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"
  where
    "a_host_shadow_root_rel h = (\<lambda>(x, y). (cast x, cast y)) ` {(host, shadow_root).
      host |\<in>| element_ptr_kinds h \<and> |h \<turnstile> get_shadow_root host|\<^sub>r = Some shadow_root}"

lemma a_host_shadow_root_rel_code [code]: "a_host_shadow_root_rel h = set (concat (map
  (\<lambda>host. (case |h \<turnstile> get_shadow_root host|\<^sub>r of
    Some shadow_root \<Rightarrow> [(cast host, cast shadow_root)] |
    None \<Rightarrow> []))
  (sorted_list_of_fset (element_ptr_kinds h)))
)"
  by(auto simp add: a_host_shadow_root_rel_def)

definition a_all_ptrs_in_heap :: "(_) heap \<Rightarrow> bool" where
  "a_all_ptrs_in_heap h = ((\<forall>host shadow_root_ptr.
    (h \<turnstile> get_shadow_root host \<rightarrow>\<^sub>r Some shadow_root_ptr) \<longrightarrow>
      shadow_root_ptr |\<in>| shadow_root_ptr_kinds h))"

definition a_distinct_lists :: "(_) heap \<Rightarrow> bool"
  where
    "a_distinct_lists h = distinct (concat (
      map (\<lambda>element_ptr. (case |h \<turnstile> get_shadow_root element_ptr|\<^sub>r of
        Some shadow_root_ptr \<Rightarrow> [shadow_root_ptr] | None \<Rightarrow> []))
        |h \<turnstile> element_ptr_kinds_M|\<^sub>r
    ))"

definition a_shadow_root_valid :: "(_) heap \<Rightarrow> bool" where
  "a_shadow_root_valid h = (\<forall>shadow_root_ptr \<in> fset (shadow_root_ptr_kinds h).
     (\<exists>host \<in> fset(element_ptr_kinds h).
       |h \<turnstile> get_tag_name host|\<^sub>r \<in> safe_shadow_root_element_types \<and>
       |h \<turnstile> get_shadow_root host|\<^sub>r = Some shadow_root_ptr))"

definition a_heap_is_wellformed :: "(_) heap \<Rightarrow> bool"
  where
    "a_heap_is_wellformed h \<longleftrightarrow> CD.a_heap_is_wellformed h \<and>
      acyclic (CD.a_parent_child_rel h \<union> a_host_shadow_root_rel h) \<and>
      a_all_ptrs_in_heap h \<and>
      a_distinct_lists h \<and>
      a_shadow_root_valid h"
end

global_interpretation l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs get_child_nodes get_child_nodes_locs
  get_disconnected_nodes get_disconnected_nodes_locs get_shadow_root get_shadow_root_locs
  get_tag_name get_tag_name_locs
  defines heap_is_wellformed = a_heap_is_wellformed
    and parent_child_rel = CD.a_parent_child_rel
    and host_shadow_root_rel = a_host_shadow_root_rel
    and all_ptrs_in_heap = a_all_ptrs_in_heap
    and distinct_lists = a_distinct_lists
    and shadow_root_valid = a_shadow_root_valid
    and heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = CD.a_heap_is_wellformed
    and parent_child_rel\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = CD.a_parent_child_rel
    and acyclic_heap\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = CD.a_acyclic_heap
    and all_ptrs_in_heap\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = CD.a_all_ptrs_in_heap
    and distinct_lists\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = CD.a_distinct_lists
    and owner_document_valid\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = CD.a_owner_document_valid
  .

interpretation i_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M: l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf
  get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs
  heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M parent_child_rel
  by (auto simp add: l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def
      heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def parent_child_rel_def instances)
declare i_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances]

lemma heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_is_l_heap_is_wellformed [instances]:
  "l_heap_is_wellformed type_wf known_ptr heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M parent_child_rel get_child_nodes
  get_disconnected_nodes"
  apply(auto simp add: l_heap_is_wellformed_def)[1]
  using i_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.heap_is_wellformed_children_in_heap apply blast
  using i_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.heap_is_wellformed_disc_nodes_in_heap apply blast
  using i_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.heap_is_wellformed_one_parent apply blast
  using i_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.heap_is_wellformed_one_disc_parent apply blast
  using i_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.heap_is_wellformed_children_disc_nodes_different apply blast
  using i_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.heap_is_wellformed_disconnected_nodes_distinct apply blast
  using i_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.heap_is_wellformed_children_distinct apply blast
  using i_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.heap_is_wellformed_children_disc_nodes apply blast
  using i_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.parent_child_rel_child apply (blast, blast)
  using i_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.parent_child_rel_finite apply blast
  using i_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.parent_child_rel_acyclic apply blast
  using i_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.parent_child_rel_node_ptr apply blast
  using i_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.parent_child_rel_parent_in_heap apply blast
  using i_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.parent_child_rel_child_in_heap apply blast
  done

locale l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M =
  l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs
  get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs
  get_shadow_root get_shadow_root_locs get_tag_name get_tag_name_locs
  + CD: l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M
  known_ptr type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes
  get_disconnected_nodes_locs heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M parent_child_rel
  + l_heap_is_wellformed_defs
  heap_is_wellformed parent_child_rel
  + l_get_host\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M
  get_shadow_root get_shadow_root_locs get_host get_host_locs type_wf
  + l_get_disconnected_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs
  get_disconnected_document get_disconnected_document_locs type_wf
  + l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs
  for get_child_nodes :: "(_::linorder) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"
    and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"
    and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"
    and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"
    and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and known_ptr :: "(_) object_ptr \<Rightarrow> bool"
    and type_wf :: "(_) heap \<Rightarrow> bool"
    and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"
    and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"
    and heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"
    and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"
    and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and get_disconnected_document :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"
    and get_disconnected_document_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" +
  assumes heap_is_wellformed_impl: "heap_is_wellformed = a_heap_is_wellformed"
begin
lemmas heap_is_wellformed_def =
  heap_is_wellformed_impl[unfolded a_heap_is_wellformed_def,
    folded CD.heap_is_wellformed_impl CD.parent_child_rel_impl]

lemma a_distinct_lists_code [code]:
  "a_all_ptrs_in_heap h = ((\<forall>host \<in> fset (element_ptr_kinds h).
    h \<turnstile> ok (get_shadow_root host) \<longrightarrow> (case |h \<turnstile> get_shadow_root host|\<^sub>r of
    Some shadow_root_ptr \<Rightarrow> shadow_root_ptr |\<in>| shadow_root_ptr_kinds h |
    None \<Rightarrow> True)))"
  apply(auto simp add: a_all_ptrs_in_heap_def split: option.splits)[1]
  by (meson is_OK_returns_result_I local.get_shadow_root_ptr_in_heap select_result_I2)

lemma get_shadow_root_shadow_root_ptr_in_heap:
  assumes "heap_is_wellformed h"
  assumes "h \<turnstile> get_shadow_root host \<rightarrow>\<^sub>r Some shadow_root_ptr"
  shows "shadow_root_ptr |\<in>| shadow_root_ptr_kinds h"
  using assms
  by(auto simp add: heap_is_wellformed_def a_all_ptrs_in_heap_def)

lemma get_host_ptr_in_heap:
  assumes "heap_is_wellformed h"
  assumes "h \<turnstile> get_host shadow_root_ptr \<rightarrow>\<^sub>r host"
  shows "shadow_root_ptr |\<in>| shadow_root_ptr_kinds h"
  using assms get_shadow_root_shadow_root_ptr_in_heap
  by(auto simp add: get_host_def
      elim!: bind_returns_result_E2
      dest!: filter_M_holds_for_result
      intro!: bind_pure_I
      split: list.splits)

lemma shadow_root_same_host:
  assumes "heap_is_wellformed h" and "type_wf h"
  assumes "h \<turnstile> get_shadow_root host \<rightarrow>\<^sub>r Some shadow_root_ptr"
  assumes "h \<turnstile> get_shadow_root host' \<rightarrow>\<^sub>r Some shadow_root_ptr"
  shows "host = host'"
proof (rule ccontr)
  assume " host \<noteq> host'"
  have "host |\<in>| element_ptr_kinds h"
    using assms(3)
    by (meson is_OK_returns_result_I local.get_shadow_root_ptr_in_heap)
  moreover
  have "host' |\<in>| element_ptr_kinds h"
    using assms(4)
    by (meson is_OK_returns_result_I local.get_shadow_root_ptr_in_heap)
  ultimately show False
    using assms
    apply(auto simp add: heap_is_wellformed_def a_distinct_lists_def)[1]
    apply(drule distinct_concat_map_E(1)[where x=host and y=host'])
       apply(simp)
      apply(simp)
    using \<open>host \<noteq> host'\<close> apply(simp)
    apply(auto)[1]
    done
qed

lemma shadow_root_host_dual:
  assumes "h \<turnstile> get_host shadow_root_ptr \<rightarrow>\<^sub>r host"
  shows "h \<turnstile> get_shadow_root host \<rightarrow>\<^sub>r Some shadow_root_ptr"
  using assms
  by(auto simp add: get_host_def
      dest: filter_M_holds_for_result
      elim!: bind_returns_result_E2
      intro!: bind_pure_I split: list.splits)

lemma disc_doc_disc_node_dual:
  assumes "h \<turnstile> get_disconnected_document disc_node \<rightarrow>\<^sub>r disc_doc"
  obtains disc_nodes where
    "h \<turnstile> get_disconnected_nodes disc_doc \<rightarrow>\<^sub>r disc_nodes" and
    "disc_node \<in> set disc_nodes"
  using assms get_disconnected_nodes_pure
  by(auto simp add: get_disconnected_document_def bind_pure_I
      dest!: filter_M_holds_for_result
      elim!: bind_returns_result_E2
      intro!: filter_M_pure_I
      split: if_splits list.splits)

lemma get_host_valid_tag_name:
  assumes "heap_is_wellformed h" and "type_wf h"
  assumes "h \<turnstile> get_host shadow_root_ptr \<rightarrow>\<^sub>r host"
  assumes "h \<turnstile> get_tag_name host \<rightarrow>\<^sub>r tag"
  shows "tag \<in> safe_shadow_root_element_types"
proof -
  obtain host' where
    "host' |\<in>| element_ptr_kinds h" and
    "|h \<turnstile> get_tag_name host'|\<^sub>r \<in> safe_shadow_root_element_types"
    and "h \<turnstile> get_shadow_root host' \<rightarrow>\<^sub>r Some shadow_root_ptr"
    using assms
    by (metis get_host_ptr_in_heap local.a_shadow_root_valid_def
        local.get_shadow_root_ok local.heap_is_wellformed_def returns_result_select_result)
  then have "host = host'"
    by (meson assms(1) assms(2) assms(3) shadow_root_host_dual shadow_root_same_host)
  then show ?thesis
    using \<open>|h \<turnstile> get_tag_name host'|\<^sub>r \<in> safe_shadow_root_element_types\<close> assms(4) by auto
qed

lemma a_host_shadow_root_rel_finite: "finite (a_host_shadow_root_rel h)"
proof -
  have "a_host_shadow_root_rel h = (\<Union>host \<in> fset (element_ptr_kinds h).
    (case |h \<turnstile> get_shadow_root host|\<^sub>r of
      Some shadow_root \<Rightarrow> {(cast host, cast shadow_root)} | None \<Rightarrow> {}))"
    by(auto simp add: a_host_shadow_root_rel_def split: option.splits)
  moreover have "finite (\<Union>host \<in> fset (element_ptr_kinds h).
    (case |h \<turnstile> get_shadow_root host|\<^sub>r of
      Some shadow_root \<Rightarrow> {(cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r host, cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root)} |
      None \<Rightarrow> {}))"
    by(auto split: option.splits)
  ultimately show ?thesis
    by auto
qed

lemma heap_is_wellformed_children_in_heap:
  "heap_is_wellformed h \<Longrightarrow> h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children \<Longrightarrow> child \<in> set children \<Longrightarrow>
  child |\<in>| node_ptr_kinds h"
  using CD.heap_is_wellformed_children_in_heap local.heap_is_wellformed_def by blast
lemma heap_is_wellformed_disc_nodes_in_heap:
  "heap_is_wellformed h \<Longrightarrow> h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes \<Longrightarrow>
  node \<in> set disc_nodes \<Longrightarrow> node |\<in>| node_ptr_kinds h"
  using CD.heap_is_wellformed_disc_nodes_in_heap local.heap_is_wellformed_def by blast
lemma heap_is_wellformed_one_parent:
  "heap_is_wellformed h \<Longrightarrow> h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children \<Longrightarrow>
  h \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children' \<Longrightarrow> set children \<inter> set children' \<noteq> {} \<Longrightarrow> ptr = ptr'"
  using CD.heap_is_wellformed_one_parent local.heap_is_wellformed_def by blast
lemma heap_is_wellformed_one_disc_parent:
  "heap_is_wellformed h \<Longrightarrow> h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes \<Longrightarrow>
  h \<turnstile> get_disconnected_nodes document_ptr' \<rightarrow>\<^sub>r disc_nodes' \<Longrightarrow> set disc_nodes \<inter> set disc_nodes' \<noteq> {}
  \<Longrightarrow> document_ptr = document_ptr'"
  using CD.heap_is_wellformed_one_disc_parent local.heap_is_wellformed_def by blast
lemma heap_is_wellformed_children_disc_nodes_different:
  "heap_is_wellformed h \<Longrightarrow> h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children \<Longrightarrow>
  h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes \<Longrightarrow> set children \<inter> set disc_nodes = {}"
  using CD.heap_is_wellformed_children_disc_nodes_different local.heap_is_wellformed_def by blast
lemma heap_is_wellformed_disconnected_nodes_distinct:
  "heap_is_wellformed h \<Longrightarrow> h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes \<Longrightarrow>
  distinct disc_nodes"
  using CD.heap_is_wellformed_disconnected_nodes_distinct local.heap_is_wellformed_def by blast
lemma heap_is_wellformed_children_distinct:
  "heap_is_wellformed h \<Longrightarrow> h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children \<Longrightarrow> distinct children"
  using CD.heap_is_wellformed_children_distinct local.heap_is_wellformed_def by blast
lemma heap_is_wellformed_children_disc_nodes:
  "heap_is_wellformed h \<Longrightarrow> node_ptr |\<in>| node_ptr_kinds h \<Longrightarrow>
  \<not>(\<exists>parent \<in> fset (object_ptr_kinds h). node_ptr \<in> set |h \<turnstile> get_child_nodes parent|\<^sub>r) \<Longrightarrow>
  (\<exists>document_ptr \<in> fset (document_ptr_kinds h).
    node_ptr \<in> set |h \<turnstile> get_disconnected_nodes document_ptr|\<^sub>r)"
  using CD.heap_is_wellformed_children_disc_nodes local.heap_is_wellformed_def by blast
lemma parent_child_rel_finite: "heap_is_wellformed h \<Longrightarrow> finite (parent_child_rel h)"
  using CD.parent_child_rel_finite by blast
lemma parent_child_rel_acyclic: "heap_is_wellformed h \<Longrightarrow> acyclic (parent_child_rel h)"
  using CD.parent_child_rel_acyclic heap_is_wellformed_def by blast
lemma parent_child_rel_child_in_heap:
  "heap_is_wellformed h \<Longrightarrow> type_wf h \<Longrightarrow> known_ptr parent \<Longrightarrow>
  (parent, child_ptr) \<in> parent_child_rel h \<Longrightarrow> child_ptr |\<in>| object_ptr_kinds h"
  using CD.parent_child_rel_child_in_heap local.heap_is_wellformed_def by blast
end

interpretation i_heap_is_wellformed?: l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes
  get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_shadow_root
  get_shadow_root_locs get_tag_name get_tag_name_locs known_ptr type_wf heap_is_wellformed
  parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host get_host_locs get_disconnected_document
  get_disconnected_document_locs
  by(auto simp add: l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def
      l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def
      heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def parent_child_rel_def heap_is_wellformed_def
      instances)
declare l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms [instances]

lemma heap_is_wellformed_is_l_heap_is_wellformed [instances]:
  "l_heap_is_wellformed ShadowRootClass.type_wf ShadowRootClass.known_ptr
    Shadow_DOM.heap_is_wellformed Shadow_DOM.parent_child_rel Shadow_DOM.get_child_nodes
    get_disconnected_nodes"
  apply(auto simp add: l_heap_is_wellformed_def instances)[1]
  using heap_is_wellformed_children_in_heap apply metis
  using heap_is_wellformed_disc_nodes_in_heap apply metis
  using heap_is_wellformed_one_parent apply blast
  using heap_is_wellformed_one_disc_parent apply blast
  using heap_is_wellformed_children_disc_nodes_different apply blast
  using heap_is_wellformed_disconnected_nodes_distinct apply metis
  using heap_is_wellformed_children_distinct apply metis
  using heap_is_wellformed_children_disc_nodes apply metis
  using i_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.parent_child_rel_child apply(blast, blast)
  using i_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.parent_child_rel_finite apply blast
  using parent_child_rel_acyclic apply blast
  using i_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.parent_child_rel_node_ptr apply blast
  using i_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.parent_child_rel_parent_in_heap apply blast
  using parent_child_rel_child_in_heap apply metis
  done

subsubsection \<open>get\_parent\<close>

interpretation i_get_parent_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M: l_get_parent_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes
  get_child_nodes_locs known_ptrs get_parent get_parent_locs heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M
  parent_child_rel get_disconnected_nodes
  by(simp add: l_get_parent_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def instances)
declare i_get_parent_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.l_get_parent_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances]

interpretation i_get_parent_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M: l_get_parent_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes
  get_child_nodes_locs known_ptrs get_parent get_parent_locs heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M
  parent_child_rel get_disconnected_nodes get_disconnected_nodes_locs
  by(auto simp add: l_get_parent_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def instances)
declare i_get_parent_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.l_get_parent_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances]

lemma get_parent_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_is_l_get_parent_wf [instances]: "l_get_parent_wf type_wf known_ptr
  known_ptrs heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M parent_child_rel get_child_nodes get_parent"
  apply(auto simp add: l_get_parent_wf_def l_get_parent_wf_axioms_def instances)[1]
  using i_get_parent_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.child_parent_dual apply fast
  using i_get_parent_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.heap_wellformed_induct apply metis
  using i_get_parent_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.heap_wellformed_induct_rev apply metis
  using i_get_parent_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.parent_child_rel_parent apply fast
  done

subsubsection \<open>get\_disconnected\_nodes\<close>

subsubsection \<open>set\_disconnected\_nodes\<close>

paragraph \<open>get\_disconnected\_nodes\<close>

interpretation i_set_disconnected_nodes_get_disconnected_nodes_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M:
  l_set_disconnected_nodes_get_disconnected_nodes_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf
  get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes
  set_disconnected_nodes_locs heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M parent_child_rel get_child_nodes
  by (simp add: l_set_disconnected_nodes_get_disconnected_nodes_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def instances)
declare i_set_disconnected_nodes_get_disconnected_nodes_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.l_set_disconnected_nodes_get_disconnected_nodes_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances]

lemma set_disconnected_nodes_get_disconnected_nodes_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_is_l_set_disconnected_nodes_get_disconnected_nodes_wf [instances]:
  "l_set_disconnected_nodes_get_disconnected_nodes_wf type_wf known_ptr heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M
    parent_child_rel get_child_nodes get_disconnected_nodes get_disconnected_nodes_locs
    set_disconnected_nodes set_disconnected_nodes_locs"
  apply(auto simp add: l_set_disconnected_nodes_get_disconnected_nodes_wf_def
      l_set_disconnected_nodes_get_disconnected_nodes_wf_axioms_def instances)[1]
  using i_set_disconnected_nodes_get_disconnected_nodes_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.remove_from_disconnected_nodes_removes
  apply fast
  done

paragraph \<open>get\_root\_node\<close>

interpretation i_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M:l_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf known_ptrs
  heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M parent_child_rel get_child_nodes get_child_nodes_locs get_disconnected_nodes
  get_disconnected_nodes_locs get_parent get_parent_locs get_ancestors get_ancestors_locs
  get_root_node get_root_node_locs
  by(simp add: l_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def instances)
declare i_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.l_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances]

lemma get_ancestors_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_is_l_get_ancestors_wf [instances]:
  "l_get_ancestors_wf heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M parent_child_rel known_ptr known_ptrs type_wf
  get_ancestors get_ancestors_locs get_child_nodes get_parent"
  apply(auto simp add: l_get_ancestors_wf_def l_get_ancestors_wf_axioms_def instances)[1]
  using i_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.get_ancestors_never_empty apply blast
  using i_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.get_ancestors_ok apply blast
  using i_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.get_ancestors_reads apply blast
  using i_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.get_ancestors_ptrs_in_heap apply blast
  using i_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.get_ancestors_remains_not_in_ancestors apply blast
  using i_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.get_ancestors_also_parent apply blast
  using i_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.get_ancestors_obtains_children apply blast
  using i_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.get_ancestors_parent_child_rel apply blast
  using i_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.get_ancestors_parent_child_rel apply blast
  done

lemma get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_is_l_get_root_node_wf [instances]:
  "l_get_root_node_wf heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_root_node type_wf known_ptr known_ptrs
    get_ancestors get_parent"
  apply(auto simp add: l_get_root_node_wf_def l_get_root_node_wf_axioms_def instances)[1]
  using i_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.get_root_node_ok apply blast
  using i_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.get_root_node_ptr_in_heap apply blast
  using i_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.get_root_node_root_in_heap apply blast
  using i_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.get_ancestors_same_root_node apply(blast, blast)
  using i_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.get_root_node_same_no_parent apply blast
    (* using i_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.get_root_node_not_node_same apply blast *)
  using i_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.get_root_node_parent_same apply (blast, blast)
  done

subsubsection \<open>to\_tree\_order\<close>

interpretation i_to_tree_order_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M: l_to_tree_order_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes
  get_child_nodes_locs to_tree_order known_ptrs get_parent get_parent_locs heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M
  parent_child_rel get_disconnected_nodes get_disconnected_nodes_locs
  apply(simp add: l_to_tree_order_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def instances)
  done
declare i_to_tree_order_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.l_to_tree_order_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms [instances]

lemma to_tree_order_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_is_l_to_tree_order_wf [instances]:
  "l_to_tree_order_wf heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M parent_child_rel type_wf known_ptr known_ptrs
  to_tree_order get_parent get_child_nodes"
  apply(auto simp add: l_to_tree_order_wf_def l_to_tree_order_wf_axioms_def instances)[1]
  using i_to_tree_order_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.to_tree_order_ok apply blast
  using i_to_tree_order_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.to_tree_order_ptrs_in_heap apply blast
  using i_to_tree_order_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.to_tree_order_parent_child_rel apply(blast, blast)
  using i_to_tree_order_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.to_tree_order_child2 apply blast
  using i_to_tree_order_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.to_tree_order_node_ptrs apply blast
  using i_to_tree_order_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.to_tree_order_child apply blast
  using i_to_tree_order_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.to_tree_order_ptr_in_result apply blast
  using i_to_tree_order_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.to_tree_order_parent apply blast
  using i_to_tree_order_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.to_tree_order_subset apply blast
  done

paragraph \<open>get\_root\_node\<close>

interpretation i_to_tree_order_wf_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M: l_to_tree_order_wf_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M
  known_ptr type_wf known_ptrs heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M parent_child_rel get_child_nodes
  get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_parent get_parent_locs
  get_ancestors get_ancestors_locs get_root_node get_root_node_locs to_tree_order
  by(auto simp add: l_to_tree_order_wf_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def instances)
declare i_to_tree_order_wf_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.l_to_tree_order_wf_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms [instances]

lemma to_tree_order_wf_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_is_l_to_tree_order_wf_get_root_node_wf [instances]:
  "l_to_tree_order_wf_get_root_node_wf type_wf known_ptr known_ptrs to_tree_order get_root_node
    heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"
  apply(auto simp add: l_to_tree_order_wf_get_root_node_wf_def
      l_to_tree_order_wf_get_root_node_wf_axioms_def instances)[1]
  using i_to_tree_order_wf_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.to_tree_order_get_root_node apply blast
  using i_to_tree_order_wf_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.to_tree_order_same_root apply blast
  done

subsubsection \<open>remove\_child\<close>

interpretation i_remove_child_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M: l_remove_child_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs
  set_child_nodes set_child_nodes_locs get_parent
  get_parent_locs get_owner_document get_disconnected_nodes get_disconnected_nodes_locs
  set_disconnected_nodes set_disconnected_nodes_locs remove_child remove_child_locs remove type_wf
  known_ptr known_ptrs heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M parent_child_rel
  by unfold_locales
declare i_remove_child_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.l_remove_child_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms [instances]

lemma remove_child_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_is_l_remove_child_wf2 [instances]: "l_remove_child_wf2 type_wf known_ptr
  known_ptrs remove_child heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes remove"
  apply(auto simp add: l_remove_child_wf2_def l_remove_child_wf2_axioms_def instances)[1]
  using i_remove_child_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.remove_child_heap_is_wellformed_preserved apply(fast, fast, fast)
  using i_remove_child_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.remove_heap_is_wellformed_preserved apply(fast, fast, fast)
  using i_remove_child_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.remove_child_removes_child apply fast
  using i_remove_child_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.remove_child_removes_first_child apply fast
  using i_remove_child_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.remove_removes_child apply fast
  using i_remove_child_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M.remove_for_all_empty_children apply fast
  done

subsection \<open>A wellformed heap\<close>

subsubsection \<open>get\_parent\<close>

interpretation i_get_parent_wf?: l_get_parent_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes
  get_child_nodes_locs known_ptrs get_parent get_parent_locs heap_is_wellformed parent_child_rel
  get_disconnected_nodes
  using instances
  by(simp add: l_get_parent_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def)
declare l_get_parent_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances]

lemma get_parent_wf_is_l_get_parent_wf [instances]:
  "l_get_parent_wf ShadowRootClass.type_wf ShadowRootClass.known_ptr ShadowRootClass.known_ptrs
  heap_is_wellformed parent_child_rel Shadow_DOM.get_child_nodes Shadow_DOM.get_parent"
  apply(auto simp add: l_get_parent_wf_def l_get_parent_wf_axioms_def instances)[1]
  using child_parent_dual apply blast
  using heap_wellformed_induct apply metis
  using heap_wellformed_induct_rev apply metis
  using parent_child_rel_parent apply metis
  done

subsubsection \<open>remove\_shadow\_root\<close>

locale l_remove_shadow_root_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M =
  l_get_tag_name +
  l_get_disconnected_nodes  +
  l_set_shadow_root_get_tag_name +
  l_get_child_nodes  +
  l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M +
  l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M +
  l_delete_shadow_root_get_disconnected_nodes +
  l_delete_shadow_root_get_child_nodes +
  l_set_shadow_root_get_disconnected_nodes +
  l_set_shadow_root_get_child_nodes +
  l_delete_shadow_root_get_tag_name +
  l_set_shadow_root_get_shadow_root +
  l_delete_shadow_root_get_shadow_root +
  l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M
begin

lemma remove_shadow_root_preserves:
  assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
  assumes "h \<turnstile> remove_shadow_root ptr \<rightarrow>\<^sub>h h'"
  shows "known_ptrs h'" and "type_wf h'" "heap_is_wellformed h'"
proof -
  obtain shadow_root_ptr h2 where
    "h \<turnstile> get_shadow_root ptr \<rightarrow>\<^sub>r Some shadow_root_ptr" and
    "h \<turnstile> get_child_nodes (cast shadow_root_ptr) \<rightarrow>\<^sub>r []" and
    h2: "h \<turnstile> set_shadow_root ptr None \<rightarrow>\<^sub>h h2" and
    h': "h2 \<turnstile> delete_M shadow_root_ptr \<rightarrow>\<^sub>h h'"
    using assms(4)
    by(auto simp add: remove_shadow_root_def
        elim!: bind_returns_heap_E bind_returns_heap_E2[rotated, OF get_shadow_root_pure, rotated]
        bind_returns_heap_E2[rotated, OF get_child_nodes_pure, rotated]
        split: option.splits if_splits)

  have "type_wf h2"
    using writes_small_big[where P="\<lambda>h h'. type_wf h \<longrightarrow> type_wf h'", OF set_shadow_root_writes h2]
    using \<open>type_wf h\<close> set_shadow_root_types_preserved
    by(auto simp add: reflp_def transp_def)
  then show "type_wf h'"
    using h' delete_shadow_root_type_wf_preserved local.type_wf_impl
    by blast

  have object_ptr_kinds_eq_h: "object_ptr_kinds h = object_ptr_kinds h2"
    apply(rule writes_small_big[where P="\<lambda>h h'. object_ptr_kinds h = object_ptr_kinds h'",
          OF set_shadow_root_writes h2])
    using set_shadow_root_pointers_preserved
      apply blast
    by (auto simp add: reflp_def transp_def)
  have node_ptr_kinds_eq_h: "node_ptr_kinds h = node_ptr_kinds h2"
    using object_ptr_kinds_eq_h
    by (simp add: node_ptr_kinds_def)
  have element_ptr_kinds_eq_h: "element_ptr_kinds h = element_ptr_kinds h2"
    using node_ptr_kinds_eq_h
    by (simp add: element_ptr_kinds_def)
  have document_ptr_kinds_eq_h: "document_ptr_kinds h = document_ptr_kinds h2"
    using object_ptr_kinds_eq_h
    by (simp add: document_ptr_kinds_def)
  have shadow_root_ptr_kinds_eq_h: "shadow_root_ptr_kinds h = shadow_root_ptr_kinds h2"
    using object_ptr_kinds_eq_h
    by (simp add: shadow_root_ptr_kinds_def)

  have "known_ptrs h2"
    using \<open>known_ptrs h\<close> object_ptr_kinds_eq_h known_ptrs_subset
    by blast

  have object_ptr_kinds_eq_h2: "object_ptr_kinds h' |\<subseteq>| object_ptr_kinds h2"
    using h' delete_shadow_root_pointers
    by auto
  have object_ptr_kinds_eq2_h2:
    "object_ptr_kinds h2 = object_ptr_kinds h' |\<union>| {|cast shadow_root_ptr|}"
    using h' delete_shadow_root_pointers
    by auto
  have node_ptr_kinds_eq_h2: "node_ptr_kinds h2 = node_ptr_kinds h'"
    using object_ptr_kinds_eq_h2
    by(auto simp add: node_ptr_kinds_def delete_shadow_root_pointers[OF h'])
  have element_ptr_kinds_eq_h2: "element_ptr_kinds h2 = element_ptr_kinds h'"
    using node_ptr_kinds_eq_h2
    by (simp add: element_ptr_kinds_def)
  have document_ptr_kinds_eq_h2: "document_ptr_kinds h2 = document_ptr_kinds h'"
    using object_ptr_kinds_eq_h2
    by(auto simp add: document_ptr_kinds_def delete_shadow_root_pointers[OF h'])
  have shadow_root_ptr_kinds_eq_h2: "shadow_root_ptr_kinds h' |\<subseteq>| shadow_root_ptr_kinds h2"
    using object_ptr_kinds_eq_h2
    by (auto simp add: shadow_root_ptr_kinds_def)
  have shadow_root_ptr_kinds_eq2_h2:
    "shadow_root_ptr_kinds h2 = shadow_root_ptr_kinds h' |\<union>| {|shadow_root_ptr|}"
    using object_ptr_kinds_eq2_h2
    by (auto simp add: shadow_root_ptr_kinds_def)

  show "known_ptrs h'"
    using object_ptr_kinds_eq_h2 \<open>known_ptrs h2\<close> known_ptrs_subset
    by blast

  have disconnected_nodes_eq_h:
    "\<And>doc_ptr disc_nodes. h \<turnstile> get_disconnected_nodes doc_ptr \<rightarrow>\<^sub>r disc_nodes =
      h2 \<turnstile> get_disconnected_nodes doc_ptr \<rightarrow>\<^sub>r disc_nodes"
    using get_disconnected_nodes_reads set_shadow_root_writes h2 set_shadow_root_get_disconnected_nodes
    by(rule reads_writes_preserved)
  then have disconnected_nodes_eq2_h:
    "\<And>doc_ptr. |h \<turnstile> get_disconnected_nodes doc_ptr|\<^sub>r = |h2 \<turnstile> get_disconnected_nodes doc_ptr|\<^sub>r"
    using select_result_eq by force

  have disconnected_nodes_eq_h2:
    "\<And>doc_ptr disc_nodes. h2 \<turnstile> get_disconnected_nodes doc_ptr \<rightarrow>\<^sub>r disc_nodes =
      h' \<turnstile> get_disconnected_nodes doc_ptr \<rightarrow>\<^sub>r disc_nodes"
    using get_disconnected_nodes_reads get_disconnected_nodes_delete_shadow_root[OF h']
    apply(auto simp add: reads_def reflp_def transp_def preserved_def)[1]
    by blast+
  then have disconnected_nodes_eq2_h2:
    "\<And>doc_ptr. |h2 \<turnstile> get_disconnected_nodes doc_ptr|\<^sub>r = |h' \<turnstile> get_disconnected_nodes doc_ptr|\<^sub>r"
    using select_result_eq by force

  have tag_name_eq_h:
    "\<And>doc_ptr disc_nodes. h \<turnstile> get_tag_name doc_ptr \<rightarrow>\<^sub>r disc_nodes =
      h2 \<turnstile> get_tag_name doc_ptr \<rightarrow>\<^sub>r disc_nodes"
    using get_tag_name_reads set_shadow_root_writes h2 set_shadow_root_get_tag_name
    by(rule reads_writes_preserved)
  then have tag_name_eq2_h: "\<And>doc_ptr. |h \<turnstile> get_tag_name doc_ptr|\<^sub>r = |h2 \<turnstile> get_tag_name doc_ptr|\<^sub>r"
    using select_result_eq by force

  have tag_name_eq_h2:
    "\<And>doc_ptr disc_nodes. h2 \<turnstile> get_tag_name doc_ptr \<rightarrow>\<^sub>r disc_nodes =
      h' \<turnstile> get_tag_name doc_ptr \<rightarrow>\<^sub>r disc_nodes"
    using get_tag_name_reads get_tag_name_delete_shadow_root[OF h']
    apply(auto simp add: reads_def reflp_def transp_def preserved_def)[1]
    by blast+
  then have tag_name_eq2_h2: "\<And>doc_ptr. |h2 \<turnstile> get_tag_name doc_ptr|\<^sub>r = |h' \<turnstile> get_tag_name doc_ptr|\<^sub>r"
    using select_result_eq by force

  have children_eq_h:
    "\<And>ptr' children. h \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children = h2 \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children"
    using get_child_nodes_reads set_shadow_root_writes h2 set_shadow_root_get_child_nodes
    by(rule reads_writes_preserved)

  then have children_eq2_h: "\<And>ptr'. |h \<turnstile> get_child_nodes ptr'|\<^sub>r = |h2 \<turnstile> get_child_nodes ptr'|\<^sub>r"
    using select_result_eq by force

  have children_eq_h2:
    "\<And>ptr' children. ptr' \<noteq> cast shadow_root_ptr \<Longrightarrow> h2 \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children =
      h' \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children"
    using get_child_nodes_reads h' get_child_nodes_delete_shadow_root
    apply(auto simp add: reads_def reflp_def transp_def preserved_def)[1]
    by blast+
  then have children_eq2_h2:
    "\<And>ptr'. ptr' \<noteq> cast shadow_root_ptr \<Longrightarrow> |h2 \<turnstile> get_child_nodes ptr'|\<^sub>r =
    |h' \<turnstile> get_child_nodes ptr'|\<^sub>r"
    using select_result_eq by force

  have "cast shadow_root_ptr |\<notin>| object_ptr_kinds h'"
    using h' delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M_ptr_not_in_heap
    by auto

  have get_shadow_root_eq_h:
    "\<And>shadow_root_opt ptr'. ptr \<noteq> ptr' \<Longrightarrow> h \<turnstile> get_shadow_root ptr' \<rightarrow>\<^sub>r shadow_root_opt =
    h2 \<turnstile> get_shadow_root ptr' \<rightarrow>\<^sub>r shadow_root_opt"
    using get_shadow_root_reads set_shadow_root_writes h2
    apply(rule reads_writes_preserved)
    using set_shadow_root_get_shadow_root_different_pointers
    by fast

  have get_shadow_root_eq_h2:
    "\<And>shadow_root_opt ptr'. h2 \<turnstile> get_shadow_root ptr' \<rightarrow>\<^sub>r shadow_root_opt =
    h' \<turnstile> get_shadow_root ptr' \<rightarrow>\<^sub>r shadow_root_opt"
    using get_shadow_root_reads get_shadow_root_delete_shadow_root[OF h']
    apply(auto simp add: reads_def reflp_def transp_def preserved_def)[1]
    by blast+
  then
  have get_shadow_root_eq2_h2: "\<And>ptr'. |h2 \<turnstile> get_shadow_root ptr'|\<^sub>r =
    |h' \<turnstile> get_shadow_root ptr'|\<^sub>r"
    using select_result_eq by force

  have "acyclic (parent_child_rel h)"
    using \<open>heap_is_wellformed h\<close>
    by (simp add: heap_is_wellformed_def CD.heap_is_wellformed_def CD.acyclic_heap_def)
  moreover
  have "parent_child_rel h = parent_child_rel h2"
    by(auto simp add: CD.parent_child_rel_def object_ptr_kinds_eq_h children_eq2_h)
  moreover
  have "parent_child_rel h' \<subseteq> parent_child_rel h2"
    using object_ptr_kinds_eq_h2
    apply(auto simp add: CD.parent_child_rel_def)[1]
    by (metis \<open>cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr |\<notin>| object_ptr_kinds h'\<close> children_eq2_h2)
  ultimately
  have "CD.a_acyclic_heap h'"
    using acyclic_subset
    by (auto simp add: heap_is_wellformed_def CD.heap_is_wellformed_def CD.acyclic_heap_def)

  moreover
  have "CD.a_all_ptrs_in_heap h"
    using \<open>heap_is_wellformed h\<close>
    by (simp add: heap_is_wellformed_def CD.heap_is_wellformed_def)
  then have "CD.a_all_ptrs_in_heap h2"
    by(auto simp add: children_eq2_h disconnected_nodes_eq2_h document_ptr_kinds_eq_h
        CD.a_all_ptrs_in_heap_def object_ptr_kinds_eq_h node_ptr_kinds_def
        children_eq_h disconnected_nodes_eq_h)
  then have "CD.a_all_ptrs_in_heap h'"
    using disconnected_nodes_eq2_h2 document_ptr_kinds_eq_h2
    apply(auto simp add: CD.a_all_ptrs_in_heap_def node_ptr_kinds_eq_h2 children_eq_h2
         object_ptr_kinds_eq_h2)[1]
    by (metis (no_types, opaque_lifting) \<open>cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr |\<notin>| object_ptr_kinds h'\<close>
        children_eq2_h2 fsubsetD in_mono object_ptr_kinds_eq_h2)

  moreover
  have "CD.a_distinct_lists h"
    using \<open>heap_is_wellformed h\<close>
    by (simp add: heap_is_wellformed_def CD.heap_is_wellformed_def)
  then have "CD.a_distinct_lists h2"
    by(auto simp add: CD.a_distinct_lists_def object_ptr_kinds_eq_h document_ptr_kinds_eq_h
        children_eq2_h disconnected_nodes_eq2_h)
  then have "CD.a_distinct_lists h'"
    apply(auto simp add: CD.a_distinct_lists_def document_ptr_kinds_eq_h2
        disconnected_nodes_eq2_h2)[1]
     apply(auto simp add:  intro!: distinct_concat_map_I)[1]
      apply(case_tac "x = cast shadow_root_ptr")
    using \<open>cast shadow_root_ptr |\<notin>| object_ptr_kinds h'\<close> apply simp
    using children_eq_h2 concat_map_all_distinct[of "(\<lambda>ptr. |h2 \<turnstile> get_child_nodes ptr|\<^sub>r)"]
      apply (metis (no_types, lifting) children_eq2_h2 finite_fset fset_mp
        object_ptr_kinds_eq_h2 set_sorted_list_of_set)
     apply(case_tac "x = cast shadow_root_ptr")
    using \<open>cast shadow_root_ptr |\<notin>| object_ptr_kinds h'\<close> apply simp
     apply(case_tac "y = cast shadow_root_ptr")
    using \<open>cast shadow_root_ptr |\<notin>| object_ptr_kinds h'\<close> apply simp
    using children_eq_h2 distinct_concat_map_E(1)[of "(\<lambda>ptr. |h2 \<turnstile> get_child_nodes ptr|\<^sub>r)"]
     apply (metis (no_types, lifting) IntI \<open>known_ptrs h'\<close> \<open>type_wf h'\<close> children_eq2_h2 empty_iff
             finite_fset is_OK_returns_result_I l_get_child_nodes.get_child_nodes_ptr_in_heap
             local.get_child_nodes_ok local.known_ptrs_known_ptr local.l_get_child_nodes_axioms
             returns_result_select_result sorted_list_of_set.set_sorted_key_list_of_set)
    apply(case_tac "xa = cast shadow_root_ptr")
    using \<open>cast shadow_root_ptr |\<notin>| object_ptr_kinds h'\<close> apply simp
    by (metis (mono_tags, lifting) CD.distinct_lists_no_parent \<open>known_ptrs h'\<close>
        \<open>local.CD.a_distinct_lists h2\<close> \<open>type_wf h'\<close> children_eq2_h2 children_eq_h2
        disconnected_nodes_eq_h2 is_OK_returns_result_I l_get_child_nodes.get_child_nodes_ptr_in_heap
        local.get_child_nodes_ok local.get_disconnected_nodes_ok local.known_ptrs_known_ptr
        local.l_get_child_nodes_axioms returns_result_select_result)
  moreover
  have "CD.a_owner_document_valid h"
    using \<open>heap_is_wellformed h\<close>
    by (simp add: heap_is_wellformed_def CD.heap_is_wellformed_def)
  then have "CD.a_owner_document_valid h2"
    by(auto simp add: CD.a_owner_document_valid_def object_ptr_kinds_eq_h document_ptr_kinds_eq_h
        node_ptr_kinds_eq_h children_eq2_h disconnected_nodes_eq2_h)
  then have "CD.a_owner_document_valid h'"
    apply(auto simp add: CD.a_owner_document_valid_def  document_ptr_kinds_eq_h2 node_ptr_kinds_eq_h2
        disconnected_nodes_eq2_h2)[1]
  proof -
    fix node_ptr
    assume 0: "\<forall>node_ptr\<in>fset (node_ptr_kinds h').
      (\<exists>document_ptr. document_ptr |\<in>| document_ptr_kinds h' \<and>
      node_ptr \<in> set |h' \<turnstile> get_disconnected_nodes document_ptr|\<^sub>r) \<or>
      (\<exists>parent_ptr. parent_ptr |\<in>| object_ptr_kinds h2 \<and>
      node_ptr \<in> set |h2 \<turnstile> get_child_nodes parent_ptr|\<^sub>r)"
      and 1: "node_ptr |\<in>| node_ptr_kinds h'"
      and 2: "\<forall>parent_ptr. parent_ptr |\<in>| object_ptr_kinds h' \<longrightarrow>
        node_ptr \<notin> set |h' \<turnstile> get_child_nodes parent_ptr|\<^sub>r"
    then have "\<forall>parent_ptr. parent_ptr |\<in>| object_ptr_kinds h2 \<longrightarrow>
      node_ptr \<notin> set |h2 \<turnstile> get_child_nodes parent_ptr|\<^sub>r"
      apply(auto)[1]
      apply(case_tac "parent_ptr = cast shadow_root_ptr")
      using \<open>h \<turnstile> get_child_nodes (cast shadow_root_ptr) \<rightarrow>\<^sub>r []\<close> children_eq_h
       apply auto[1]
      using children_eq2_h2 object_ptr_kinds_eq_h2 h' delete_shadow_root_pointers
      by (metis fempty_iff finsert_iff funionE)
    then show "\<exists>document_ptr. document_ptr |\<in>| document_ptr_kinds h' \<and>
      node_ptr \<in> set |h' \<turnstile> get_disconnected_nodes document_ptr|\<^sub>r"
      using 0 1
      by auto
  qed

  ultimately have "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h'"
    by(simp add: CD.heap_is_wellformed_def)

  moreover
  have "acyclic (parent_child_rel h \<union> a_host_shadow_root_rel h)"
    using \<open>heap_is_wellformed h\<close>
    by(simp add: heap_is_wellformed_def)
  then
  have "acyclic (parent_child_rel h2 \<union> a_host_shadow_root_rel h2)"
  proof -
    have "a_host_shadow_root_rel h2 \<subseteq> a_host_shadow_root_rel h"
      apply(auto simp add: a_host_shadow_root_rel_def element_ptr_kinds_eq_h)[1]
      apply(case_tac "aa = ptr")
       apply(simp)
       apply (metis (no_types, lifting) \<open>type_wf h2\<close> assms(2) h2 local.get_shadow_root_ok
          local.type_wf_impl option.distinct(1) returns_result_eq
          returns_result_select_result set_shadow_root_get_shadow_root)
      using get_shadow_root_eq_h
      by (metis (mono_tags, lifting) \<open>type_wf h2\<close> image_eqI is_OK_returns_result_E
          local.get_shadow_root_ok mem_Collect_eq prod.simps(2) select_result_I2)
    then show ?thesis
      using \<open>parent_child_rel h = parent_child_rel h2\<close>
      by (metis (no_types, opaque_lifting) \<open>acyclic (parent_child_rel h \<union> a_host_shadow_root_rel h)\<close>
          acyclic_subset order_refl sup_mono)
  qed
  then
  have "acyclic (parent_child_rel h' \<union> a_host_shadow_root_rel h')"
  proof -
    have "a_host_shadow_root_rel h' \<subseteq> a_host_shadow_root_rel h2"
      by(auto simp add: a_host_shadow_root_rel_def element_ptr_kinds_eq_h2 get_shadow_root_eq2_h2)
    then show ?thesis
      using \<open>parent_child_rel h' \<subseteq> parent_child_rel h2\<close>
        \<open>acyclic (parent_child_rel h2 \<union> a_host_shadow_root_rel h2)\<close>
      using acyclic_subset sup_mono
      by (metis (no_types, opaque_lifting))
  qed

  moreover
  have "a_all_ptrs_in_heap h"
    using \<open>heap_is_wellformed h\<close>
    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 shadow_root_ptr_kinds_eq_h)[1]
    apply(case_tac "host = ptr")
     apply(simp)
     apply (metis assms(2) h2 local.type_wf_impl option.distinct(1) returns_result_eq
        set_shadow_root_get_shadow_root)
    using get_shadow_root_eq_h
    by fastforce
  then
  have "a_all_ptrs_in_heap h'"
    apply(auto simp add: a_all_ptrs_in_heap_def get_shadow_root_eq_h2)[1]
    apply(auto simp add: shadow_root_ptr_kinds_eq2_h2)[1]
    by (metis (no_types, lifting) \<open>h \<turnstile> get_shadow_root ptr \<rightarrow>\<^sub>r Some shadow_root_ptr\<close> assms(1)
        assms(2) get_shadow_root_eq_h get_shadow_root_eq_h2 h2 local.shadow_root_same_host
        local.type_wf_impl option.distinct(1) select_result_I2 set_shadow_root_get_shadow_root)

  moreover
  have "a_distinct_lists h"
    using \<open>heap_is_wellformed h\<close>
    by(simp add: heap_is_wellformed_def)
  then
  have "a_distinct_lists h2"
    apply(auto simp add: a_distinct_lists_def element_ptr_kinds_eq_h)[1]
    apply(auto intro!: distinct_concat_map_I split: option.splits)[1]
    apply(case_tac "x = ptr")
     apply(simp)
     apply (metis (no_types, opaque_lifting) assms(2) h2 is_OK_returns_result_I
        l_set_shadow_root_get_shadow_root.set_shadow_root_get_shadow_root
        l_set_shadow_root_get_shadow_root_axioms local.type_wf_impl option.discI
        returns_result_eq returns_result_select_result)

    apply(case_tac "y = ptr")
     apply(simp)
     apply (metis (no_types, opaque_lifting) assms(2) h2 is_OK_returns_result_I
        l_set_shadow_root_get_shadow_root.set_shadow_root_get_shadow_root
        l_set_shadow_root_get_shadow_root_axioms local.type_wf_impl option.discI
        returns_result_eq returns_result_select_result)
    by (metis \<open>type_wf h2\<close> assms(1) assms(2) get_shadow_root_eq_h local.get_shadow_root_ok
        local.shadow_root_same_host returns_result_select_result)

  then
  have "a_distinct_lists h'"
    by(auto simp add: a_distinct_lists_def element_ptr_kinds_eq_h2 get_shadow_root_eq2_h2)

  moreover
  have "a_shadow_root_valid h"
    using \<open>heap_is_wellformed h\<close>
    by(simp add: heap_is_wellformed_def)
  then
  have "a_shadow_root_valid h'"
    by (smt (verit) \<open>h \<turnstile> get_shadow_root ptr \<rightarrow>\<^sub>r Some shadow_root_ptr\<close> assms(2)
        delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M_ptr_not_in_heap element_ptr_kinds_eq_h element_ptr_kinds_eq_h2
        finsert_iff funion_finsert_right get_shadow_root_eq2_h2 get_shadow_root_eq_h h'
        local.a_shadow_root_valid_def local.get_shadow_root_ok object_ptr_kinds_eq2_h2
        object_ptr_kinds_eq_h option.sel returns_result_select_result select_result_I2
        shadow_root_ptr_kinds_commutes sup_bot.right_neutral tag_name_eq2_h tag_name_eq2_h2)
  ultimately show "heap_is_wellformed h'"
    by(simp add: heap_is_wellformed_def)
qed
end

interpretation i_remove_shadow_root_wf?: l_remove_shadow_root_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M
  type_wf get_tag_name get_tag_name_locs get_disconnected_nodes get_disconnected_nodes_locs
  set_shadow_root set_shadow_root_locs known_ptr get_child_nodes get_child_nodes_locs get_shadow_root
  get_shadow_root_locs heap_is_wellformed parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host
  get_host_locs get_disconnected_document get_disconnected_document_locs remove_shadow_root
  remove_shadow_root_locs known_ptrs get_parent get_parent_locs
  by(auto simp add: l_remove_shadow_root_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def instances)
declare l_remove_shadow_root_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms [instances]

subsubsection \<open>get\_root\_node\<close>

interpretation i_get_root_node_wf?: l_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf known_ptrs
  heap_is_wellformed parent_child_rel get_child_nodes get_child_nodes_locs get_disconnected_nodes
  get_disconnected_nodes_locs get_parent get_parent_locs get_ancestors get_ancestors_locs
  get_root_node get_root_node_locs
  by(simp add: l_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def instances)
declare l_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms[instances]

lemma get_ancestors_wf_is_l_get_ancestors_wf [instances]:
  "l_get_ancestors_wf heap_is_wellformed parent_child_rel known_ptr known_ptrs type_wf get_ancestors
  get_ancestors_locs get_child_nodes get_parent"
  apply(auto simp add: l_get_ancestors_wf_def l_get_ancestors_wf_axioms_def instances)[1]
  using get_ancestors_never_empty apply blast
  using get_ancestors_ok apply blast
  using get_ancestors_reads apply blast
  using get_ancestors_ptrs_in_heap apply blast
  using get_ancestors_remains_not_in_ancestors apply blast
  using get_ancestors_also_parent apply blast
  using get_ancestors_obtains_children apply blast
  using get_ancestors_parent_child_rel apply blast
  using get_ancestors_parent_child_rel apply blast
  done

lemma get_root_node_wf_is_l_get_root_node_wf [instances]:
  "l_get_root_node_wf heap_is_wellformed get_root_node type_wf known_ptr known_ptrs get_ancestors
    get_parent"
  using known_ptrs_is_l_known_ptrs
  apply(auto simp add: l_get_root_node_wf_def l_get_root_node_wf_axioms_def)[1]
  using get_root_node_ok apply blast
  using get_root_node_ptr_in_heap apply blast
  using get_root_node_root_in_heap apply blast
  using get_ancestors_same_root_node apply(blast, blast)
  using get_root_node_same_no_parent apply blast
    (* using get_root_node_not_node_same apply blast *)
  using get_root_node_parent_same apply (blast, blast)
  done

subsubsection \<open>get\_parent\_get\_host\<close>

locale l_get_parent_get_host_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M =
  l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M +
  l_get_parent_wf +
  l_get_shadow_root +
  l_get_host +
  l_get_child_nodes
begin
lemma host_shadow_root_rel_finite: "finite (a_host_shadow_root_rel h)"
proof -
  have "a_host_shadow_root_rel h = (\<Union>host \<in> fset (element_ptr_kinds h).
    (case |h \<turnstile> get_shadow_root host|\<^sub>r of
    Some shadow_root \<Rightarrow> {(cast host, cast shadow_root)} | None \<Rightarrow> {}))"
    by(auto simp add: a_host_shadow_root_rel_def split: option.splits)
  moreover have "finite (\<Union>host \<in> fset (element_ptr_kinds h).
    (case |h \<turnstile> get_shadow_root host|\<^sub>r of
    Some shadow_root \<Rightarrow> {(cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r host, cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root)} |
    None \<Rightarrow> {}))"
    by(auto split: option.splits)
  ultimately show ?thesis
    by auto
qed

lemma host_shadow_root_rel_shadow_root:
  "h \<turnstile> get_shadow_root host \<rightarrow>\<^sub>r shadow_root_option \<Longrightarrow>
   shadow_root_option = Some shadow_root \<longleftrightarrow> ((cast host, cast shadow_root) \<in> a_host_shadow_root_rel h)"
  apply(auto simp add: a_host_shadow_root_rel_def)[1]
  by(metis (mono_tags, lifting) case_prodI is_OK_returns_result_I
      l_get_shadow_root.get_shadow_root_ptr_in_heap local.l_get_shadow_root_axioms
      mem_Collect_eq pair_imageI select_result_I2)

lemma host_shadow_root_rel_host:
  "heap_is_wellformed h \<Longrightarrow> h \<turnstile> get_host shadow_root \<rightarrow>\<^sub>r host \<Longrightarrow>
  (cast host, cast shadow_root) \<in> a_host_shadow_root_rel h"
  apply(auto simp add: a_host_shadow_root_rel_def)[1]
  using shadow_root_host_dual
  by (metis (no_types, lifting) Collect_cong host_shadow_root_rel_shadow_root
      local.a_host_shadow_root_rel_def split_cong)

lemma heap_wellformed_induct_si [consumes 1, case_names step]:
  assumes "heap_is_wellformed h"
  assumes "\<And>parent. (\<And>children child. h \<turnstile> get_child_nodes parent \<rightarrow>\<^sub>r children \<Longrightarrow>
    child \<in> set children \<Longrightarrow> P (cast child)) \<Longrightarrow> (\<And>shadow_root host. parent = cast host \<Longrightarrow>
      h \<turnstile> get_shadow_root host \<rightarrow>\<^sub>r Some shadow_root \<Longrightarrow>  P (cast shadow_root)) \<Longrightarrow> P parent"
  shows "P ptr"
proof -
  fix ptr
  have "finite (parent_child_rel h \<union> a_host_shadow_root_rel h)"
    using host_shadow_root_rel_finite
    using local.CD.parent_child_rel_finite local.CD.parent_child_rel_impl
    by auto
  then
  have "wf ((parent_child_rel h \<union> a_host_shadow_root_rel h)\<inverse>)"
    using assms(1)
    apply(simp add: heap_is_wellformed_def)
    by (simp add: finite_acyclic_wf_converse local.CD.parent_child_rel_impl)
  then show "?thesis"
  proof (induct rule: wf_induct_rule)
    case (less parent)
    then show ?case
      apply(auto)[1]
      using assms host_shadow_root_rel_shadow_root local.CD.parent_child_rel_child
      by blast
  qed
qed

lemma heap_wellformed_induct_rev_si [consumes 1, case_names step]:
  assumes "heap_is_wellformed h"
  assumes "\<And>child. (\<And>parent child_node. child = cast child_node \<Longrightarrow>
    h \<turnstile> get_parent child_node \<rightarrow>\<^sub>r Some parent \<Longrightarrow> P parent) \<Longrightarrow>
    (\<And>host shadow_root. child = cast shadow_root \<Longrightarrow> h \<turnstile> get_host shadow_root \<rightarrow>\<^sub>r host \<Longrightarrow>
    P (cast host)) \<Longrightarrow> P child"
  shows "P ptr"
proof -
  fix ptr
  have "finite (parent_child_rel h \<union> a_host_shadow_root_rel h)"
    using host_shadow_root_rel_finite
    using local.CD.parent_child_rel_finite local.CD.parent_child_rel_impl
    by auto
  then
  have "wf (parent_child_rel h \<union> a_host_shadow_root_rel h)"
    using assms(1)
    apply(simp add: heap_is_wellformed_def)
    by (simp add: finite_acyclic_wf)
  then show "?thesis"
  proof (induct rule: wf_induct_rule)
    case (less parent)
    then show ?case
      apply(auto)[1]
      using parent_child_rel_parent host_shadow_root_rel_host
      using assms(1) assms(2) by auto
  qed
qed
end

interpretation i_get_parent_get_host_wf?: l_get_parent_get_host_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M
  get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs
  get_shadow_root get_shadow_root_locs get_tag_name get_tag_name_locs known_ptr type_wf
  heap_is_wellformed parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host get_host_locs
  get_disconnected_document get_disconnected_document_locs known_ptrs get_parent get_parent_locs
  by(auto simp add: l_get_parent_get_host_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def instances)
declare l_get_parent_get_host_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms [instances]

locale l_get_parent_get_host_wf =
  l_heap_is_wellformed_defs +
  l_get_parent_defs +
  l_get_shadow_root_defs +
  l_get_host_defs +
  l_get_child_nodes_defs +
  assumes heap_wellformed_induct_si [consumes 1, case_names step]:
    "heap_is_wellformed h
    \<Longrightarrow> (\<And>parent. (\<And>children child. h \<turnstile> get_child_nodes parent \<rightarrow>\<^sub>r children \<Longrightarrow> child \<in> set children
      \<Longrightarrow> P (cast child))
        \<Longrightarrow> (\<And>shadow_root host. parent = cast host \<Longrightarrow> h \<turnstile> get_shadow_root host \<rightarrow>\<^sub>r Some shadow_root
    \<Longrightarrow>  P (cast shadow_root))
      \<Longrightarrow> P parent)
    \<Longrightarrow> P ptr"
  assumes heap_wellformed_induct_rev_si [consumes 1, case_names step]:
    "heap_is_wellformed h
    \<Longrightarrow> (\<And>child. (\<And>parent child_node. child = cast child_node \<Longrightarrow>
      h \<turnstile> get_parent child_node \<rightarrow>\<^sub>r Some parent \<Longrightarrow> P parent)
      \<Longrightarrow> (\<And>host shadow_root. child = cast shadow_root \<Longrightarrow> h \<turnstile> get_host shadow_root \<rightarrow>\<^sub>r host
    \<Longrightarrow> P (cast host))
      \<Longrightarrow> P child)
    \<Longrightarrow> P ptr"

lemma l_get_parent_get_host_wf_is_get_parent_get_host_wf [instances]:
  "l_get_parent_get_host_wf heap_is_wellformed get_parent get_shadow_root get_host get_child_nodes"
  apply(auto simp add: l_get_parent_get_host_wf_def instances)[1]
  using heap_wellformed_induct_si apply metis
  using heap_wellformed_induct_rev_si apply blast
  done

subsubsection \<open>get\_host\<close>

locale l_get_host_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M =
  l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M  get_child_nodes get_child_nodes_locs get_disconnected_nodes
  get_disconnected_nodes_locs get_shadow_root get_shadow_root_locs get_tag_name get_tag_name_locs
  known_ptr type_wf heap_is_wellformed parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host
  get_host_locs +
  l_type_wf type_wf +
  l_get_host\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_shadow_root get_shadow_root_locs get_host get_host_locs type_wf +
  l_get_shadow_root type_wf get_shadow_root get_shadow_root_locs
  for known_ptr :: "(_::linorder) object_ptr \<Rightarrow> bool"
    and known_ptrs :: "(_) heap \<Rightarrow> bool"
    and type_wf :: "(_) heap \<Rightarrow> bool"
    and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"
    and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"
    and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and get_child_nodes :: "(_::linorder) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"
    and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"
    and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"
    and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"
    and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"
    and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"
    and heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"
begin

lemma get_host_ok [simp]:
  assumes "heap_is_wellformed h"
  assumes "type_wf h"
  assumes "known_ptrs h"
  assumes "shadow_root_ptr |\<in>| shadow_root_ptr_kinds h"
  shows "h \<turnstile> ok (get_host shadow_root_ptr)"
proof -
  obtain host where host: "host |\<in>| element_ptr_kinds h"
    and "|h \<turnstile> get_tag_name host|\<^sub>r \<in> safe_shadow_root_element_types"
    and shadow_root: "h \<turnstile> get_shadow_root host \<rightarrow>\<^sub>r Some shadow_root_ptr"
    using assms(1) assms(4) get_shadow_root_ok assms(2)
    apply(auto simp add: heap_is_wellformed_def a_shadow_root_valid_def)[1]
    by (smt returns_result_select_result)

  obtain host_candidates where
    host_candidates: "h \<turnstile> filter_M (\<lambda>element_ptr. Heap_Error_Monad.bind (get_shadow_root element_ptr)
      (\<lambda>shadow_root_opt. return (shadow_root_opt = Some shadow_root_ptr)))
                (sorted_list_of_set (fset (element_ptr_kinds h)))
          \<rightarrow>\<^sub>r host_candidates"
    apply(drule is_OK_returns_result_E[rotated])
    using get_shadow_root_ok assms(2)
    by(auto intro!: filter_M_is_OK_I bind_pure_I bind_is_OK_I2)
  then have "host_candidates = [host]"
    apply(rule filter_M_ex1)
        apply (simp add: host)
       apply (smt (verit) assms(1) assms(2) bind_pure_returns_result_I2 finite_fset
               host local.get_shadow_root_ok local.get_shadow_root_pure local.shadow_root_same_host
               return_returns_result returns_result_eq shadow_root sorted_list_of_set(1))
    by (simp_all add: assms(2) bind_pure_I bind_pure_returns_result_I2 host local.get_shadow_root_ok
        returns_result_eq shadow_root)
  then
  show ?thesis
    using host_candidates host assms(1) get_shadow_root_ok
    apply(auto simp add: get_host_def known_ptrs_known_ptr
        intro!: bind_is_OK_pure_I filter_M_pure_I filter_M_is_OK_I bind_pure_I split: list.splits)[1]
    using assms(2) apply blast
     apply (meson list.distinct(1) returns_result_eq)
    by (meson list.distinct(1) list.inject returns_result_eq)
qed
end

interpretation i_get_host_wf?: l_get_host_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_document
  get_disconnected_document_locs known_ptr known_ptrs type_wf get_host get_host_locs get_shadow_root
  get_shadow_root_locs get_child_nodes get_child_nodes_locs get_disconnected_nodes
  get_disconnected_nodes_locs get_tag_name get_tag_name_locs heap_is_wellformed parent_child_rel
  heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M
  by(auto simp add: l_get_host_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def instances)
declare l_get_host_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms [instances]

locale l_get_host_wf = l_heap_is_wellformed_defs + l_known_ptrs + l_type_wf + l_get_host_defs +
  assumes get_host_ok:
    "heap_is_wellformed h \<Longrightarrow> known_ptrs h \<Longrightarrow> type_wf h \<Longrightarrow>
    shadow_root_ptr |\<in>| shadow_root_ptr_kinds h \<Longrightarrow> h \<turnstile> ok (get_host shadow_root_ptr)"

lemma get_host_wf_is_l_get_host_wf [instances]:
  "l_get_host_wf heap_is_wellformed known_ptr known_ptrs type_wf get_host"
  by(auto simp add: l_get_host_wf_def l_get_host_wf_axioms_def instances)

subsubsection \<open>get\_root\_node\_si\<close>

locale l_get_root_node_si_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M =
  l_get_root_node_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M +
  l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M +
  l_get_parent_wf +
  l_get_parent_get_host_wf +
  l_get_host_wf
begin
lemma get_root_node_si_ptr_in_heap:
  assumes "h \<turnstile> ok (get_root_node_si ptr)"
  shows "ptr |\<in>| object_ptr_kinds h"
  using assms
  unfolding get_root_node_si_def
  using get_ancestors_si_ptr_in_heap
  by auto

lemma get_ancestors_si_ok:
  assumes "heap_is_wellformed h" and "known_ptrs h" and "type_wf h"
    and "ptr |\<in>| object_ptr_kinds h"
  shows "h \<turnstile> ok (get_ancestors_si ptr)"
proof (insert assms(1) assms(4), induct rule: heap_wellformed_induct_rev_si)
  case (step child)
  then show ?case
    using assms(2) assms(3)
    apply(auto simp add: get_ancestors_si_def[of child] assms(1) get_parent_parent_in_heap
        intro!: bind_is_OK_pure_I
        split: option.splits)[1]
    using local.get_parent_ok apply blast
    using get_host_ok assms(1) apply blast
    by (meson assms(1) is_OK_returns_result_I local.get_shadow_root_ptr_in_heap
        local.shadow_root_host_dual)
qed

lemma get_ancestors_si_remains_not_in_ancestors:
  assumes "heap_is_wellformed h"
    and "heap_is_wellformed h'"
    and "h \<turnstile> get_ancestors_si ptr \<rightarrow>\<^sub>r ancestors"
    and "h' \<turnstile> get_ancestors_si ptr \<rightarrow>\<^sub>r ancestors'"
    and "\<And>p children children'. h \<turnstile> get_child_nodes p \<rightarrow>\<^sub>r children
        \<Longrightarrow> h' \<turnstile> get_child_nodes p \<rightarrow>\<^sub>r children' \<Longrightarrow> set children' \<subseteq> set children"
    and "\<And>p shadow_root_option shadow_root_option'. h \<turnstile> get_shadow_root p \<rightarrow>\<^sub>r shadow_root_option \<Longrightarrow>
        h' \<turnstile> get_shadow_root p \<rightarrow>\<^sub>r shadow_root_option' \<Longrightarrow> (if shadow_root_option = None
        then shadow_root_option' = None else shadow_root_option' = None \<or>
          shadow_root_option' = shadow_root_option)"
    and "node \<notin> set ancestors"
    and object_ptr_kinds_eq3: "object_ptr_kinds h = object_ptr_kinds h'"
    and known_ptrs: "known_ptrs h"
    and type_wf: "type_wf h"
    and type_wf': "type_wf h'"
  shows "node \<notin> set ancestors'"
proof -
  have object_ptr_kinds_M_eq:
    "\<And>ptrs. h \<turnstile> object_ptr_kinds_M \<rightarrow>\<^sub>r ptrs = h' \<turnstile> object_ptr_kinds_M \<rightarrow>\<^sub>r ptrs"
    using object_ptr_kinds_eq3
    by(simp add: object_ptr_kinds_M_defs)
  then have object_ptr_kinds_eq: "|h \<turnstile> object_ptr_kinds_M|\<^sub>r = |h' \<turnstile> object_ptr_kinds_M|\<^sub>r"
    by(simp)

  show ?thesis
  proof (insert assms(1) assms(3) assms(4) assms(7), induct ptr arbitrary: ancestors ancestors'
      rule: heap_wellformed_induct_rev_si)
    case (step child)

    obtain ancestors_remains where ancestors_remains:
      "ancestors = child # ancestors_remains"
      using \<open>h \<turnstile> get_ancestors_si child \<rightarrow>\<^sub>r ancestors\<close> get_ancestors_si_never_empty
      by(auto simp add: get_ancestors_si_def[of child]
          elim!: bind_returns_result_E2
          split: option.splits)
    obtain ancestors_remains' where ancestors_remains':
      "ancestors' = child # ancestors_remains'"
      using \<open>h' \<turnstile> get_ancestors_si child \<rightarrow>\<^sub>r ancestors'\<close> get_ancestors_si_never_empty
      by(auto simp add: get_ancestors_si_def[of child]
          elim!: bind_returns_result_E2
          split: option.splits)
    have "child |\<in>| object_ptr_kinds h"
      using local.get_ancestors_si_ptr_in_heap object_ptr_kinds_eq3 step.prems(2) by fastforce

    have "node \<noteq> child"
      using ancestors_remains step.prems(3) by auto

    have 1: "\<And>p parent. h' \<turnstile> get_parent p \<rightarrow>\<^sub>r Some parent \<Longrightarrow> h \<turnstile> get_parent p \<rightarrow>\<^sub>r Some parent"
    proof -
      fix p parent
      assume "h' \<turnstile> get_parent p \<rightarrow>\<^sub>r Some parent"
      then obtain children' where
        children': "h' \<turnstile> get_child_nodes parent \<rightarrow>\<^sub>r children'" and
        p_in_children': "p \<in> set children'"
        using get_parent_child_dual by blast
      obtain children where children: "h \<turnstile> get_child_nodes parent \<rightarrow>\<^sub>r children"
        using get_child_nodes_ok assms(1) get_child_nodes_ptr_in_heap object_ptr_kinds_eq children'
          known_ptrs
        using  type_wf type_wf'
        by (metis \<open>h' \<turnstile> get_parent p \<rightarrow>\<^sub>r Some parent\<close> get_parent_parent_in_heap is_OK_returns_result_E
            local.known_ptrs_known_ptr object_ptr_kinds_eq3)
      have "p \<in> set children"
        using assms(5) children children' p_in_children'
        by blast
      then show "h \<turnstile> get_parent p \<rightarrow>\<^sub>r Some parent"
        using child_parent_dual assms(1) children known_ptrs  type_wf  by blast
    qed

    have 2: "\<And>p host. h' \<turnstile> get_host p \<rightarrow>\<^sub>r host \<Longrightarrow> h \<turnstile> get_host p \<rightarrow>\<^sub>r host"
    proof -
      fix p host
      assume "h' \<turnstile> get_host p \<rightarrow>\<^sub>r host"
      then have "h' \<turnstile> get_shadow_root host \<rightarrow>\<^sub>r Some p"
        using local.shadow_root_host_dual by blast
      then have "h \<turnstile> get_shadow_root host \<rightarrow>\<^sub>r Some p"
        by (metis assms(6) element_ptr_kinds_commutes is_OK_returns_result_I local.get_shadow_root_ok
            local.get_shadow_root_ptr_in_heap node_ptr_kinds_commutes object_ptr_kinds_eq3
            option.distinct(1) returns_result_select_result type_wf)
      then show "h \<turnstile> get_host p \<rightarrow>\<^sub>r host"
        by (metis assms(1) is_OK_returns_result_E known_ptrs local.get_host_ok
            local.get_shadow_root_shadow_root_ptr_in_heap local.shadow_root_host_dual
            local.shadow_root_same_host type_wf)

    qed

    show ?case
    proof (cases "cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r child")
      case None
      then show ?thesis
        using step(3) step(4) \<open>node \<noteq> child\<close>
        apply(auto simp add: get_ancestors_si_def[of child]
            elim!: bind_returns_result_E2
            split: option.splits)[1]
        by (metis "2" assms(1) l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M.shadow_root_same_host list.set_intros(2)
            local.l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms local.shadow_root_host_dual step.hyps(2)
            step.prems(3) type_wf)
    next
      case (Some node_child)
      then
      show ?thesis
        using step(3) step(4) \<open>node \<noteq> child\<close>
        apply(auto simp add: get_ancestors_si_def[of child]
            elim!: bind_returns_result_E2
            split: option.splits)[1]
         apply (meson "1" option.distinct(1) returns_result_eq)
        by (metis "1" list.set_intros(2) option.inject returns_result_eq step.hyps(1) step.prems(3))
    qed
  qed
qed

lemma get_ancestors_si_ptrs_in_heap:
  assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
  assumes "h \<turnstile> get_ancestors_si ptr \<rightarrow>\<^sub>r ancestors"
  assumes "ptr' \<in> set ancestors"
  shows "ptr' |\<in>| object_ptr_kinds h"
proof (insert assms(4) assms(5), induct ancestors arbitrary: ptr)
  case Nil
  then show ?case
    by(auto)
next
  case (Cons a ancestors)
  then obtain x where x: "h \<turnstile> get_ancestors_si x \<rightarrow>\<^sub>r a # ancestors"
    by(auto simp add: get_ancestors_si_def[of a] elim!: bind_returns_result_E2 split: option.splits)
  then have "x = a"
    by(auto simp add: get_ancestors_si_def[of x] elim!: bind_returns_result_E2 split: option.splits)
  then show ?case
  proof (cases "ptr' = a")
    case True
    then show ?thesis
      using Cons.hyps Cons.prems(2) get_ancestors_si_ptr_in_heap x
      using \<open>x = a\<close> by blast
  next
    case False
    obtain ptr'' where ptr'': "h \<turnstile> get_ancestors_si ptr'' \<rightarrow>\<^sub>r ancestors"
      using \<open> h \<turnstile> get_ancestors_si x \<rightarrow>\<^sub>r a # ancestors\<close> Cons.prems(2) False
      by(auto simp add: get_ancestors_si_def[of x] elim!: bind_returns_result_E2 split: option.splits)
    then show ?thesis
      using Cons.hyps Cons.prems(2) False by auto
  qed
qed

lemma get_ancestors_si_reads:
  assumes "heap_is_wellformed h"
  shows "reads get_ancestors_si_locs (get_ancestors_si node_ptr) h h'"
proof (insert assms(1), induct rule: heap_wellformed_induct_rev_si)
  case (step child)
  then show ?case
    using [[simproc del: Product_Type.unit_eq]] get_parent_reads[unfolded reads_def]
      get_host_reads[unfolded reads_def]
    apply(simp (no_asm) add: get_ancestors_si_def)
    by(auto simp add: get_ancestors_si_locs_def get_parent_reads_pointers
        intro!: reads_bind_pure reads_subset[OF check_in_heap_reads]
        reads_subset[OF return_reads] reads_subset[OF get_parent_reads]
        reads_subset[OF get_child_nodes_reads] reads_subset[OF get_host_reads]
        split: option.splits)
qed

lemma get_ancestors_si_subset:
  assumes "heap_is_wellformed h"
    and "h \<turnstile> get_ancestors_si ptr \<rightarrow>\<^sub>r ancestors"
    and "ancestor \<in> set ancestors"
    and "h \<turnstile> get_ancestors_si ancestor \<rightarrow>\<^sub>r ancestor_ancestors"
    and type_wf: "type_wf h"
    and known_ptrs: "known_ptrs h"
  shows "set ancestor_ancestors \<subseteq> set ancestors"
proof (insert assms(1) assms(2) assms(3), induct ptr arbitrary: ancestors
    rule: heap_wellformed_induct_rev_si)
  case (step child)
  have "child |\<in>| object_ptr_kinds h"
    using get_ancestors_si_ptr_in_heap step(3) by auto
      (*  then have "h \<turnstile> check_in_heap child \<rightarrow>\<^sub>r ()"
    using returns_result_select_result by force *)

  obtain tl_ancestors where tl_ancestors: "ancestors = child # tl_ancestors"
    using step(3)
    by(auto simp add: get_ancestors_si_def[of child] intro!: bind_pure_I
        elim!: bind_returns_result_E2 split: option.splits)
  show ?case
  proof (induct "cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r child")
    case None
    show ?case
    proof (induct "cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child")
      case None
      then show ?case
        using step(3) \<open>None = cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r child\<close>
        apply(auto simp add: get_ancestors_si_def[of child] elim!: bind_returns_result_E2)[1]
        by (metis (no_types, lifting) assms(4) empty_iff empty_set select_result_I2 set_ConsD
            step.prems(1) step.prems(2))
    next
      case (Some shadow_root_child)
      then
      have "shadow_root_child |\<in>| shadow_root_ptr_kinds h"
        using  \<open>child |\<in>| object_ptr_kinds h\<close>
        by (metis (no_types, lifting) shadow_root_ptr_casts_commute shadow_root_ptr_kinds_commutes)
      obtain host where host: "h \<turnstile> get_host shadow_root_child \<rightarrow>\<^sub>r host"
        using get_host_ok assms
        by (meson \<open>shadow_root_child |\<in>| shadow_root_ptr_kinds h\<close> is_OK_returns_result_E)
      then
      have "h \<turnstile> get_ancestors_si (cast host) \<rightarrow>\<^sub>r tl_ancestors"
        using Some step(3) tl_ancestors None
        by(auto simp add: get_ancestors_si_def[of child] intro!: bind_pure_returns_result_I
            elim!: bind_returns_result_E2 split: option.splits dest: returns_result_eq)
      then
      show ?case
        using step(2) Some host step(4) tl_ancestors
        by (metis (no_types, lifting) assms(4) dual_order.trans eq_iff returns_result_eq set_ConsD
            set_subset_Cons shadow_root_ptr_casts_commute step.prems(1))
    qed
  next
    case (Some child_node)
    note s1 = Some
    obtain parent_opt where parent_opt: "h \<turnstile> get_parent child_node \<rightarrow>\<^sub>r parent_opt"
      using \<open>child |\<in>| object_ptr_kinds h\<close> assms(1) Some[symmetric]
        get_parent_ok[OF type_wf known_ptrs]
      by (metis (no_types, lifting) is_OK_returns_result_E known_ptrs get_parent_ok
          l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms node_ptr_casts_commute node_ptr_kinds_commutes)
    then show ?case
    proof (induct parent_opt)
      case None
      then have "ancestors = [child]"
        using step(3) s1
        apply(simp add: get_ancestors_si_def)
        by(auto elim!: bind_returns_result_E2 split: option.splits dest: returns_result_eq)
      show ?case
        using step(3) step(4)
        apply(auto simp add: \<open>ancestors = [child]\<close>)[1]
        using assms(4) returns_result_eq by fastforce
    next
      case (Some parent)
      then
      have "h \<turnstile> get_ancestors_si parent \<rightarrow>\<^sub>r tl_ancestors"
        using s1 tl_ancestors step(3)
        by(auto simp add: get_ancestors_si_def[of child]
            elim!: bind_returns_result_E2
            split: option.splits dest: returns_result_eq)
      show ?case
        by (metis (no_types, lifting) Some.prems \<open>h \<turnstile> get_ancestors_si parent \<rightarrow>\<^sub>r tl_ancestors\<close>
            assms(4) eq_iff node_ptr_casts_commute order_trans s1 select_result_I2 set_ConsD
            set_subset_Cons step.hyps(1) step.prems(1) step.prems(2) tl_ancestors)
    qed
  qed
qed

lemma get_ancestors_si_also_parent:
  assumes "heap_is_wellformed h"
    and "h \<turnstile> get_ancestors_si some_ptr \<rightarrow>\<^sub>r ancestors"
    and "cast child \<in> set ancestors"
    and "h \<turnstile> get_parent child \<rightarrow>\<^sub>r Some parent"
    and type_wf: "type_wf h"
    and known_ptrs: "known_ptrs h"
  shows "parent \<in> set ancestors"
proof -
  obtain child_ancestors where child_ancestors:
    "h \<turnstile> get_ancestors_si (cast child) \<rightarrow>\<^sub>r child_ancestors"
    by (meson assms(1) assms(4) get_ancestors_si_ok is_OK_returns_result_I known_ptrs
        local.get_parent_ptr_in_heap node_ptr_kinds_commutes returns_result_select_result
        type_wf)
  then have "parent \<in> set child_ancestors"
    apply(simp add: get_ancestors_si_def)
    by(auto elim!: bind_returns_result_E2 split: option.splits dest!: returns_result_eq[OF assms(4)]
        get_ancestors_si_ptr)
  then show ?thesis
    using assms child_ancestors get_ancestors_si_subset by blast
qed

lemma get_ancestors_si_also_host:
  assumes "heap_is_wellformed h"
    and "h \<turnstile> get_ancestors_si some_ptr \<rightarrow>\<^sub>r ancestors"
    and "cast shadow_root \<in> set ancestors"
    and "h \<turnstile> get_host shadow_root \<rightarrow>\<^sub>r host"
    and type_wf: "type_wf h"
    and known_ptrs: "known_ptrs h"
  shows "cast host \<in> set ancestors"
proof -
  obtain child_ancestors where child_ancestors:
    "h \<turnstile> get_ancestors_si (cast shadow_root) \<rightarrow>\<^sub>r child_ancestors"
    by (meson assms(1) assms(2) assms(3) get_ancestors_si_ok get_ancestors_si_ptrs_in_heap
        is_OK_returns_result_E known_ptrs type_wf)
  then have "cast host \<in> set child_ancestors"
    apply(simp add: get_ancestors_si_def)
    by(auto elim!: bind_returns_result_E2 split: option.splits dest!: returns_result_eq[OF assms(4)]
        get_ancestors_si_ptr)
  then show ?thesis
    using assms child_ancestors get_ancestors_si_subset by blast
qed

lemma get_ancestors_si_parent_child_rel:
  assumes "heap_is_wellformed h" and "known_ptrs h" and "type_wf h"
  assumes "h \<turnstile> get_ancestors_si child \<rightarrow>\<^sub>r ancestors"
  assumes "((ptr, child) \<in> (parent_child_rel h)\<^sup>*)"
  shows "ptr \<in> set ancestors"
proof (insert assms(5), induct ptr rule: heap_wellformed_induct_si[OF assms(1)])
  case (1 ptr)
  then show ?case
  proof (cases "ptr = child")
    case True
    then show ?thesis
      using assms(4) local.get_ancestors_si_ptr by blast
  next
    case False
    obtain ptr_child where
      ptr_child: "(ptr, ptr_child) \<in> (parent_child_rel h) \<and> (ptr_child, child) \<in> (parent_child_rel h)\<^sup>*"
      using converse_rtranclE[OF 1(3)] \<open>ptr \<noteq> child\<close>
      by metis
    then obtain ptr_child_node
      where ptr_child_ptr_child_node: "ptr_child = cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr_child_node"
      using ptr_child node_ptr_casts_commute3 CD.parent_child_rel_node_ptr
      by (metis )
    then obtain children where
      children: "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children" and
      ptr_child_node: "ptr_child_node \<in> set children"
    proof -
      assume a1: "\<And>children. \<lbrakk>h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children; ptr_child_node \<in> set children\<rbrakk>
                   \<Longrightarrow> thesis"

      have "ptr |\<in>| object_ptr_kinds h"
        using CD.parent_child_rel_parent_in_heap ptr_child by blast
      moreover have "ptr_child_node \<in> set |h \<turnstile> get_child_nodes ptr|\<^sub>r"
        by (metis calculation \<open>known_ptrs h\<close> local.get_child_nodes_ok local.known_ptrs_known_ptr
            CD.parent_child_rel_child ptr_child ptr_child_ptr_child_node
            returns_result_select_result \<open>type_wf h\<close>)
      ultimately show ?thesis
        using a1 get_child_nodes_ok \<open>type_wf h\<close> \<open>known_ptrs h\<close>
        by (meson local.known_ptrs_known_ptr returns_result_select_result)
    qed
    moreover have "(cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr_child_node, child) \<in> (parent_child_rel h)\<^sup>*"
      using ptr_child ptr_child_ptr_child_node by auto
    ultimately have "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr_child_node \<in> set ancestors"
      using 1 by auto
    moreover have "h \<turnstile> get_parent ptr_child_node \<rightarrow>\<^sub>r Some ptr"
      using assms(1) children ptr_child_node child_parent_dual
      using \<open>known_ptrs h\<close> \<open>type_wf h\<close> by blast
    ultimately show ?thesis
      using get_ancestors_si_also_parent assms \<open>type_wf h\<close> by blast
  qed
qed

lemma get_ancestors_si_parent_child_host_shadow_root_rel:
  assumes "heap_is_wellformed h" and "known_ptrs h" and "type_wf h"
  assumes "h \<turnstile> get_ancestors_si child \<rightarrow>\<^sub>r ancestors"
  assumes "((ptr, child) \<in> (parent_child_rel h \<union> a_host_shadow_root_rel h)\<^sup>*)"
  shows "ptr \<in> set ancestors"
proof (insert assms(5), induct ptr rule: heap_wellformed_induct_si[OF assms(1)])
  case (1 ptr)
  then show ?case
  proof (cases "ptr = child")
    case True
    then show ?thesis
      using assms(4) local.get_ancestors_si_ptr by blast
  next
    case False

    obtain ptr_child where
      ptr_child: "(ptr, ptr_child) \<in> (parent_child_rel h \<union> local.a_host_shadow_root_rel h) \<and>
          (ptr_child, child) \<in> (parent_child_rel h \<union> local.a_host_shadow_root_rel h)\<^sup>*"
      using converse_rtranclE[OF 1(3)] \<open>ptr \<noteq> child\<close>
      by metis
    then show ?thesis
    proof(cases "(ptr, ptr_child) \<in> parent_child_rel h")
      case True

      then obtain ptr_child_node
        where ptr_child_ptr_child_node: "ptr_child = cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr_child_node"
        using ptr_child node_ptr_casts_commute3 CD.parent_child_rel_node_ptr
        by (metis)
      then obtain children where
        children: "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children" and
        ptr_child_node: "ptr_child_node \<in> set children"
      proof -
        assume a1: "\<And>children. \<lbrakk>h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children; ptr_child_node \<in> set children\<rbrakk>
                   \<Longrightarrow> thesis"

        have "ptr |\<in>| object_ptr_kinds h"
          using CD.parent_child_rel_parent_in_heap True by blast
        moreover have "ptr_child_node \<in> set |h \<turnstile> get_child_nodes ptr|\<^sub>r"
          by (metis True assms(2) assms(3) calculation local.CD.parent_child_rel_child
              local.get_child_nodes_ok local.known_ptrs_known_ptr ptr_child_ptr_child_node
              returns_result_select_result)
        ultimately show ?thesis
          using a1 get_child_nodes_ok \<open>type_wf h\<close> \<open>known_ptrs h\<close>
          by (meson local.known_ptrs_known_ptr returns_result_select_result)
      qed
      moreover have "(cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr_child_node, child) \<in>
          (parent_child_rel h \<union> local.a_host_shadow_root_rel h)\<^sup>*"
        using ptr_child True ptr_child_ptr_child_node by auto
      ultimately have "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr_child_node \<in> set ancestors"
        using 1 by auto
      moreover have "h \<turnstile> get_parent ptr_child_node \<rightarrow>\<^sub>r Some ptr"
        using assms(1) children ptr_child_node child_parent_dual
        using \<open>known_ptrs h\<close> \<open>type_wf h\<close> by blast
      ultimately show ?thesis
        using get_ancestors_si_also_parent assms \<open>type_wf h\<close> by blast
    next
      case False
      then
      obtain host where host: "ptr = cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r host"
        using ptr_child
        by(auto simp add: a_host_shadow_root_rel_def)
      then obtain shadow_root where shadow_root: "h \<turnstile> get_shadow_root host \<rightarrow>\<^sub>r Some shadow_root"
        and ptr_child_shadow_root: "ptr_child = cast shadow_root"
        using ptr_child False
        apply(auto simp add: a_host_shadow_root_rel_def)[1]
        by (metis (no_types, lifting) assms(3) local.get_shadow_root_ok select_result_I)

      moreover have "(cast shadow_root, child) \<in> (parent_child_rel h \<union> local.a_host_shadow_root_rel h)\<^sup>*"
        using ptr_child ptr_child_shadow_root by blast
      ultimately have "cast shadow_root \<in> set ancestors"
        using "1.hyps"(2) host by blast
      moreover have "h \<turnstile> get_host shadow_root \<rightarrow>\<^sub>r host"
        by (metis assms(1) assms(2) assms(3) is_OK_returns_result_E local.get_host_ok
            local.get_shadow_root_shadow_root_ptr_in_heap local.shadow_root_host_dual
            local.shadow_root_same_host shadow_root)
      ultimately show ?thesis
        using get_ancestors_si_also_host assms(1) assms(2) assms(3) assms(4) host
        by blast
    qed
  qed
qed

lemma get_root_node_si_ok:
  assumes "heap_is_wellformed h" and "known_ptrs h" and "type_wf h"
    and "ptr |\<in>| object_ptr_kinds h"
  shows "h \<turnstile> ok (get_root_node_si ptr)"
  using assms get_ancestors_si_ok
  by(auto simp add: get_root_node_si_def)

lemma get_root_node_si_root_in_heap:
  assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
  assumes "h \<turnstile> get_root_node_si ptr \<rightarrow>\<^sub>r root"
  shows "root |\<in>| object_ptr_kinds h"
  using assms
  apply(auto simp add: get_root_node_si_def elim!: bind_returns_result_E2)[1]
  by (simp add: get_ancestors_si_never_empty get_ancestors_si_ptrs_in_heap)

lemma get_root_node_si_same_no_parent:
  assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
  assumes "h \<turnstile> get_root_node_si ptr \<rightarrow>\<^sub>r cast child"
  shows "h \<turnstile> get_parent child \<rightarrow>\<^sub>r None"
proof (insert assms(1) assms(4), induct ptr rule: heap_wellformed_induct_rev_si)
  case (step c)
  then show ?case
  proof (cases "cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r c")
    case None
    then show ?thesis
      using step(3)
      by(auto simp add: get_root_node_si_def get_ancestors_si_def[of c]
          elim!: bind_returns_result_E2
          split: if_splits option.splits
          intro!: step(2) bind_pure_returns_result_I)
  next
    case (Some child_node)
    note s = this
    then obtain parent_opt where parent_opt: "h \<turnstile> get_parent child_node \<rightarrow>\<^sub>r parent_opt"
      using step(3)
      apply(auto simp add: get_root_node_si_def get_ancestors_si_def
          intro!: bind_pure_I
          elim!: bind_returns_result_E2)[1]
      by(auto split: option.splits)
    then show ?thesis
    proof(induct parent_opt)
      case None
      then show ?case
        using Some get_root_node_si_no_parent returns_result_eq step.prems by fastforce
    next
      case (Some parent)
      then show ?case
        using step(3) s
        apply(auto simp add: get_root_node_si_def get_ancestors_si_def[of c]
            elim!: bind_returns_result_E2 split: option.splits list.splits if_splits)[1]
        using assms(1) get_ancestors_si_never_empty apply blast
        by(auto simp add: get_root_node_si_def
            dest: returns_result_eq
            intro!: step(1) bind_pure_returns_result_I)
    qed
  qed
qed
end

interpretation i_get_root_node_si_wf?: l_get_root_node_si_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr known_ptrs
  get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_host get_host_locs
  get_ancestors_si get_ancestors_si_locs get_root_node_si get_root_node_si_locs
  get_disconnected_nodes get_disconnected_nodes_locs get_shadow_root get_shadow_root_locs
  get_tag_name get_tag_name_locs heap_is_wellformed parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M
  get_disconnected_document get_disconnected_document_locs
  by(auto simp add: instances l_get_root_node_si_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def)
declare l_get_root_node_si_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms [instances]

subsubsection \<open>get\_disconnected\_document\<close>

locale l_get_disconnected_document_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M =
  l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M +
  l_get_disconnected_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M +
  l_get_parent_wf +
  l_get_parent
begin

lemma get_disconnected_document_ok:
  assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
  assumes "h \<turnstile> get_parent node_ptr \<rightarrow>\<^sub>r None"
  shows "h \<turnstile> ok (get_disconnected_document node_ptr)"
proof -
  have "node_ptr |\<in>| node_ptr_kinds h"
    by (meson assms(4) is_OK_returns_result_I local.get_parent_ptr_in_heap)
  have "\<not>(\<exists>parent \<in> fset (object_ptr_kinds h). node_ptr \<in> set |h \<turnstile> get_child_nodes parent|\<^sub>r)"
    apply(auto)[1]
    using assms(4) child_parent_dual[OF assms(1)]
      assms(1) assms(2) assms(3) known_ptrs_known_ptr option.simps(3)
      returns_result_eq returns_result_select_result
    by (metis (no_types, lifting) CD.get_child_nodes_ok)
  then
  have "(\<exists>document_ptr \<in> fset (document_ptr_kinds h). node_ptr \<in>
    set |h \<turnstile> get_disconnected_nodes document_ptr|\<^sub>r)"
    using heap_is_wellformed_children_disc_nodes
    using \<open>node_ptr |\<in>| node_ptr_kinds h\<close> assms(1) by blast
  then obtain some_owner_document where
    "some_owner_document \<in> set (sorted_list_of_set (fset (document_ptr_kinds h)))" and
    "node_ptr \<in> set |h \<turnstile> get_disconnected_nodes some_owner_document|\<^sub>r"
    by auto

  have h5: "\<exists>!x. x \<in> set (sorted_list_of_set (fset (document_ptr_kinds h))) \<and>
    h \<turnstile> Heap_Error_Monad.bind (get_disconnected_nodes x)
                  (\<lambda>children. return (node_ptr \<in> set children)) \<rightarrow>\<^sub>r True"
    apply(auto intro!: bind_pure_returns_result_I)[1]
     apply (smt (verit, best) CD.get_disconnected_nodes_ok CD.get_disconnected_nodes_pure
             \<open>\<exists>document_ptr\<in>fset (document_ptr_kinds h). node_ptr \<in> set |h \<turnstile> get_disconnected_nodes
             document_ptr|\<^sub>r\<close> assms(2) bind_pure_returns_result_I return_returns_result
             returns_result_select_result)
    apply(auto elim!: bind_returns_result_E2 intro!: bind_pure_returns_result_I)[1]
    using heap_is_wellformed_one_disc_parent assms(1)
    by blast
  let ?filter_M = "filter_M
             (\<lambda>document_ptr.
                 Heap_Error_Monad.bind (get_disconnected_nodes document_ptr)
                  (\<lambda>disconnected_nodes. return (node_ptr \<in> set disconnected_nodes)))
             (sorted_list_of_set (fset (document_ptr_kinds h)))"
  have "h \<turnstile> ok (?filter_M)"
    using CD.get_disconnected_nodes_ok
    by (smt (verit) CD.get_disconnected_nodes_pure DocumentMonad.ptr_kinds_M_ptr_kinds
        DocumentMonad.ptr_kinds_ptr_kinds_M assms(2) bind_is_OK_pure_I bind_pure_I
        document_ptr_kinds_M_def filter_M_is_OK_I l_ptr_kinds_M.ptr_kinds_M_ok return_ok
        return_pure returns_result_select_result)
  then
  obtain candidates where candidates: "h \<turnstile> filter_M
             (\<lambda>document_ptr.
                 Heap_Error_Monad.bind (get_disconnected_nodes document_ptr)
                  (\<lambda>disconnected_nodes. return (node_ptr \<in> set disconnected_nodes)))
             (sorted_list_of_set (fset (document_ptr_kinds h)))
       \<rightarrow>\<^sub>r candidates"
    by auto
  have "candidates = [some_owner_document]"
    apply(rule filter_M_ex1[OF candidates \<open>some_owner_document \<in>
               set (sorted_list_of_set (fset (document_ptr_kinds h)))\<close> h5])
    using \<open>node_ptr \<in> set |h \<turnstile> get_disconnected_nodes some_owner_document|\<^sub>r\<close>
      \<open>some_owner_document \<in> set (sorted_list_of_set (fset (document_ptr_kinds h)))\<close>
    by(auto simp add: CD.get_disconnected_nodes_ok assms(2)
        intro!: bind_pure_I
        intro!: bind_pure_returns_result_I)
  then show ?thesis
    using candidates  \<open>node_ptr |\<in>| node_ptr_kinds h\<close>
    apply(auto simp add: get_disconnected_document_def
        intro!: bind_is_OK_pure_I filter_M_pure_I bind_pure_I
        split: list.splits)[1]
     apply (meson not_Cons_self2 returns_result_eq)
    by (meson list.distinct(1) list.inject returns_result_eq)
qed
end

interpretation i_get_disconnected_document_wf?: l_get_disconnected_document_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M
  get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs
  get_shadow_root get_shadow_root_locs get_tag_name get_tag_name_locs known_ptr type_wf
  heap_is_wellformed parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host get_host_locs
  get_disconnected_document get_disconnected_document_locs known_ptrs get_parent get_parent_locs
  by(auto simp add: l_get_disconnected_document_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def instances)
declare l_get_disconnected_document_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms [instances]

subsubsection \<open>get\_owner\_document\<close>

locale l_get_owner_document_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M =
  l_get_disconnected_nodes +
  l_get_child_nodes +
  l_get_owner_document\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M +
  l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M +
  l_get_parent_wf +
  l_known_ptrs +
  l_get_root_node_si_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M +
  l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M +
  assumes known_ptr_impl: "known_ptr = ShadowRootClass.known_ptr"
begin
lemma get_owner_document_disconnected_nodes:
  assumes "heap_is_wellformed h"
  assumes "h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes"
  assumes "node_ptr \<in> set disc_nodes"
  assumes known_ptrs: "known_ptrs h"
  assumes type_wf: "type_wf h"
  shows "h \<turnstile> get_owner_document (cast node_ptr) \<rightarrow>\<^sub>r document_ptr"
proof -
  have 2: "node_ptr |\<in>| node_ptr_kinds h"
    using assms
    apply(auto simp add: heap_is_wellformed_def CD.heap_is_wellformed_def CD.a_all_ptrs_in_heap_def)[1]
    using assms(1) local.heap_is_wellformed_disc_nodes_in_heap by blast
  have 3: "document_ptr |\<in>| document_ptr_kinds h"
    using assms(2) get_disconnected_nodes_ptr_in_heap by blast
  then have 4: "\<not>(\<exists>parent_ptr. parent_ptr |\<in>| object_ptr_kinds h \<and>
    node_ptr \<in> set |h \<turnstile> get_child_nodes parent_ptr|\<^sub>r)"
    using CD.distinct_lists_no_parent assms
    unfolding heap_is_wellformed_def CD.heap_is_wellformed_def by simp
  moreover have "(\<exists>document_ptr. document_ptr |\<in>| document_ptr_kinds h \<and>
        node_ptr \<in> set |h \<turnstile> get_disconnected_nodes document_ptr|\<^sub>r) \<or>
       (\<exists>parent_ptr. parent_ptr |\<in>| object_ptr_kinds h \<and>
        node_ptr \<in> set |h \<turnstile> get_child_nodes parent_ptr|\<^sub>r)"
    using assms(1) 2 "3" assms(2) assms(3) by auto
  ultimately have 0: "\<exists>!document_ptr\<in>set |h \<turnstile> document_ptr_kinds_M|\<^sub>r.
    node_ptr \<in> set |h \<turnstile> get_disconnected_nodes document_ptr|\<^sub>r"
    by (meson DocumentMonad.ptr_kinds_ptr_kinds_M assms(1) disjoint_iff
        local.get_disconnected_nodes_ok local.heap_is_wellformed_one_disc_parent
        returns_result_select_result type_wf)

  have "h \<turnstile> get_parent node_ptr \<rightarrow>\<^sub>r None"
    by (metis (mono_tags, lifting) "2" "4" known_ptrs local.get_parent_child_dual
        local.get_parent_ok local.get_parent_parent_in_heap returns_result_select_result
        select_result_I2 split_option_ex type_wf)

  then have 4: "h \<turnstile> get_root_node_si (cast node_ptr) \<rightarrow>\<^sub>r cast node_ptr"
    using get_root_node_si_no_parent
    by simp
  obtain document_ptrs where document_ptrs: "h \<turnstile> document_ptr_kinds_M \<rightarrow>\<^sub>r document_ptrs"
    by simp
  then have "h \<turnstile> ok (filter_M (\<lambda>document_ptr. do {
          disconnected_nodes \<leftarrow> get_disconnected_nodes document_ptr;
          return (((cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr)) \<in> cast ` set disconnected_nodes)
        }) document_ptrs)"
    using assms(1) get_disconnected_nodes_ok type_wf
    by(auto intro!: bind_is_OK_I2 filter_M_is_OK_I bind_pure_I)
  then obtain candidates where
    candidates: "h \<turnstile> filter_M (\<lambda>document_ptr. do {
          disconnected_nodes \<leftarrow> get_disconnected_nodes document_ptr;
          return (((cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr)) \<in> cast ` set disconnected_nodes)
        }) document_ptrs \<rightarrow>\<^sub>r candidates"
    by auto

  have filter: "filter (\<lambda>document_ptr. |h \<turnstile> do {
          disconnected_nodes \<leftarrow> get_disconnected_nodes document_ptr;
          return (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr \<in> cast ` set disconnected_nodes)
        }|\<^sub>r) document_ptrs = [document_ptr]"
    apply(rule filter_ex1)
    using 0 document_ptrs
       apply (smt (verit) DocumentMonad.ptr_kinds_ptr_kinds_M bind_pure_returns_result_I2
               local.get_disconnected_nodes_ok local.get_disconnected_nodes_pure node_ptr_inclusion
               return_returns_result select_result_I2 type_wf)
    using assms(2) assms(3)
      apply (metis (no_types, lifting) bind_pure_returns_result_I2 is_OK_returns_result_I
        local.get_disconnected_nodes_pure node_ptr_inclusion return_returns_result select_result_I2)
    using document_ptrs  3 apply(simp)
    using document_ptrs
    by simp
  have "h \<turnstile> filter_M (\<lambda>document_ptr. do {
          disconnected_nodes \<leftarrow> get_disconnected_nodes document_ptr;
          return (((cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr)) \<in> cast ` set disconnected_nodes)
        }) document_ptrs \<rightarrow>\<^sub>r [document_ptr]"
    apply(rule filter_M_filter2)
    using get_disconnected_nodes_ok document_ptrs 3  assms(1)  type_wf filter
    by(auto intro: bind_pure_I bind_is_OK_I2)

  with 4 document_ptrs have "h \<turnstile> CD.a_get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr () \<rightarrow>\<^sub>r document_ptr"
    by(auto simp add: CD.a_get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def
        intro!: bind_pure_returns_result_I filter_M_pure_I bind_pure_I split: option.splits)
  moreover have "known_ptr (cast node_ptr)"
    using known_ptrs_known_ptr[OF known_ptrs, where ptr="cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr"] 2
      known_ptrs_implies
    by(simp)
  ultimately show ?thesis
    using 2
    apply(auto simp add: CD.a_get_owner_document_tups_def get_owner_document_def
        a_get_owner_document_tups_def known_ptr_impl)[1]
    apply(split invoke_splits, (rule conjI | rule impI)+)+
        apply(drule(1) known_ptr_not_shadow_root_ptr)
        apply(drule(1) known_ptr_not_document_ptr)
        apply(drule(1)  known_ptr_not_character_data_ptr)
        apply(drule(1)  known_ptr_not_element_ptr)
        apply(simp add: NodeClass.known_ptr_defs)
    by(auto split: option.splits intro!: bind_pure_returns_result_I)
qed

lemma in_disconnected_nodes_no_parent:
  assumes "heap_is_wellformed h"
  assumes "h \<turnstile> get_parent node_ptr \<rightarrow>\<^sub>r None"
  assumes "h \<turnstile> get_owner_document (cast node_ptr) \<rightarrow>\<^sub>r owner_document"
  assumes "h \<turnstile> get_disconnected_nodes owner_document \<rightarrow>\<^sub>r disc_nodes"
  assumes "known_ptrs h"
  assumes "type_wf h"
  shows "node_ptr \<in> set disc_nodes"
proof -
  have "\<And>parent. parent |\<in>| object_ptr_kinds h \<Longrightarrow> node_ptr \<notin> set |h \<turnstile> get_child_nodes parent|\<^sub>r"
    using assms(2)
    by (meson get_child_nodes_ok assms(1) assms(5) assms(6) local.child_parent_dual
        local.known_ptrs_known_ptr option.distinct(1) returns_result_eq returns_result_select_result)
  then show ?thesis
    by (metis (no_types, lifting) assms(1) assms(2) assms(3) assms(4) assms(5) assms(6)
        is_OK_returns_result_I local.get_disconnected_nodes_ok
        local.get_owner_document_disconnected_nodes local.get_parent_ptr_in_heap
        local.heap_is_wellformed_children_disc_nodes returns_result_eq returns_result_select_result)
qed

lemma get_owner_document_owner_document_in_heap_node:
  assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
  assumes "h \<turnstile> CD.a_get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr () \<rightarrow>\<^sub>r owner_document"
  shows "owner_document |\<in>| document_ptr_kinds h"
proof -
  obtain root where
    root: "h \<turnstile> get_root_node_si (cast node_ptr) \<rightarrow>\<^sub>r root"
    using assms(4)
    by(auto simp add: CD.a_get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def
        elim!: bind_returns_result_E2
        split: option.splits)

  then show ?thesis
  proof (cases "is_document_ptr root")
    case True
    then show ?thesis
      using assms(4) root
      apply(auto simp add: CD.a_get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def elim!:  bind_returns_result_E2
          intro!: filter_M_pure_I bind_pure_I split: option.splits)[1]
       apply(drule(1) returns_result_eq) apply(auto)[1]
      using assms document_ptr_kinds_commutes get_root_node_si_root_in_heap
      by blast
  next
    case False
    have "known_ptr root"
      using assms local.get_root_node_si_root_in_heap local.known_ptrs_known_ptr root by blast
    have "root |\<in>| object_ptr_kinds h"
      using root
      using assms local.get_root_node_si_root_in_heap
      by blast

    have "\<not>is_shadow_root_ptr root"
      using root
      using local.get_root_node_si_root_not_shadow_root by blast
    then have "is_node_ptr_kind root"
      using False \<open>known_ptr root\<close> \<open>root |\<in>| object_ptr_kinds h\<close>
      apply(simp add: known_ptr_impl known_ptr_defs DocumentClass.known_ptr_defs
          CharacterDataClass.known_ptr_defs ElementClass.known_ptr_defs NodeClass.known_ptr_defs)
      using is_node_ptr_kind_none by force
    then
    have "(∃document_ptr ∈ fset (document_ptr_kinds h).
root ∈ cast ` set |h ⊨ get_disconnected_nodes document_ptr|_r)"
      using local.child_parent_dual local.get_child_nodes_ok local.get_root_node_si_same_no_parent
        local.heap_is_wellformed_children_disc_nodes local.known_ptrs_known_ptr node_ptr_casts_commute3
        node_ptr_inclusion node_ptr_kinds_commutes option.distinct(1) returns_result_eq
        returns_result_select_result root
      by (metis (no_types, opaque_lifting) assms ‹root |∈| object_ptr_kinds h›)
    then obtain some_owner_document where
      "some_owner_document |∈| document_ptr_kinds h" and
      "root ∈ cast ` set |h ⊨ get_disconnected_nodes some_owner_document|_r"
      by auto
    then
    obtain candidates where
      candidates: "h ⊨ filter_M
             (λdocument_ptr.
                 Heap_Error_Monad.bind (get_disconnected_nodes document_ptr)
                  (λdisconnected_nodes. return (root ∈ cast_n_o_d_e_{_}_p_t_r₂_o_b_j_e_c_t_{_}_p_t_r ` set disconnected_nodes)))
             (sorted_list_of_set (fset (document_ptr_kinds h)))
       →_r candidates"
      by (metis (no_types, lifting) assms bind_is_OK_I2 bind_pure_I filter_M_is_OK_I finite_fset
          is_OK_returns_result_E local.get_disconnected_nodes_ok local.get_disconnected_nodes_pure
          return_ok return_pure sorted_list_of_set(1))
    then have "some_owner_document ∈ set candidates"
      apply(rule filter_M_in_result_if_ok)
      using ‹some_owner_document |∈| document_ptr_kinds h›
        ‹root ∈ cast ` set |h ⊨ get_disconnected_nodes some_owner_document|_r›
        apply(auto intro!: bind_pure_I bind_pure_returns_result_I)[1]
      using ‹some_owner_document |∈| document_ptr_kinds h›
        ‹root ∈ cast ` set |h ⊨ get_disconnected_nodes some_owner_document|_r›
       apply(auto intro!: bind_pure_I bind_pure_returns_result_I)[1]
      using ‹some_owner_document |∈| document_ptr_kinds h›
        ‹root ∈ cast ` set |h ⊨ get_disconnected_nodes some_owner_document|_r›
      apply(auto simp add: assms local.get_disconnected_nodes_ok
          intro!: bind_pure_I bind_pure_returns_result_I)[1]
      done
    then have "candidates ≠ []"
      by auto
    then have "owner_document ∈ set candidates"
      using assms(4) root
      apply(auto simp add: CD.a_get_owner_document_n_o_d_e_{_}_p_t_{r_def} elim!:  bind_returns_result_E2
          intro!: filter_M_pure_I bind_pure_I split: option.splits)[1]
       apply (metis candidates list.set_sel(1) returns_result_eq)
      by (metis ‹is_node_ptr_kind root› node_ptr_no_document_ptr_cast returns_result_eq)

    then show ?thesis
      using candidates
      by (meson bind_pure_I bind_returns_result_E2 filter_M_holds_for_result is_OK_returns_result_I
          local.get_disconnected_nodes_ptr_in_heap local.get_disconnected_nodes_pure return_pure)
  qed
qed

lemma get_owner_document_owner_document_in_heap:
  assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
  assumes "h ⊨ get_owner_document ptr →_r owner_document"
  shows "owner_document |∈| document_ptr_kinds h"
  using assms
  apply(auto simp add: get_owner_document_def a_get_owner_document_tups_def
      CD.a_get_owner_document_tups_def)[1]
  apply(split invoke_split_asm)+
proof -
  assume "h ⊨ invoke [] ptr () →_r owner_document"
  then show "owner_document |∈| document_ptr_kinds h"
    by (meson invoke_empty is_OK_returns_result_I)
next
  assume "h ⊨ Heap_Error_Monad.bind (check_in_heap ptr)
          (λ_. (CD.a_get_owner_document_d_o_c_u_m_e_n_t_{_}_p_t_r ∘ the ∘ cast_o_b_j_e_c_t_{_}_p_t_r₂_d_o_c_u_m_e_n_t_{_}_p_t_r) ptr ())
    →_r owner_document"
  then show "owner_document |∈| document_ptr_kinds h"
    by(auto simp add: CD.a_get_owner_document_d_o_c_u_m_e_n_t_{_}_p_t_{r_def}
        elim!: bind_returns_result_E2
        split: if_splits)
next
  assume 0: "heap_is_wellformed h"
    and 1: "type_wf h"
    and 2: "known_ptrs h"
    and 3: "¬ is_element_ptr_o_b_j_e_c_t_{_}_p_t_r ptr"
    and 4: "is_character_data_ptr_o_b_j_e_c_t_{_}_p_t_r ptr"
    and 5: "h ⊨ Heap_Error_Monad.bind (check_in_heap ptr)
          (λ_. (CD.a_get_owner_document_n_o_d_e_{_}_p_t_r ∘ the ∘ cast_o_b_j_e_c_t_{_}_p_t_r₂_n_o_d_e_{_}_p_t_r) ptr ()) →_r owner_document"
  then show ?thesis
    by (metis bind_returns_result_E2 check_in_heap_pure comp_apply
        get_owner_document_owner_document_in_heap_node)
next
  assume 0: "heap_is_wellformed h"
    and 1: "type_wf h"
    and 2: "known_ptrs h"
    and 3: "is_element_ptr_o_b_j_e_c_t_{_}_p_t_r ptr"
    and 4: "h ⊨ Heap_Error_Monad.bind (check_in_heap ptr)
          (λ_. (CD.a_get_owner_document_n_o_d_e_{_}_p_t_r ∘ the ∘ cast_o_b_j_e_c_t_{_}_p_t_r₂_n_o_d_e_{_}_p_t_r) ptr ()) →_r owner_document"
  then show ?thesis
    by (metis bind_returns_result_E2 check_in_heap_pure comp_apply
        get_owner_document_owner_document_in_heap_node)
next
  assume 0: "heap_is_wellformed h"
    and 1: "type_wf h"
    and 2: "known_ptrs h"
    and 3: "¬ is_element_ptr_o_b_j_e_c_t_{_}_p_t_r ptr"
    and 4: "¬ is_character_data_ptr_o_b_j_e_c_t_{_}_p_t_r ptr"
    and 5: "¬ is_document_ptr_o_b_j_e_c_t_{_}_p_t_r ptr"
    and 6: "is_shadow_root_ptr_o_b_j_e_c_t_{_}_p_t_r ptr"
    and 7: "h ⊨ Heap_Error_Monad.bind (check_in_heap ptr)
            (λ_. (local.a_get_owner_document_s_h_a_d_o_w_{_}_r_o_o_t_{_}_p_t_r ∘ the ∘ cast_o_b_j_e_c_t_{_}_p_t_r₂_s_h_a_d_o_w_{_}_r_o_o_t_{_}_p_t_r) ptr ())
             →_r owner_document"
  then show "owner_document |∈| document_ptr_kinds h"
    apply(auto simp add: CD.a_get_owner_document_d_o_c_u_m_e_n_t_{_}_p_t_{r_def} a_get_owner_document_s_h_a_d_o_w_{_}_r_o_o_t_{_}_p_t_{r_def}
        intro!: filter_M_pure_I bind_pure_I
        elim!: bind_returns_result_E2
        split: if_splits option.splits)[1]
    using get_owner_document_owner_document_in_heap_node by blast
qed

lemma get_owner_document_ok:
  assumes "heap_is_wellformed h"  "known_ptrs h" "type_wf h"
  assumes "ptr |∈| object_ptr_kinds h"
  shows "h ⊨ ok (get_owner_document ptr)"
proof -
  have "known_ptr ptr"
    using assms(2) assms(4) local.known_ptrs_known_ptr
    by blast
  then show ?thesis
    apply(simp add: get_owner_document_def a_get_owner_document_tups_def CD.a_get_owner_document_tups_def)
    apply(split invoke_splits, (rule conjI | rule impI)+)+
  proof -
    assume 0: "known_ptr ptr"
      and 1: "¬ is_element_ptr_o_b_j_e_c_t_{_}_p_t_r ptr"
      and 2: "¬ is_character_data_ptr_o_b_j_e_c_t_{_}_p_t_r ptr"
      and 3: "¬ is_document_ptr_o_b_j_e_c_t_{_}_p_t_r ptr"
      and 4: "¬ is_shadow_root_ptr_o_b_j_e_c_t_{_}_p_t_r ptr"
    then show "h ⊨ ok invoke [] ptr ()"
      using NodeClass.a_known_ptr_def known_ptr_not_character_data_ptr known_ptr_not_document_ptr
        known_ptr_not_shadow_root_ptr known_ptr_not_element_ptr known_ptr_impl
      by blast
  next
    assume 0: "known_ptr ptr"
      and 1: "¬ is_element_ptr_o_b_j_e_c_t_{_}_p_t_r ptr"
      and 2: "¬ is_character_data_ptr_o_b_j_e_c_t_{_}_p_t_r ptr"
      and 3: "¬ is_document_ptr_o_b_j_e_c_t_{_}_p_t_r ptr"
    then show "is_shadow_root_ptr_o_b_j_e_c_t_{_}_p_t_r ptr ⟶ h ⊨ ok Heap_Error_Monad.bind (check_in_heap ptr)
    (λ_. (local.a_get_owner_document_s_h_a_d_o_w_{_}_r_o_o_t_{_}_p_t_r ∘ the ∘ cast_o_b_j_e_c_t_{_}_p_t_r₂_s_h_a_d_o_w_{_}_r_o_o_t_{_}_p_t_r) ptr ())"
      using assms(1) assms(2) assms(3) assms(4)
      by(auto simp add: local.get_host_ok get_root_node_def
          CD.a_get_owner_document_d_o_c_u_m_e_n_t_{_}_p_t_{r_def} CD.a_get_owner_document_n_o_d_e_{_}_p_t_{r_def}
          a_get_owner_document_s_h_a_d_o_w_{_}_r_o_o_t_{_}_p_t_{r_def}
          intro!: bind_is_OK_pure_I filter_M_pure_I bind_pure_I filter_M_is_OK_I get_root_node_si_ok
          get_disconnected_nodes_ok
          intro!: local.get_shadow_root_ptr_in_heap local.shadow_root_host_dual
          split: option.splits)
  next
    show "is_document_ptr_o_b_j_e_c_t_{_}_p_t_r ptr ⟶ h ⊨ ok Heap_Error_Monad.bind (check_in_heap ptr)
    (λ_. (local.CD.a_get_owner_document_d_o_c_u_m_e_n_t_{_}_p_t_r ∘ the ∘ cast_o_b_j_e_c_t_{_}_p_t_r₂_d_o_c_u_m_e_n_t_{_}_p_t_r) ptr ())"
      using assms(4)
      by(auto simp add: CD.a_get_owner_document_d_o_c_u_m_e_n_t_{_}_p_t_{r_def} split: option.splits)
  next
    show "is_character_data_ptr_o_b_j_e_c_t_{_}_p_t_r ptr ⟶ h ⊨ ok Heap_Error_Monad.bind (check_in_heap ptr)
    (λ_. (local.CD.a_get_owner_document_n_o_d_e_{_}_p_t_r ∘ the ∘ cast_o_b_j_e_c_t_{_}_p_t_r₂_n_o_d_e_{_}_p_t_r) ptr ())"
      using assms(1) assms(2) assms(3) assms(4)
      by(auto simp add: local.get_host_ok get_root_node_def
          CD.a_get_owner_document_d_o_c_u_m_e_n_t_{_}_p_t_{r_def} CD.a_get_owner_document_n_o_d_e_{_}_p_t_{r_def}
          a_get_owner_document_s_h_a_d_o_w_{_}_r_o_o_t_{_}_p_t_{r_def}
          intro!: bind_is_OK_pure_I filter_M_pure_I bind_pure_I filter_M_is_OK_I get_root_node_si_ok
          get_disconnected_nodes_ok
          intro!: local.get_shadow_root_ptr_in_heap local.shadow_root_host_dual split: option.splits)
  next
    show "is_element_ptr_o_b_j_e_c_t_{_}_p_t_r ptr ⟶ h ⊨ ok Heap_Error_Monad.bind (check_in_heap ptr)
    (λ_. (local.CD.a_get_owner_document_n_o_d_e_{_}_p_t_r ∘ the ∘ cast_o_b_j_e_c_t_{_}_p_t_r₂_n_o_d_e_{_}_p_t_r) ptr ())"
      using assms(1) assms(2) assms(3) assms(4)
      by(auto simp add: local.get_host_ok get_root_node_def
          CD.a_get_owner_document_d_o_c_u_m_e_n_t_{_}_p_t_{r_def} CD.a_get_owner_document_n_o_d_e_{_}_p_t_{r_def}
          a_get_owner_document_s_h_a_d_o_w_{_}_r_o_o_t_{_}_p_t_{r_def}
          intro!: bind_is_OK_pure_I filter_M_pure_I bind_pure_I filter_M_is_OK_I get_root_node_si_ok
          get_disconnected_nodes_ok
          intro!: local.get_shadow_root_ptr_in_heap local.shadow_root_host_dual
          split: option.splits)
  qed
qed
end
interpretation i_get_owner_document_wf?: l_get_owner_document_wf_S_h_a_d_o_w_{_}_D_O_M
  type_wf get_disconnected_nodes get_disconnected_nodes_locs known_ptr
  get_child_nodes get_child_nodes_locs DocumentClass.known_ptr get_parent get_parent_locs
  get_root_node_si get_root_node_si_locs CD.a_get_owner_document get_host get_host_locs
  get_owner_document get_shadow_root get_shadow_root_locs get_tag_name get_tag_name_locs
  heap_is_wellformed parent_child_rel heap_is_wellformed_C_o_r_e_{_}_D_O_M get_disconnected_document
  get_disconnected_document_locs known_ptrs get_ancestors_si get_ancestors_si_locs
  by(auto simp add: l_get_owner_document_wf_S_h_a_d_o_w_{_}_D_O_{M_def} l_get_owner_document_wf_S_h_a_d_o_w_{_}_D_O_{M_axioms_def}
      instances)
declare l_get_owner_document_wf_S_h_a_d_o_w_{_}_D_O_{M_axioms} [instances]

lemma get_owner_document_wf_is_l_get_owner_document_wf [instances]: "l_get_owner_document_wf
  heap_is_wellformed type_wf known_ptr known_ptrs get_disconnected_nodes get_owner_document
  get_parent"
  apply(auto simp add: l_get_owner_document_wf_def l_get_owner_document_wf_axioms_def instances)[1]
  using get_owner_document_disconnected_nodes apply fast
  using in_disconnected_nodes_no_parent apply fast
  using get_owner_document_owner_document_in_heap apply fast
  using get_owner_document_ok apply fast
  done

subsubsection ‹remove\_child›

locale l_remove_child_wf2_S_h_a_d_o_w_{_}_D_O_M =
  l_set_disconnected_nodes_get_disconnected_nodes +
  l_get_child_nodes +
  l_heap_is_wellformed_S_h_a_d_o_w_{_}_D_O_M +
  l_get_owner_document_wf_S_h_a_d_o_w_{_}_D_O_M +
  l_remove_child_C_o_r_e_{_}_D_O_M +
  l_set_child_nodes_get_shadow_root +
  l_set_disconnected_nodes_get_shadow_root +
  l_set_child_nodes_get_tag_name +
  l_set_disconnected_nodes_get_tag_name +
  CD: l_remove_child_wf2_C_o_r_e_{_}_D_O_M _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ heap_is_wellformed_C_o_r_e_{_}_D_O_M
begin
lemma remove_child_preserves_type_wf:
  assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
  assumes "h ⊨ remove_child ptr child →_h h'"
  shows "type_wf h'"
  using CD.remove_child_heap_is_wellformed_preserved(1) assms
  unfolding heap_is_wellformed_def
  by auto

lemma remove_child_preserves_known_ptrs:
  assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
  assumes "h ⊨ remove_child ptr child →_h h'"
  shows "known_ptrs h'"
  using CD.remove_child_heap_is_wellformed_preserved(2) assms
  unfolding heap_is_wellformed_def
  by auto

lemma remove_child_heap_is_wellformed_preserved:
  assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
  assumes "h ⊨ remove_child ptr child →_h h'"
  shows "heap_is_wellformed h'"
proof -
  have "heap_is_wellformed_C_o_r_e_{_}_D_O_M h'"
    using CD.remove_child_heap_is_wellformed_preserved(3) assms
    unfolding heap_is_wellformed_def
    by auto

  have shadow_root_eq: "∧ptr' shadow_root_ptr_opt. h ⊨ get_shadow_root ptr' →_r shadow_root_ptr_opt =
    h' ⊨ get_shadow_root ptr' →_r shadow_root_ptr_opt"
    using get_shadow_root_reads remove_child_writes assms(4)
    apply(rule reads_writes_preserved)
    by(auto simp add: remove_child_locs_def set_child_nodes_get_shadow_root
        set_disconnected_nodes_get_shadow_root)
  then
  have shadow_root_eq2: "∧ptr'. |h ⊨ get_shadow_root ptr'|_r = |h' ⊨ get_shadow_root ptr'|_r"
    by (meson select_result_eq)

  have tag_name_eq: "∧ptr' tag. h ⊨ get_tag_name ptr' →_r tag = h' ⊨ get_tag_name ptr' →_r tag"
    using get_tag_name_reads remove_child_writes assms(4)
    apply(rule reads_writes_preserved)
    by(auto simp add: remove_child_locs_def set_child_nodes_get_tag_name
        set_disconnected_nodes_get_tag_name)
  then
  have tag_name_eq2: "∧ptr'. |h ⊨ get_tag_name ptr'|_r = |h' ⊨ get_tag_name ptr'|_r"
    by (meson select_result_eq)

  have object_ptr_kinds_eq: "object_ptr_kinds h = object_ptr_kinds h'"
    apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h = object_ptr_kinds h'",
          OF remove_child_writes assms(4)])
    unfolding remove_child_locs_def
    using set_disconnected_nodes_pointers_preserved set_child_nodes_pointers_preserved
    by (auto simp add: reflp_def transp_def)

  have shadow_root_ptr_kinds_eq: "shadow_root_ptr_kinds h = shadow_root_ptr_kinds h'"
    using object_ptr_kinds_eq
    by(auto simp add: shadow_root_ptr_kinds_def)
  have element_ptr_kinds_eq: "element_ptr_kinds h = element_ptr_kinds h'"
    using object_ptr_kinds_eq
    by(auto simp add: element_ptr_kinds_def node_ptr_kinds_def)

  have "parent_child_rel h' ⊆ parent_child_rel h"
    using ‹heap_is_wellformed h› heap_is_wellformed_def
    using CD.remove_child_parent_child_rel_subset
    using ‹known_ptrs h› ‹type_wf h› assms(4)
    by simp

  show ?thesis
    using ‹heap_is_wellformed h›
    using ‹heap_is_wellformed_C_o_r_e_{_}_D_O_M h'› ‹parent_child_rel h' ⊆ parent_child_rel h›
    apply(auto simp add: heap_is_wellformed_def CD.heap_is_wellformed_def CD.acyclic_heap_def
        a_host_shadow_root_rel_def a_all_ptrs_in_heap_def a_distinct_lists_def a_shadow_root_valid_def
        object_ptr_kinds_eq element_ptr_kinds_eq shadow_root_ptr_kinds_eq  shadow_root_eq shadow_root_eq2
        tag_name_eq tag_name_eq2)[1]
    by (meson acyclic_subset order_refl sup_mono)
qed

lemma remove_preserves_type_wf:
  assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
  assumes "h ⊨ remove child →_h h'"
  shows "type_wf h'"
  using CD.remove_heap_is_wellformed_preserved(1) assms
  unfolding heap_is_wellformed_def
  by auto

lemma remove_preserves_known_ptrs:
  assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
  assumes "h ⊨ remove child →_h h'"
  shows "known_ptrs h'"
  using CD.remove_heap_is_wellformed_preserved(2) assms
  unfolding heap_is_wellformed_def
  by auto

lemma remove_heap_is_wellformed_preserved:
  assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
  assumes "h ⊨ remove child →_h h'"
  shows "heap_is_wellformed h'"
  using assms
  by(auto simp add: remove_def elim!: bind_returns_heap_E2
      intro: remove_child_heap_is_wellformed_preserved
      split: option.splits)

lemma remove_child_removes_child:
  "heap_is_wellformed h ==> h ⊨ remove_child ptr' child →_h h'
    ==> h' ⊨ get_child_nodes ptr →_r children
    ==> known_ptrs h ==> type_wf h
    ==> child ∉ set children"
  using CD.remove_child_removes_child local.heap_is_wellformed_def by blast
lemma remove_child_removes_first_child: "heap_is_wellformed h ==> type_wf h ==> known_ptrs h ==>
    h ⊨ get_child_nodes ptr →_r node_ptr # children ==> h ⊨ remove_child ptr node_ptr →_h h' ==>
    h' ⊨ get_child_nodes ptr →_r children"
  using CD.remove_child_removes_first_child local.heap_is_wellformed_def by blast
lemma remove_removes_child: "heap_is_wellformed h ==> type_wf h ==> known_ptrs h ==>
    h ⊨ get_child_nodes ptr →_r node_ptr # children ==> h ⊨ remove node_ptr →_h h' ==>
    h' ⊨ get_child_nodes ptr →_r children"
  using CD.remove_removes_child local.heap_is_wellformed_def by blast
lemma remove_for_all_empty_children: "heap_is_wellformed h ==> type_wf h ==> known_ptrs h ==>
    h ⊨ get_child_nodes ptr →_r children ==> h ⊨ forall_M remove children →_h h' ==>
    h' ⊨ get_child_nodes ptr →_r []"
  using CD.remove_for_all_empty_children local.heap_is_wellformed_def by blast
end

interpretation i_remove_child_wf2?: l_remove_child_wf2_S_h_a_d_o_w_{_}_D_O_M
  type_wf get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes
  set_disconnected_nodes_locs known_ptr get_child_nodes get_child_nodes_locs get_shadow_root
  get_shadow_root_locs get_tag_name get_tag_name_locs heap_is_wellformed parent_child_rel
  heap_is_wellformed_C_o_r_e_{_}_D_O_M get_host get_host_locs get_disconnected_document
  get_disconnected_document_locs DocumentClass.known_ptr get_parent get_parent_locs get_root_node_si
  get_root_node_si_locs CD.a_get_owner_document get_owner_document known_ptrs get_ancestors_si
  get_ancestors_si_locs set_child_nodes set_child_nodes_locs remove_child remove_child_locs remove
  by(auto simp add: l_remove_child_wf2_S_h_a_d_o_w_{_}_D_O_{M_def} instances)
declare l_remove_child_wf2_S_h_a_d_o_w_{_}_D_O_{M_axioms} [instances]

lemma remove_child_wf2_is_l_remove_child_wf2 [instances]:
  "l_remove_child_wf2 type_wf known_ptr known_ptrs remove_child heap_is_wellformed get_child_nodes remove"
  apply(auto simp add: l_remove_child_wf2_def l_remove_child_wf2_axioms_def instances)[1]
  using remove_child_preserves_type_wf apply fast
  using remove_child_preserves_known_ptrs apply fast
  using remove_child_heap_is_wellformed_preserved apply (fast)
  using remove_preserves_type_wf apply fast
  using remove_preserves_known_ptrs apply fast
  using remove_heap_is_wellformed_preserved apply (fast)
  using remove_child_removes_child apply fast
  using remove_child_removes_first_child apply fast
  using remove_removes_child apply fast
  using remove_for_all_empty_children apply fast
  done

subsubsection ‹adopt\_node›

locale l_adopt_node_wf2_S_h_a_d_o_w_{_}_D_O_M =
  l_get_child_nodes +
  l_get_disconnected_nodes +
  l_set_child_nodes_get_shadow_root +
  l_set_disconnected_nodes_get_shadow_root +
  l_set_child_nodes_get_tag_name +
  l_set_disconnected_nodes_get_tag_name +
  l_heap_is_wellformed_S_h_a_d_o_w_{_}_D_O_M +
  l_get_root_node +
  l_set_disconnected_nodes_get_child_nodes +
  l_get_owner_document_wf +
  l_remove_child_wf2 +
  l_adopt_node_wf_C_o_r_e_{_}_D_O_M +
  l_adopt_node_C_o_r_e_{_}_D_O_M +
  l_get_parent_wf_C_o_r_e_{_}_D_O_M
begin

lemma adopt_node_removes_child:
  assumes wellformed: "heap_is_wellformed h"
    and adopt_node: "h ⊨ adopt_node owner_document node_ptr →_h h2"
    and children: "h2 ⊨ get_child_nodes ptr →_r children"
    and known_ptrs: "known_ptrs h"
    and type_wf: "type_wf h"
  shows "node_ptr ∉ set children"
proof -
  obtain old_document parent_opt h' where
    old_document: "h ⊨ get_owner_document (cast node_ptr) →_r old_document" and
    parent_opt: "h ⊨ get_parent node_ptr →_r parent_opt" and
    h': "h ⊨ (case parent_opt of Some parent ==> remove_child parent node_ptr | None ==> return () ) →_h h'"
    using adopt_node
    by(auto simp add: adopt_node_def elim!: bind_returns_heap_E bind_returns_heap_E2[rotated,
          OF get_owner_document_pure, rotated] bind_returns_heap_E2[rotated, OF get_parent_pure, rotated]
        bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated] split: if_splits)

  then have "h' ⊨ get_child_nodes ptr →_r children"
    using adopt_node
    apply(auto simp add: adopt_node_def  dest!: bind_returns_heap_E3[rotated, OF old_document, rotated]
        bind_returns_heap_E3[rotated, OF parent_opt, rotated] elim!: bind_returns_heap_E4[rotated, OF h', rotated])[1]
    apply(auto split: if_splits elim!: bind_returns_heap_E
        bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated])[1]
     apply (simp add: set_disconnected_nodes_get_child_nodes children
        reads_writes_preserved[OF get_child_nodes_reads set_disconnected_nodes_writes])
    using children by blast
  show ?thesis
  proof(insert parent_opt h', induct parent_opt)
    case None
    then show ?case
      using child_parent_dual wellformed known_ptrs type_wf
        ‹h' ⊨ get_child_nodes ptr →_r children› returns_result_eq by fastforce
  next
    case (Some option)
    then show ?case
      using remove_child_removes_child ‹h' ⊨ get_child_nodes ptr →_r children› known_ptrs type_wf
        wellformed
      by auto
  qed
qed

lemma adopt_node_preserves_wellformedness:
  assumes "heap_is_wellformed h"
    and "h ⊨ adopt_node document_ptr child →_h h'"
    and known_ptrs: "known_ptrs h"
    and type_wf: "type_wf h"
  shows "heap_is_wellformed h'" and "known_ptrs h'" and "type_wf h'"
proof -
  obtain old_document parent_opt h2 where
    old_document: "h ⊨ get_owner_document (cast child) →_r old_document" and
    parent_opt: "h ⊨ get_parent child →_r parent_opt" and
    h2: "h ⊨ (case parent_opt of Some parent ==> remove_child parent child | None ==> return ()) →_h h2" and
    h': "h2 ⊨ (if document_ptr ≠ old_document then do {
        old_disc_nodes ← get_disconnected_nodes old_document;
        set_disconnected_nodes old_document (remove1 child old_disc_nodes);
        disc_nodes ← get_disconnected_nodes document_ptr;
        set_disconnected_nodes document_ptr (child # disc_nodes)
      } else do {
        return ()
      }) →_h h'"
    using assms(2)
    by(auto simp add: adopt_node_def elim!: bind_returns_heap_E
        dest!: pure_returns_heap_eq[rotated, OF get_owner_document_pure]
        pure_returns_heap_eq[rotated, OF get_parent_pure])

  have object_ptr_kinds_h_eq3: "object_ptr_kinds h = object_ptr_kinds h2"
    using h2 apply(simp split: option.splits)
    apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h = object_ptr_kinds h'",
          OF remove_child_writes])
    using remove_child_pointers_preserved
    by (auto simp add: reflp_def transp_def)
  then have object_ptr_kinds_M_eq_h:
    "∧ptrs. h ⊨ object_ptr_kinds_M →_r ptrs = h2 ⊨ object_ptr_kinds_M →_r ptrs"
    unfolding object_ptr_kinds_M_defs by simp
  then have object_ptr_kinds_eq_h: "|h ⊨ object_ptr_kinds_M|_r = |h2 ⊨ object_ptr_kinds_M|_r"
    by simp
  then have node_ptr_kinds_eq_h: "|h ⊨ node_ptr_kinds_M|_r = |h2 ⊨ node_ptr_kinds_M|_r"
    using node_ptr_kinds_M_eq by blast

  have wellformed_h2: "heap_is_wellformed h2"
    using h2 remove_child_heap_is_wellformed_preserved known_ptrs type_wf
    by (metis (no_types, lifting) assms(1) option.case_eq_if pure_returns_heap_eq return_pure)
  have "type_wf h2"
    using h2 remove_child_preserves_type_wf assms
    by(auto split: option.splits)
  have "known_ptrs h2"
    using h2 remove_child_preserves_known_ptrs assms
    by(auto split: option.splits)

  then have "heap_is_wellformed h' ∧ known_ptrs h' ∧ type_wf h'"
  proof(cases "document_ptr = old_document")
    case True
    then show "heap_is_wellformed h' ∧ known_ptrs h' ∧ type_wf h'"
      using h' wellformed_h2 ‹known_ptrs h2› ‹type_wf h2› by auto
  next
    case False
    then obtain h3 old_disc_nodes disc_nodes_document_ptr_h3 where
      docs_neq: "document_ptr ≠ old_document" and
      old_disc_nodes: "h2 ⊨ get_disconnected_nodes old_document →_r old_disc_nodes" and
      h3: "h2 ⊨ set_disconnected_nodes old_document (remove1 child old_disc_nodes) →_h h3" and
      disc_nodes_document_ptr_h3: "h3 ⊨ get_disconnected_nodes document_ptr →_r disc_nodes_document_ptr_h3" and
      h': "h3 ⊨ set_disconnected_nodes document_ptr (child # disc_nodes_document_ptr_h3) →_h h'"
      using h'
      by(auto elim!: bind_returns_heap_E
          bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated] )

    have object_ptr_kinds_h2_eq3: "object_ptr_kinds h2 = 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 h3])
      using set_disconnected_nodes_pointers_preserved set_child_nodes_pointers_preserved
      by (auto simp add: reflp_def transp_def)
    then have object_ptr_kinds_M_eq_h2:
      "∧ptrs. h2 ⊨ object_ptr_kinds_M →_r ptrs = h3 ⊨ object_ptr_kinds_M →_r ptrs"
      by(simp add: object_ptr_kinds_M_defs)
    then have object_ptr_kinds_eq_h2: "|h2 ⊨ object_ptr_kinds_M|_r = |h3 ⊨ object_ptr_kinds_M|_r"
      by(simp)
    then have node_ptr_kinds_eq_h2: "|h2 ⊨ node_ptr_kinds_M|_r = |h3 ⊨ node_ptr_kinds_M|_r"
      using node_ptr_kinds_M_eq by blast
    then have node_ptr_kinds_eq3_h2: "node_ptr_kinds h2 = node_ptr_kinds h3"
      by auto
    have document_ptr_kinds_eq2_h2: "|h2 ⊨ document_ptr_kinds_M|_r = |h3 ⊨ document_ptr_kinds_M|_r"
      using object_ptr_kinds_eq_h2 document_ptr_kinds_M_eq by auto
    then have document_ptr_kinds_eq3_h2: "document_ptr_kinds h2 = document_ptr_kinds h3"
      using object_ptr_kinds_eq_h2 document_ptr_kinds_M_eq by auto
    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_disconnected_nodes_writes h3
      apply(rule reads_writes_preserved)
      by (simp add: set_disconnected_nodes_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 object_ptr_kinds_h3_eq3: "object_ptr_kinds h3 = object_ptr_kinds h'"
      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 set_child_nodes_pointers_preserved
      by (auto simp add: reflp_def transp_def)
    then have object_ptr_kinds_M_eq_h3:
      "∧ptrs. h3 ⊨ object_ptr_kinds_M →_r ptrs = h' ⊨ object_ptr_kinds_M →_r ptrs"
      by(simp add: object_ptr_kinds_M_defs)
    then have object_ptr_kinds_eq_h3: "|h3 ⊨ object_ptr_kinds_M|_r = |h' ⊨ object_ptr_kinds_M|_r"
      by(simp)
    then have node_ptr_kinds_eq_h3: "|h3 ⊨ node_ptr_kinds_M|_r = |h' ⊨ node_ptr_kinds_M|_r"
      using node_ptr_kinds_M_eq by blast
    then have node_ptr_kinds_eq3_h3: "node_ptr_kinds h3 = node_ptr_kinds h'"
      by auto
    have document_ptr_kinds_eq2_h3: "|h3 ⊨ document_ptr_kinds_M|_r = |h' ⊨ document_ptr_kinds_M|_r"
      using object_ptr_kinds_eq_h3 document_ptr_kinds_M_eq by auto
    then have document_ptr_kinds_eq3_h3: "document_ptr_kinds h3 = document_ptr_kinds h'"
      using object_ptr_kinds_eq_h3 document_ptr_kinds_M_eq by auto
    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 (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_h2: "∧doc_ptr disc_nodes. old_document ≠ doc_ptr ==>
      h2 ⊨ get_disconnected_nodes doc_ptr →_r disc_nodes =
      h3 ⊨ get_disconnected_nodes doc_ptr →_r disc_nodes"
      using get_disconnected_nodes_reads set_disconnected_nodes_writes h3
      apply(rule reads_writes_preserved)
      by (simp add: set_disconnected_nodes_get_disconnected_nodes_different_pointers)
    then have disconnected_nodes_eq2_h2: "∧doc_ptr. old_document ≠ doc_ptr ==>
      |h2 ⊨ get_disconnected_nodes doc_ptr|_r = |h3 ⊨ get_disconnected_nodes doc_ptr|_r"
      using select_result_eq by force
    obtain disc_nodes_old_document_h2 where disc_nodes_old_document_h2:
      "h2 ⊨ get_disconnected_nodes old_document →_r disc_nodes_old_document_h2"
      using old_disc_nodes by blast
    then have disc_nodes_old_document_h3:
      "h3 ⊨ get_disconnected_nodes old_document →_r remove1 child disc_nodes_old_document_h2"
      using h3 old_disc_nodes returns_result_eq set_disconnected_nodes_get_disconnected_nodes
      by fastforce
    have "distinct disc_nodes_old_document_h2"
      using disc_nodes_old_document_h2 local.heap_is_wellformed_disconnected_nodes_distinct
        wellformed_h2 by blast

    have "type_wf h2"
    proof (insert h2, induct  parent_opt)
      case None
      then show ?case
        using type_wf by simp
    next
      case (Some option)
      then show ?case
        using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'", OF remove_child_writes]
          type_wf remove_child_types_preserved
        by (simp add: reflp_def transp_def)
    qed
    then have "type_wf h3"
      using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'",
          OF set_disconnected_nodes_writes h3]
      using  set_disconnected_nodes_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 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 (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_document_ptr_h3"
      using disconnected_nodes_eq_h2 docs_neq disc_nodes_document_ptr_h3 by auto
    have disc_nodes_document_ptr_h':
      "h' ⊨ get_disconnected_nodes document_ptr →_r child # disc_nodes_document_ptr_h3"
      using h' disc_nodes_document_ptr_h3
      using set_disconnected_nodes_get_disconnected_nodes by blast

    have document_ptr_in_heap: "document_ptr |∈| document_ptr_kinds h2"
      using disc_nodes_document_ptr_h3 document_ptr_kinds_eq2_h2 get_disconnected_nodes_ok assms(1)
      unfolding heap_is_wellformed_def
      using disc_nodes_document_ptr_h2 get_disconnected_nodes_ptr_in_heap by blast
    have old_document_in_heap: "old_document |∈| document_ptr_kinds h2"
      using disc_nodes_old_document_h3 document_ptr_kinds_eq2_h2 get_disconnected_nodes_ok assms(1)
      unfolding heap_is_wellformed_def
      using get_disconnected_nodes_ptr_in_heap old_disc_nodes by blast

    have "child ∈ set disc_nodes_old_document_h2"
    proof (insert parent_opt h2, induct parent_opt)
      case None
      then have "h = h2"
        by(auto)
      moreover have "CD.a_owner_document_valid h"
        using assms(1)  by(simp add: heap_is_wellformed_def CD.heap_is_wellformed_def)
      ultimately show ?case
        using old_document disc_nodes_old_document_h2 None(1) child_parent_dual[OF assms(1)]
          in_disconnected_nodes_no_parent assms(1) known_ptrs type_wf by blast
    next
      case (Some option)
      then show ?case
        apply(simp split: option.splits)
        using assms(1) disc_nodes_old_document_h2 old_document remove_child_in_disconnected_nodes
          known_ptrs by blast
    qed
    have "child ∉ set (remove1 child disc_nodes_old_document_h2)"
      using disc_nodes_old_document_h3 h3 known_ptrs wellformed_h2
        ‹distinct disc_nodes_old_document_h2› by auto
    have "child ∉ set disc_nodes_document_ptr_h3"
    proof -
      have "CD.a_distinct_lists h2"
        using heap_is_wellformed_def CD.heap_is_wellformed_def wellformed_h2 by blast
      then have 0: "distinct (concat (map (λdocument_ptr.
        |h2 ⊨ get_disconnected_nodes document_ptr|_r) |h2 ⊨ document_ptr_kinds_M|_r))"
        by(simp add: CD.a_distinct_lists_def)
      show ?thesis
        using distinct_concat_map_E(1)[OF 0] ‹child ∈ set disc_nodes_old_document_h2›
          disc_nodes_old_document_h2 disc_nodes_document_ptr_h2
        by (meson ‹type_wf h2› docs_neq known_ptrs local.get_owner_document_disconnected_nodes
            local.known_ptrs_preserved object_ptr_kinds_h_eq3 returns_result_eq wellformed_h2)
    qed

    have child_in_heap: "child |∈| node_ptr_kinds h"
      using get_owner_document_ptr_in_heap[OF is_OK_returns_result_I[OF old_document]]
        node_ptr_kinds_commutes by blast
    have "CD.a_acyclic_heap h2"
      using wellformed_h2 by (simp add: heap_is_wellformed_def CD.heap_is_wellformed_def)
    have "parent_child_rel h' ⊆ parent_child_rel h2"
    proof
      fix x
      assume "x ∈ parent_child_rel h'"
      then show "x ∈ parent_child_rel h2"
        using object_ptr_kinds_h2_eq3  object_ptr_kinds_h3_eq3 children_eq2_h2 children_eq2_h3
          mem_Collect_eq object_ptr_kinds_M_eq_h3 select_result_eq split_cong
        unfolding CD.parent_child_rel_def
        by(simp)
    qed
    then have " CD.a_acyclic_heap h'"
      using ‹ CD.a_acyclic_heap h2› CD.acyclic_heap_def acyclic_subset by blast

    moreover have " CD.a_all_ptrs_in_heap h2"
      using wellformed_h2 by (simp add: heap_is_wellformed_def CD.heap_is_wellformed_def)
    then have "CD.a_all_ptrs_in_heap h3"
      apply(auto simp add: CD.a_all_ptrs_in_heap_def node_ptr_kinds_eq3_h2 children_eq_h2)[1]
       apply (metis ‹type_wf h'› children_eq2_h3 children_eq_h2 children_eq_h3 known_ptrs
          l_heap_is_wellformed.heap_is_wellformed_children_in_heap local.get_child_nodes_ok
          local.known_ptrs_known_ptr local.l_heap_is_wellformed_axioms node_ptr_kinds_eq3_h2
          object_ptr_kinds_h2_eq3 object_ptr_kinds_h3_eq3 object_ptr_kinds_h_eq3
          returns_result_select_result wellformed_h2)
      by (metis (no_types, opaque_lifting) disc_nodes_old_document_h2 disc_nodes_old_document_h3
          disconnected_nodes_eq2_h2 document_ptr_kinds_eq3_h2 select_result_I2
          set_remove1_subset subsetD)
    then have "CD.a_all_ptrs_in_heap h'"
      apply(auto simp add: CD.a_all_ptrs_in_heap_def node_ptr_kinds_eq3_h3 children_eq_h3)[1]
       apply (metis (no_types, opaque_lifting) children_eq2_h3 object_ptr_kinds_h3_eq3 subsetD)
      by (metis (no_types, opaque_lifting) ‹child ∈ set disc_nodes_old_document_h2› disc_nodes_document_ptr_h'
          disc_nodes_document_ptr_h2 disc_nodes_old_document_h2 disconnected_nodes_eq2_h3
          document_ptr_kinds_eq3_h3 local.heap_is_wellformed_disc_nodes_in_heap
          node_ptr_kinds_eq3_h2 node_ptr_kinds_eq3_h3 select_result_I2 set_ConsD subsetD wellformed_h2)

    moreover have "CD.a_owner_document_valid h2"
      using wellformed_h2 by (simp add: heap_is_wellformed_def CD.heap_is_wellformed_def)
    then have "CD.a_owner_document_valid h'"
      apply(simp add: CD.a_owner_document_valid_def node_ptr_kinds_eq_h2 node_ptr_kinds_eq3_h3
          object_ptr_kinds_eq_h2 object_ptr_kinds_eq_h3 document_ptr_kinds_eq2_h2 document_ptr_kinds_eq2_h3
          children_eq2_h2 children_eq2_h3 )
      by (metis (no_types) disc_nodes_document_ptr_h' disc_nodes_document_ptr_h2
          disc_nodes_old_document_h2 disc_nodes_old_document_h3 disconnected_nodes_eq2_h2
          disconnected_nodes_eq2_h3 document_ptr_in_heap document_ptr_kinds_eq3_h2 document_ptr_kinds_eq3_h3
          in_set_remove1 list.set_intros(1) list.set_intros(2) node_ptr_kinds_eq3_h2 node_ptr_kinds_eq3_h3
          object_ptr_kinds_h2_eq3 object_ptr_kinds_h3_eq3 select_result_I2)

    have a_distinct_lists_h2: "CD.a_distinct_lists h2"
      using wellformed_h2 by (simp add: heap_is_wellformed_def CD.heap_is_wellformed_def)
    then have "CD.a_distinct_lists h'"
      apply(auto simp add: CD.a_distinct_lists_def object_ptr_kinds_eq_h3 object_ptr_kinds_eq_h2
          children_eq2_h2 children_eq2_h3)[1]
    proof -
      assume 1: "distinct (concat (map (λptr. |h' ⊨ get_child_nodes ptr|_r)
        (sorted_list_of_set (fset (object_ptr_kinds h')))))"
        and 2: "distinct (concat (map (λdocument_ptr. |h2 ⊨ get_disconnected_nodes document_ptr|_r)
        (sorted_list_of_set (fset (document_ptr_kinds h2)))))"
        and 3: "(∪x∈fset (object_ptr_kinds h'). set |h' ⊨ get_child_nodes x|_r) ∩
        (∪x∈fset (document_ptr_kinds h2). set |h2 ⊨ get_disconnected_nodes x|_r) = {}"
      show "distinct (concat (map (λdocument_ptr. |h' ⊨ get_disconnected_nodes document_ptr|_r)
        (sorted_list_of_set (fset (document_ptr_kinds h')))))"
      proof(rule distinct_concat_map_I)
        show "distinct (sorted_list_of_set (fset (document_ptr_kinds h')))"
          by(auto simp add: document_ptr_kinds_M_def )
      next
        fix x
        assume a1: "x ∈ set (sorted_list_of_set (fset (document_ptr_kinds h')))"
        have 4: "distinct |h2 ⊨ get_disconnected_nodes x|_r"
          using a_distinct_lists_h2 "2" a1 concat_map_all_distinct document_ptr_kinds_eq2_h2
            document_ptr_kinds_eq2_h3 by fastforce
        then show "distinct |h' ⊨ get_disconnected_nodes x|_r"
        proof (cases "old_document ≠ x")
          case True
          then show ?thesis
          proof (cases "document_ptr ≠ x")
            case True
            then show ?thesis
              using disconnected_nodes_eq2_h2[OF ‹old_document ≠ x›]
                disconnected_nodes_eq2_h3[OF ‹document_ptr ≠ x›] 4
              by(auto)
          next
            case False
            then show ?thesis
              using  disc_nodes_document_ptr_h3 disc_nodes_document_ptr_h' 4
                ‹child ∉ set disc_nodes_document_ptr_h3›
              by(auto simp add: disconnected_nodes_eq2_h2[OF ‹old_document ≠ x›] )
          qed
        next
          case False
          then show ?thesis
            by (metis (no_types, opaque_lifting) ‹distinct disc_nodes_old_document_h2›
                disc_nodes_old_document_h3 disconnected_nodes_eq2_h3 distinct_remove1 docs_neq select_result_I2)
        qed
      next
        fix x y
        assume a0: "x ∈ set (sorted_list_of_set (fset (document_ptr_kinds h')))"
          and a1: "y ∈ set (sorted_list_of_set (fset (document_ptr_kinds h')))"
          and a2: "x ≠ y"

        moreover have 5: "set |h2 ⊨ get_disconnected_nodes x|_r ∩
          set |h2 ⊨ get_disconnected_nodes y|_r = {}"
          using 2 calculation
          by (auto simp add: document_ptr_kinds_eq3_h2 document_ptr_kinds_eq3_h3
              dest: distinct_concat_map_E(1))
        ultimately show "set |h' ⊨ get_disconnected_nodes x|_r ∩
          set |h' ⊨ get_disconnected_nodes y|_r = {}"
        proof(cases "old_document = x")
          case True
          have "old_document ≠ y"
            using ‹x ≠ y› ‹old_document = x› by simp
          have "document_ptr ≠ x"
            using docs_neq ‹old_document = x› by auto
          show ?thesis
          proof(cases "document_ptr = y")
            case True
            then show ?thesis
              using 5 True select_result_I2[OF disc_nodes_document_ptr_h']
                select_result_I2[OF disc_nodes_document_ptr_h2]
                select_result_I2[OF disc_nodes_old_document_h2]
                select_result_I2[OF disc_nodes_old_document_h3] ‹old_document = x›
              by (metis (no_types, lifting) ‹child ∉ set (remove1 child disc_nodes_old_document_h2)›
                  ‹document_ptr ≠ x› disconnected_nodes_eq2_h3 disjoint_iff_not_equal notin_set_remove1 set_ConsD)
          next
            case False
            then show ?thesis
              using 5 select_result_I2[OF disc_nodes_document_ptr_h']
                select_result_I2[OF disc_nodes_document_ptr_h2]
                select_result_I2[OF disc_nodes_old_document_h2]
                select_result_I2[OF disc_nodes_old_document_h3]
                disconnected_nodes_eq2_h2 disconnected_nodes_eq2_h3 ‹old_document = x› docs_neq
                ‹old_document ≠ y›
              by (metis (no_types, lifting) disjoint_iff_not_equal notin_set_remove1)
          qed
        next
          case False
          then show ?thesis
          proof(cases "old_document = y")
            case True
            then show ?thesis
            proof(cases "document_ptr = x")
              case True
              show ?thesis
                using 5 select_result_I2[OF disc_nodes_document_ptr_h']
                  select_result_I2[OF disc_nodes_document_ptr_h2]
                  select_result_I2[OF disc_nodes_old_document_h2]
                  select_result_I2[OF disc_nodes_old_document_h3]
                  ‹old_document ≠ x› ‹old_document = y› ‹document_ptr = x›
                apply(simp)
                by (metis (no_types, lifting)
                    ‹child ∉ set (remove1 child disc_nodes_old_document_h2)›
                    disconnected_nodes_eq2_h3 disjoint_iff_not_equal notin_set_remove1)
            next
              case False
              then show ?thesis
                using 5 select_result_I2[OF disc_nodes_document_ptr_h']
                  select_result_I2[OF disc_nodes_document_ptr_h2]
                  select_result_I2[OF disc_nodes_old_document_h2]
                  select_result_I2[OF disc_nodes_old_document_h3] ‹old_document ≠ x›
                  ‹old_document = y› ‹document_ptr ≠ x›
                by (metis (no_types, lifting) disconnected_nodes_eq2_h2
                    disconnected_nodes_eq2_h3 disjoint_iff_not_equal docs_neq notin_set_remove1)
            qed
          next
            case False
            have "set |h2 ⊨ get_disconnected_nodes y|_r ∩ set disc_nodes_old_document_h2 = {}"
              by (metis DocumentMonad.ptr_kinds_M_ok DocumentMonad.ptr_kinds_M_ptr_kinds False
                  ‹type_wf h2› a1 disc_nodes_old_document_h2 document_ptr_kinds_M_def document_ptr_kinds_eq2_h2
                  document_ptr_kinds_eq2_h3 l_ptr_kinds_M.ptr_kinds_ptr_kinds_M local.get_disconnected_nodes_ok
                  local.heap_is_wellformed_one_disc_parent returns_result_select_result wellformed_h2)
            then show ?thesis
            proof(cases "document_ptr = x")
              case True
              then have "document_ptr ≠ y"
                using ‹x ≠ y› by auto
              have "set |h2 ⊨ get_disconnected_nodes y|_r ∩ set disc_nodes_old_document_h2 = {}"
                using ‹set |h2 ⊨ get_disconnected_nodes y|_r ∩ set disc_nodes_old_document_h2 = {}›
                by blast
              then show ?thesis
                using 5 select_result_I2[OF disc_nodes_document_ptr_h']
                  select_result_I2[OF disc_nodes_document_ptr_h2]
                  select_result_I2[OF disc_nodes_old_document_h2]
                  select_result_I2[OF disc_nodes_old_document_h3] ‹old_document ≠ x›
                  ‹old_document ≠ y› ‹document_ptr = x› ‹document_ptr ≠ y›
                  ‹child ∈ set disc_nodes_old_document_h2› disconnected_nodes_eq2_h2
                  disconnected_nodes_eq2_h3 ‹set |h2 ⊨ get_disconnected_nodes y|_r ∩ set disc_nodes_old_document_h2 = {}›
                by(auto)
            next
              case False
              then show ?thesis
              proof(cases "document_ptr = y")
                case True
                have f1: "set |h2 ⊨ get_disconnected_nodes x|_r ∩ set disc_nodes_document_ptr_h3 = {}"
                  using 2 a1 document_ptr_in_heap document_ptr_kinds_eq2_h2
                    document_ptr_kinds_eq2_h3 ‹document_ptr ≠ x›
                    select_result_I2[OF disc_nodes_document_ptr_h3, symmetric]
                    disconnected_nodes_eq2_h2[OF docs_neq[symmetric], symmetric]
                  by (simp add: "5" True)
                moreover have f1: "set |h2 ⊨ get_disconnected_nodes x|_r ∩
set |h2 ⊨ get_disconnected_nodes old_document|_r = {}"
                  using 2 a1 old_document_in_heap document_ptr_kinds_eq2_h2
                    document_ptr_kinds_eq2_h3 ‹old_document ≠ x›
                  by (metis (no_types, lifting) a0 distinct_concat_map_E(1)
                      document_ptr_kinds_eq3_h2 document_ptr_kinds_eq3_h3 finite_fset
                      set_sorted_list_of_set)
                ultimately show ?thesis
                  using 5 select_result_I2[OF disc_nodes_document_ptr_h']
                    select_result_I2[OF disc_nodes_old_document_h2] ‹old_document ≠ x›
                    ‹document_ptr ≠ x› ‹document_ptr = y›
                    ‹child ∈ set disc_nodes_old_document_h2› disconnected_nodes_eq2_h2
                    disconnected_nodes_eq2_h3
                  by auto
              next
                case False
                then show ?thesis
                  using 5
                    select_result_I2[OF disc_nodes_old_document_h2] ‹old_document ≠ x›
                    ‹document_ptr ≠ x› ‹document_ptr ≠ y›
                    <>child set disc_nodes_old_document_h2›
                    disconnected_nodes_eq2_h3
                  by (metis java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
                  set disc_nodes_old_document_h2 = {}\<close> empty_iff inf.idem)

            java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15

        qed
      qed
    next
      ix x xa xb
      assume 0: "stinctct(ncat(<>ptr| <>get_child_nodes ptr|\<^sub>r)
          (sorted_list_of_setfset(ject_ptr_kindst_ptr_kinds)))"
        and 1: "distinct (concat (map(\<lambdadocument_ptr. |h2 \<turnstile> get_disconnected_nodes cument_ptrsub>r)
        _ocument_ptr_kinds
        and 2: "        by (metis Suc_eq_plus1 lessThan_iff proj_classical_set_elem)
          (\<Union>x\<in>fset document_ptr_kindskinds h2set <>get_disconnected_nodes x|\<^sub>r) = {}"
        and 3: "xa |      case False
        and 4: "x \<in> set |h' \<turnstile> get_child_nodes xa|\<^sub>r"
        and:|<>| ument_ptr_kindsr_kinds'
        and 6: "x \<in> set |h' \<turnstile> get_disconnected_nodes xb|\<r"
      n e
        using \<open>child \<in et_es_old_document_h2ocument_h2 disc_nodes_document_ptr_h'
          disc_nodes_document_ptr_h2 disc_nodes_old_document_h2 disc_nodes_old_document_h3
          disconnected_nodes_eq2_h2have <>i<d+1. \<Psi>s' i \<otimes>\<^sub>s ket (0::nat)) = N' *\<subV_untfoldingeq16
          document_ptr_kinds_eq2_h3  old_document_in_heap
        apply(auto)[1]
        applyasesb ld_document
      proof -
        assume a1 ormrun_B normun_B\Phi>s d \<otimes>\<^sub>s ket empty))\<^sup>2"
        assume a2: "h2 \<turnstile> get_disconnected_nodes old_document \<rightarrow>\<^sub>r isc_nodes_old_document_h2old_document_h2
        assume a3: "h3 \<turnstile> get_disconnected_nodes old_document \<rightarrow>\<^sub>r remove1 child disc_nodes_old_document_h2"
e set |h' \<turnstile> get_child_nodes xa|\<subjava.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
        assume "document_ptr_kinds h2 = document_ptr_kinds h'"
        assume a5:
          (\<Union>x\<in>fset (document_ptr_kinds h'). set |h2 \<turnstile> get_disconnected_nodes x|\<^sub>r) = {}"
        f6ment<> document_ptr_kindshjava.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
          using a1\openxb |\<in>| document_ptr_kinds h'\<close> by blast
        have f7: "|h2 \<turnstile>get_disconnected_nodesnected_nodesnodescumentnt<sub>r =isc_nodes_old_document_h2ld_document_h2ent_h2
          using a2 by simp
      set disc_nodes_old_document_h2"
          using f6 a3 a1 by (metis (no_types) \<pentype_wf h'\<close>
              \<open>x \<n eth\turnstile>get_disconnected_nodes xb|\<^sub>r\<close> disconnected_nodes_eq_h3 docs_neq
              get_disconnected_nodes_ok returns_result_eq returns_result_select_result set_remove1_subset subsetCE)
        then have "set |h' \<turnstile> get_child_nodesure_o2h_sqrt>norm ((run_A \<otimes>\<^sub>s ket empty) - run_B) \<le> sqrt ((d+1) * Pfind' + d * P_nonterm)\>
          using f7 f6 a5 a4 \<open>xa |\<in>| object_ptr_kinds h'\<close>
          by fastforce
        then show ?thesis
          using \<open>x \<in> set disc_nodes_old_document_h2\<close> a1 a4 f7 by blast
      next
        assume a1: "xb \<noteq> old_document"
        assume a2: "h2 \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes_document_ptr_h3"
        assume a3: "h2 \<turnstile> get_disconnected_nodes old_document \<rightarrow>\<^sub>r disc_nodes_old_document_h2"
        assume a4: "xa |\<in>| object_ptr_kinds h'"
        assume a5: "h' \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r child # disc_nodes_document_ptr_h3"
        assume a6: "old_document |\<in>| document_ptr_kinds h'"
        assume a7: "x \<in> set |h' \<turnstile> get_disconnected_nodes xb|\<^sub>r"
        assume a8: "x \<in> set |h' \<turnstile> get_child_nodes xa|\<^sub>r"
        assume a9: "document_ptr_kinds h2 = document_ptr_kinds h'"
        assume a10: "\<And>doc_ptr. old_document \<noteq> doc_ptr \<Longrightarrow>
          |h2 \<turnstile> get_disconnected_nodes doc_ptr|\<^sub>r = |h3 \<turnstile> get_disconnected_nodes doc_ptr|\<^sub>r"
        assume a11: "\<And>doc_ptr. document_ptr \<noteq> doc_ptr \<Longrightarrow> |h3 \<turnstile> get_disconnected_nodes doc_ptr|\<^sub>r =
          |h' \<turnstile> get_disconnected_nodes doc_ptr|\<^sub>r"
        assume a12: "(\<Union>x\<in>fset (object_ptr_kinds h'). set |h' \<turnstile> get_child_nodes x|\<^sub>r) \<inter>
          (\<Union>x\<in>fset (document_ptr_kinds h'). set |h2 \<turnstile> get_disconnected_nodes x|\<^sub>r) = {}"
        have f13: "\<And>d. d \<notin> set |h' \<turnstile> document_ptr_kinds_M|\<^sub>r \<or> h2 \<turnstile> ok get_disconnected_nodes d"
          using a9 \<open>type_wf h2\<close> get_disconnected_nodes_ok
          by simp
        then have f14: "|h2 \<turnstile> get_disconnected_nodes old_document|\<^sub>r = disc_nodes_old_document_h2"
          using a6 a3 by simp
        have "x \<notin> set |h2 \<turnstile> get_disconnected_nodes xb|\<^sub>r"
          using a12 a8 a4 \<open>xb |\<in>| document_ptr_kinds h'\<close>
          by (meson UN_I disjoint_iff_not_equal)
        then have "x = child"
          using f13 a11 a10 a7 a5 a2 a1
          by (metis (no_types, lifting) select_result_I2 set_ConsD)
        then have "child \<notin> set disc_nodes_old_document_h2"
          using f14 a12 a8 a6 a4
          by (metis \<open>type_wf h'\<close> adopt_node_removes_child assms(1) assms(2) type_wf
              get_child_nodes_ok known_ptrs local.known_ptrs_known_ptr object_ptr_kinds_h2_eq3
              object_ptr_kinds_h3_eq3 object_ptr_kinds_h_eq3 returns_result_select_result)
        then show ?thesis
          using \<open>child \<in> set disc_nodes_old_document_h2\<close> by fastforce
      qed
    qed
    ultimately have "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h'"
      using \<open>CD.a_owner_document_valid h'\<close> CD.heap_is_wellformed_def
      by simp

    have shadow_root_eq_h2:
      "\<And>ptr' shadow_root_ptr_opt. h2 \<turnstile> get_shadow_root ptr' \<rightarrow>\<^sub>r shadow_root_ptr_opt =
  h3 \<turnstile> get_shadow_root ptr' \<rightarrow>\<^sub>r shadow_root_ptr_opt"
      using get_shadow_root_reads set_disconnected_nodes_writes h3
      apply(rule reads_writes_preserved)
      by(auto simp add: adopt_node_locs_def remove_child_locs_def set_child_nodes_get_shadow_root
          set_disconnected_nodes_get_shadow_root)
    then
    have shadow_root_eq2_h2: "\<And>ptr'. |h2 \<turnstile> get_shadow_root ptr'|\<^sub>r = |h3 \<turnstile> get_shadow_root ptr'|\<^sub>r"
      by (meson select_result_eq)

    have shadow_root_eq_h3:
      "\<And>ptr' shadow_root_ptr_opt. h3 \<turnstile> get_shadow_root ptr' \<rightarrow>\<^sub>r shadow_root_ptr_opt =
h' \<turnstile> get_shadow_root ptr' \<rightarrow>\<^sub>r shadow_root_ptr_opt"
      using get_shadow_root_reads set_disconnected_nodes_writes h'
      apply(rule reads_writes_preserved)
      by(auto simp add: adopt_node_locs_def remove_child_locs_def set_child_nodes_get_shadow_root
          set_disconnected_nodes_get_shadow_root)
    then
    have shadow_root_eq2_h3: "\<And>ptr'. |h3 \<turnstile> get_shadow_root ptr'|\<^sub>r = |h' \<turnstile> get_shadow_root ptr'|\<^sub>r"
      by (meson select_result_eq)

    have tag_name_eq_h2: "\<And>ptr' tag. h2 \<turnstile> get_tag_name ptr' \<rightarrow>\<^sub>r tag = h3 \<turnstile> get_tag_name ptr' \<rightarrow>\<^sub>r tag"
      using get_tag_name_reads set_disconnected_nodes_writes h3
      apply(rule reads_writes_preserved)
      by(auto simp add: adopt_node_locs_def remove_child_locs_def set_child_nodes_get_tag_name
          set_disconnected_nodes_get_tag_name)
    then
    have tag_name_eq2_h2: "\<And>ptr'. |h2 \<turnstile> get_tag_name ptr'|\<^sub>r = |h3 \<turnstile> get_tag_name ptr'|\<^sub>r"
      by (meson select_result_eq)

    have tag_name_eq_h3: "\<And>ptr' tag. h3 \<turnstile> get_tag_name ptr' \<rightarrow>\<^sub>r tag = h' \<turnstile> get_tag_name ptr' \<rightarrow>\<^sub>r tag"
      using get_tag_name_reads set_disconnected_nodes_writes h'
      apply(rule reads_writes_preserved)
      by(auto simp add: adopt_node_locs_def remove_child_locs_def set_child_nodes_get_tag_name
          set_disconnected_nodes_get_tag_name)
    then
    have tag_name_eq2_h3: "\<And>ptr'. |h3 \<turnstile> get_tag_name ptr'|\<^sub>r = |h' \<turnstile> get_tag_name ptr'|\<^sub>r"
      by (meson select_result_eq)

    have object_ptr_kinds_eq_h2: "object_ptr_kinds h2 = object_ptr_kinds h3"
      apply(rule writes_small_big[where P="\<lambda>h h'. object_ptr_kinds h = object_ptr_kinds h'",
            OF set_disconnected_nodes_writes h3])
      unfolding adopt_node_locs_def remove_child_locs_def
      using set_disconnected_nodes_pointers_preserved set_child_nodes_pointers_preserved
      by (auto simp add: reflp_def transp_def  split: if_splits)

    have object_ptr_kinds_eq_h3: "object_ptr_kinds h3 = object_ptr_kinds h'"
      apply(rule writes_small_big[where P="\<lambda>h h'. object_ptr_kinds h = object_ptr_kinds h'",
            OF set_disconnected_nodes_writes h'])
      unfolding adopt_node_locs_def remove_child_locs_def
      using set_disconnected_nodes_pointers_preserved set_child_nodes_pointers_preserved
      by (auto simp add: reflp_def transp_def  split: if_splits)

    have shadow_root_ptr_kinds_eq_h2: "shadow_root_ptr_kinds h2 = shadow_root_ptr_kinds h3"
      using object_ptr_kinds_eq_h2
      by(auto simp add: shadow_root_ptr_kinds_def)
    have element_ptr_kinds_eq_h2: "element_ptr_kinds h2 = element_ptr_kinds h3"
      using object_ptr_kinds_eq_h2
      by(auto simp add: element_ptr_kinds_def node_ptr_kinds_def)

    have shadow_root_ptr_kinds_eq_h3: "shadow_root_ptr_kinds h3 = shadow_root_ptr_kinds h'"
      using object_ptr_kinds_eq_h3
      by(auto simp add: shadow_root_ptr_kinds_def)
    have element_ptr_kinds_eq_h3: "element_ptr_kinds h3 = element_ptr_kinds h'"
      using object_ptr_kinds_eq_h3
      by(auto simp add: element_ptr_kinds_def node_ptr_kinds_def)

    have "known_ptrs h3"
      using known_ptrs local.known_ptrs_preserved object_ptr_kinds_h2_eq3 object_ptr_kinds_h_eq3
      by blast
    then have "known_ptrs h'"
      using local.known_ptrs_preserved object_ptr_kinds_h3_eq3 by blast

    show "heap_is_wellformed h' \<and> known_ptrs h' \<and> type_wf h'"
      using \<open>heap_is_wellformed h2\<close>
      using \<open>heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h'\<close> \<open>known_ptrs h'\<close> \<open>type_wf h'\<close>
      using \<open>parent_child_rel h' \<subseteq> parent_child_rel h2\<close>
      apply(auto simp add: heap_is_wellformed_def CD.heap_is_wellformed_def CD.acyclic_heap_def
          a_host_shadow_root_rel_def a_all_ptrs_in_heap_def a_distinct_lists_def a_shadow_root_valid_def
          object_ptr_kinds_eq_h2 object_ptr_kinds_eq_h3 element_ptr_kinds_eq_h2 element_ptr_kinds_eq_h3
          shadow_root_ptr_kinds_eq_h2 shadow_root_ptr_kinds_eq_h3  shadow_root_eq_h2 shadow_root_eq_h3
          shadow_root_eq2_h2 shadow_root_eq2_h3 tag_name_eq_h2 tag_name_eq_h3 tag_name_eq2_h2
          tag_name_eq2_h3 CD.parent_child_rel_def children_eq2_h2 children_eq2_h3 object_ptr_kinds_h2_eq3
          object_ptr_kinds_h3_eq3)[1]
      done
  qed
  then show "heap_is_wellformed h'" and "known_ptrs h'" and "type_wf h'"
    by auto
qed

lemma adopt_node_node_in_disconnected_nodes:
  assumes wellformed: "heap_is_wellformed h"
    and adopt_node: "h \<turnstile> adopt_node owner_document node_ptr \<rightarrow>\<^sub>h h'"
    and "h' \<turnstile> get_disconnected_nodes owner_document \<rightarrow>\<^sub>r disc_nodes"
    and known_ptrs: "known_ptrs h"
    and type_wf: "type_wf h"
  shows "node_ptr \<in> set disc_nodes"
proof -
  obtain old_document parent_opt h2 where
    old_document: "h \<turnstile> get_owner_document (cast node_ptr) \<rightarrow>\<^sub>r old_document" and
    parent_opt: "h \<turnstile> get_parent node_ptr \<rightarrow>\<^sub>r parent_opt" and
    h2: "h \<turnstile> (case parent_opt of Some parent \<Rightarrow> remove_child parent node_ptr | None \<Rightarrow> return ()) \<rightarrow>\<^sub>h h2" and
    h': "h2 \<turnstile> (if owner_document \<noteq> old_document then do {
        old_disc_nodes \<leftarrow> get_disconnected_nodes old_document;
        set_disconnected_nodes old_document (remove1 node_ptr old_disc_nodes);
        disc_nodes \<leftarrow> get_disconnected_nodes owner_document;
        set_disconnected_nodes owner_document (node_ptr # disc_nodes)
      } else do {
        return ()
      }) \<rightarrow>\<^sub>h h'"
    using assms(2)
    by(auto simp add: adopt_node_def elim!: bind_returns_heap_E
        dest!: pure_returns_heap_eq[rotated, OF get_owner_document_pure]
        pure_returns_heap_eq[rotated, OF get_parent_pure])

  show ?thesis
  proof (cases "owner_document = old_document")
    case True
    then show ?thesis
    proof (insert parent_opt h2, induct parent_opt)
      case None
      then have "h = h'"
        using h2 h' by(auto)
      then show ?case
        using in_disconnected_nodes_no_parent assms None old_document by blast
    next
      case (Some parent)
      then show ?case
        using remove_child_in_disconnected_nodes known_ptrs True h' assms(3) old_document
        by auto
    qed
  next
    case False
    then show ?thesis
      using assms(3)  h' list.set_intros(1) select_result_I2
        set_disconnected_nodes_get_disconnected_nodes
      apply(auto elim!: bind_returns_heap_E
          bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated])[1]
    proof -
      fix x  and h'a and xb
      assume a1: "h' \<turnstile> get_disconnected_nodes owner_document \<rightarrow>\<^sub>r disc_nodes"
      assume a2: "\<And>h document_ptr disc_nodes h'.
        h \<turnstile> set_disconnected_nodes document_ptr disc_nodes \<rightarrow>\<^sub>h h' \<Longrightarrow>
        h' \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes"
      assume "h'a \<turnstile> set_disconnected_nodes owner_document (node_ptr # xb) \<rightarrow>\<^sub>h h'"
      then have "node_ptr # xb = disc_nodes"
        using a2 a1 by (meson returns_result_eq)
      then show ?thesis
        by (meson list.set_intros(1))
    qed
  qed
qed
end

interpretation l_adopt_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M Shadow_DOM.get_owner_document Shadow_DOM.get_parent
  Shadow_DOM.get_parent_locs Shadow_DOM.remove_child Shadow_DOM.remove_child_locs
  get_disconnected_nodes get_disconnected_nodes_locs
  set_disconnected_nodes set_disconnected_nodes_locs Shadow_DOM.adopt_node Shadow_DOM.adopt_node_locs
  ShadowRootClass.known_ptr ShadowRootClass.type_wf Shadow_DOM.get_child_nodes
  Shadow_DOM.get_child_nodes_locs
  ShadowRootClass.known_ptrs Shadow_DOM.set_child_nodes Shadow_DOM.set_child_nodes_locs
  Shadow_DOM.remove Shadow_DOM.heap_is_wellformed Shadow_DOM.parent_child_rel
  by(auto simp add: l_adopt_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def instances)
declare l_adopt_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms [instances]

interpretation i_adopt_node_wf2?: l_adopt_node_wf2\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M
  type_wf known_ptr get_child_nodes get_child_nodes_locs get_disconnected_nodes
  get_disconnected_nodes_locs set_child_nodes set_child_nodes_locs get_shadow_root get_shadow_root_locs
  set_disconnected_nodes set_disconnected_nodes_locs get_tag_name get_tag_name_locs heap_is_wellformed
  parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host get_host_locs get_disconnected_document
  get_disconnected_document_locs get_root_node get_root_node_locs get_parent get_parent_locs known_ptrs
  get_owner_document remove_child remove_child_locs remove adopt_node adopt_node_locs
  by(auto simp add: l_adopt_node_wf2\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def instances)
declare l_adopt_node_wf2\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms [instances]

lemma adopt_node_wf_is_l_adopt_node_wf [instances]:
  "l_adopt_node_wf type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes
  get_disconnected_nodes known_ptrs adopt_node"
  apply(auto simp add: l_adopt_node_wf_def l_adopt_node_wf_axioms_def instances)[1]
  using adopt_node_preserves_wellformedness apply blast
  using adopt_node_removes_child apply blast
  using adopt_node_node_in_disconnected_nodes apply blast
  using adopt_node_removes_first_child apply blast
  using adopt_node_document_in_heap apply blast
  using adopt_node_preserves_wellformedness apply blast
  using adopt_node_preserves_wellformedness apply blast
  done

subsubsection \<open>insert\_before\<close>

locale l_insert_before_wf2\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M =
  l_get_child_nodes +
  l_get_disconnected_nodes +
  l_set_child_nodes_get_shadow_root +
  l_set_disconnected_nodes_get_shadow_root +
  l_set_child_nodes_get_tag_name +
  l_set_disconnected_nodes_get_tag_name +
  l_set_disconnected_nodes_get_disconnected_nodes +
  l_set_child_nodes_get_disconnected_nodes +
  l_set_disconnected_nodes_get_disconnected_nodes_wf +
  l_set_disconnected_nodes_get_ancestors_si +
  l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M _ _ _ _ _ _ get_ancestors_si get_ancestors_si_locs +
  l_get_root_node_si_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M  +
  l_get_owner_document  +
  l_adopt_node  +
  l_adopt_node_wf  +
  l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M +
  l_adopt_node_get_shadow_root
begin
lemma insert_before_child_preserves:
  assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h"
  assumes "h \<turnstile> insert_before ptr node child \<rightarrow>\<^sub>h h'"
  shows "type_wf h'" and "known_ptrs h'" and "heap_is_wellformed h'"
proof -
  obtain ancestors reference_child owner_document h2 h3 disconnected_nodes_h2 where
    ancestors: "h \<turnstile> get_ancestors_si ptr \<rightarrow>\<^sub>r ancestors" and
    node_not_in_ancestors: "cast node \<notin> set ancestors" and
    reference_child:
    "h \<turnstile> (if Some node = child then a_next_sibling node else return child) \<rightarrow>\<^sub>r reference_child" and
    owner_document: "h \<turnstile> get_owner_document ptr \<rightarrow>\<^sub>r owner_document" and
    h2: "h \<turnstile> adopt_node owner_document node \<rightarrow>\<^sub>h h2" and
    disconnected_nodes_h2: "h2 \<turnstile> get_disconnected_nodes owner_document \<rightarrow>\<^sub>r disconnected_nodes_h2" and
    h3: "h2 \<turnstile> set_disconnected_nodes owner_document (remove1 node disconnected_nodes_h2) \<rightarrow>\<^sub>h h3" and
    h': "h3 \<turnstile> a_insert_node ptr node reference_child \<rightarrow>\<^sub>h h'"
        (* children: "h3 \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children" and *)
    (* h': "h3 \<turnstile> set_child_nodes ptr (insert_before_list node reference_child children) \<rightarrow>\<^sub>h h'" *)
    using assms(4)
    by(auto simp add: insert_before_def a_ensure_pre_insertion_validity_def
        elim!: bind_returns_heap_E bind_returns_result_E
        bind_returns_heap_E2[rotated, OF get_parent_pure, rotated]
        bind_returns_heap_E2[rotated, OF get_child_nodes_pure, rotated]
        bind_returns_heap_E2[rotated, OF get_disconnected_nodes_pure, rotated]
        bind_returns_heap_E2[rotated, OF get_ancestors_pure, rotated]
        bind_returns_heap_E2[rotated, OF next_sibling_pure, rotated]
        bind_returns_heap_E2[rotated, OF get_owner_document_pure, rotated]
        split: if_splits option.splits)

  have "type_wf h2"
    using ‹type_wf h›
    using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'", OF adopt_node_writes h2]
    using adopt_node_types_preserved
    by(auto simp add: reflp_def transp_def)
  then have "type_wf h3"
    using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'", OF set_disconnected_nodes_writes h3]
    using set_disconnected_nodes_types_preserved
    by(auto simp add: reflp_def transp_def)
  then show "type_wf h'"
    using writes_small_big[where P="λh h'. type_wf h ⟶ type_wf h'", OF insert_node_writes h']
    using set_child_nodes_types_preserved
    by(auto simp add: reflp_def transp_def)

  have "object_ptr_kinds h = object_ptr_kinds h2"
    using adopt_node_writes h2
    apply(rule writes_small_big)
    using adopt_node_pointers_preserved
    by(auto simp add: reflp_def transp_def)
  moreover have "… = object_ptr_kinds h3"
    using set_disconnected_nodes_writes h3
    apply(rule writes_small_big)
    using set_disconnected_nodes_pointers_preserved
    by(auto simp add: reflp_def transp_def)
  moreover have "… = object_ptr_kinds h'"
    using insert_node_writes h'
    apply(rule writes_small_big)
    using set_child_nodes_pointers_preserved
    by(auto simp add: reflp_def transp_def)

  ultimately
  show "known_ptrs h'"
    using ‹known_ptrs h› known_ptrs_preserved
    by blast

  have "known_ptrs h2"
    using ‹known_ptrs h› known_ptrs_preserved ‹object_ptr_kinds h = object_ptr_kinds h2›
    by blast
  then
  have "known_ptrs h3"
    using known_ptrs_preserved ‹object_ptr_kinds h2 = object_ptr_kinds h3›
    by blast

  have "known_ptr ptr"
    by (meson get_owner_document_ptr_in_heap is_OK_returns_result_I ‹known_ptrs h›
        l_known_ptrs.known_ptrs_known_ptr l_known_ptrs_axioms owner_document)

  have object_ptr_kinds_M_eq3_h: "object_ptr_kinds h = object_ptr_kinds h2"
    apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h = object_ptr_kinds h'",
          OF adopt_node_writes h2])
    using adopt_node_pointers_preserved
      apply blast
    by (auto simp add: reflp_def transp_def)
  then have object_ptr_kinds_M_eq_h:
    "∧ptrs. h ⊨ object_ptr_kinds_M →_r ptrs = h2 ⊨ object_ptr_kinds_M →_r ptrs"
    by(simp add: object_ptr_kinds_M_defs )
  then have object_ptr_kinds_M_eq2_h: "|h ⊨ object_ptr_kinds_M|_r = |h2 ⊨ object_ptr_kinds_M|_r"
    by simp
  then have node_ptr_kinds_eq2_h: "|h ⊨ node_ptr_kinds_M|_r = |h2 ⊨ node_ptr_kinds_M|_r"
    using node_ptr_kinds_M_eq by blast

  have wellformed_h2: "heap_is_wellformed h2"
    using adopt_node_preserves_wellformedness[OF ‹heap_is_wellformed h› h2] ‹known_ptrs h›
      ‹type_wf h›
    .

  have object_ptr_kinds_M_eq3_h2: "object_ptr_kinds h2 = 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 h3])
    unfolding a_remove_child_locs_def
    using set_disconnected_nodes_pointers_preserved
    by (auto simp add: reflp_def transp_def)
  then have object_ptr_kinds_M_eq_h2:
    "∧ptrs. h2 ⊨ object_ptr_kinds_M →_r ptrs = h3 ⊨ object_ptr_kinds_M →_r ptrs"
    by(simp add: object_ptr_kinds_M_defs)
  then have object_ptr_kinds_M_eq2_h2: "|h2 ⊨ object_ptr_kinds_M|_r = |h3 ⊨ object_ptr_kinds_M|_r"
    by simp
  then have node_ptr_kinds_eq2_h2: "|h2 ⊨ node_ptr_kinds_M|_r = |h3 ⊨ node_ptr_kinds_M|_r"
    using node_ptr_kinds_M_eq by blast
  have document_ptr_kinds_eq2_h2: "|h2 ⊨ document_ptr_kinds_M|_r = |h3 ⊨ document_ptr_kinds_M|_r"
    using object_ptr_kinds_M_eq2_h2 document_ptr_kinds_M_eq by auto

  have object_ptr_kinds_M_eq3_h': "object_ptr_kinds h3 = object_ptr_kinds h'"
    apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h = object_ptr_kinds h'",
          OF insert_node_writes h'])  unfolding a_remove_child_locs_def
    using set_child_nodes_pointers_preserved
    by (auto simp add: reflp_def transp_def)
  then have object_ptr_kinds_M_eq_h3:
    "∧ptrs. h3 ⊨ object_ptr_kinds_M →_r ptrs = h' ⊨ object_ptr_kinds_M →_r ptrs"
    by(simp add: object_ptr_kinds_M_defs)
  then have object_ptr_kinds_M_eq2_h3: "|h3 ⊨ object_ptr_kinds_M|_r = |h' ⊨ object_ptr_kinds_M|_r"
    by simp
  then have node_ptr_kinds_eq2_h3: "|h3 ⊨ node_ptr_kinds_M|_r = |h' ⊨ node_ptr_kinds_M|_r"
    using node_ptr_kinds_M_eq by blast
  have document_ptr_kinds_eq2_h3: "|h3 ⊨ document_ptr_kinds_M|_r = |h' ⊨ document_ptr_kinds_M|_r"
    using object_ptr_kinds_M_eq2_h3 document_ptr_kinds_M_eq by auto

  have shadow_root_eq_h2:
    "∧ptr' shadow_root. h ⊨ get_shadow_root ptr' →_r shadow_root =
  h2 ⊨ get_shadow_root ptr' →_r shadow_root"
    using get_shadow_root_reads adopt_node_writes h2
    apply(rule reads_writes_preserved)
    using local.adopt_node_get_shadow_root by blast

  have disconnected_nodes_eq_h2:
    "∧doc_ptr disc_nodes. owner_document ≠ doc_ptr ==>
      h2 ⊨ get_disconnected_nodes doc_ptr →_r disc_nodes =
      h3 ⊨ get_disconnected_nodes doc_ptr →_r disc_nodes"
    using get_disconnected_nodes_reads set_disconnected_nodes_writes h3
    apply(rule reads_writes_preserved)
    by (auto simp add: set_disconnected_nodes_get_disconnected_nodes_different_pointers)
  then have disconnected_nodes_eq2_h2:
    "∧doc_ptr. doc_ptr ≠ owner_document ==>
      |h2 ⊨ get_disconnected_nodes doc_ptr|_r =
      |h3 ⊨ get_disconnected_nodes doc_ptr|_r"
    using select_result_eq by force
  have disconnected_nodes_h3:
    "h3 ⊨ get_disconnected_nodes owner_document →_r remove1 node disconnected_nodes_h2"
    using h3 set_disconnected_nodes_get_disconnected_nodes
    by blast

  have disconnected_nodes_eq_h3:
    "∧doc_ptr disc_nodes. h3 ⊨ get_disconnected_nodes doc_ptr →_r disc_nodes =
      h' ⊨ get_disconnected_nodes doc_ptr →_r disc_nodes"
    using get_disconnected_nodes_reads insert_node_writes  h'
    apply(rule reads_writes_preserved)
    using set_child_nodes_get_disconnected_nodes by fast
  then have disconnected_nodes_eq2_h3:
    "∧doc_ptr. |h3 ⊨ get_disconnected_nodes doc_ptr|_r = |h' ⊨ 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_disconnected_nodes_writes h3
    apply(rule reads_writes_preserved)
    by (auto simp add: set_disconnected_nodes_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 children_eq_h3:
    "∧ptr' children. ptr ≠ ptr' ==>
       h3 ⊨ get_child_nodes ptr' →_r children = h' ⊨ get_child_nodes ptr' →_r children"
    using get_child_nodes_reads insert_node_writes h'
    apply(rule reads_writes_preserved)
    by (auto simp add: set_child_nodes_get_child_nodes_different_pointers)
  then have children_eq2_h3:
    "∧ptr'. ptr ≠ ptr' ==> |h3 ⊨ get_child_nodes ptr'|_r = |h' ⊨ get_child_nodes ptr'|_r"
    using select_result_eq by force
  obtain children_h3 where children_h3: "h3 ⊨ get_child_nodes ptr →_r children_h3"
    using h' a_insert_node_def by auto
  have children_h': "h' ⊨ get_child_nodes ptr →_r insert_before_list node reference_child children_h3"
    using h' ‹type_wf h3› ‹known_ptr ptr›
    by(auto simp add: a_insert_node_def elim!: bind_returns_heap_E2
        dest!: set_child_nodes_get_child_nodes returns_result_eq[OF children_h3])

  have ptr_in_heap: "ptr |∈| object_ptr_kinds h3"
    using children_h3 get_child_nodes_ptr_in_heap by blast
  have node_in_heap: "node |∈| node_ptr_kinds h"
    using h2 adopt_node_child_in_heap by fast
  have child_not_in_any_children:
    "∧p children. h2 ⊨ get_child_nodes p →_r children ==> node ∉ set children"
    using ‹heap_is_wellformed h› h2 adopt_node_removes_child ‹type_wf h› ‹known_ptrs h› by auto
  have "node ∈ set disconnected_nodes_h2"
    using disconnected_nodes_h2 h2 adopt_node_node_in_disconnected_nodes assms(1)
      ‹type_wf h› ‹known_ptrs h› by blast
  have node_not_in_disconnected_nodes:
    "∧d. d |∈| document_ptr_kinds h3 ==> node ∉ set |h3 ⊨ get_disconnected_nodes d|_r"
  proof -
    fix d
    assume "d |∈| document_ptr_kinds h3"
    show "node ∉ set |h3 ⊨ get_disconnected_nodes d|_r"
    proof (cases "d = owner_document")
      case True
      then show ?thesis
        using disconnected_nodes_h2 wellformed_h2 h3 remove_from_disconnected_nodes_removes
          wellformed_h2 ‹d |∈| document_ptr_kinds h3› disconnected_nodes_h3
        by fastforce
    next
      case False
      then have "set |h2 ⊨ get_disconnected_nodes d|_r ∩ set |h2 ⊨ get_disconnected_nodes owner_document|_r = {}"
        using distinct_concat_map_E(1) wellformed_h2
        by (metis (no_types, lifting) ‹d |∈| document_ptr_kinds h3› ‹type_wf h2›
            disconnected_nodes_h2 document_ptr_kinds_M_def document_ptr_kinds_eq2_h2
            l_ptr_kinds_M.ptr_kinds_ptr_kinds_M local.get_disconnected_nodes_ok
            local.heap_is_wellformed_one_disc_parent returns_result_select_result select_result_I2)
      then show ?thesis
        using disconnected_nodes_eq2_h2[OF False] ‹node ∈ set disconnected_nodes_h2›
          disconnected_nodes_h2 by fastforce
    qed
  qed

  have "cast node ≠ ptr"
    using ancestors node_not_in_ancestors get_ancestors_ptr
    by fast

  obtain ancestors_h2 where ancestors_h2: "h2 ⊨ get_ancestors_si ptr →_r ancestors_h2"
    using get_ancestors_si_ok
    by (metis ‹known_ptrs h2› ‹type_wf h2› is_OK_returns_result_E
        object_ptr_kinds_M_eq3_h2 ptr_in_heap wellformed_h2)
  have ancestors_h3: "h3 ⊨ get_ancestors_si ptr →_r ancestors_h2"
    using get_ancestors_si_reads set_disconnected_nodes_writes h3
    apply(rule reads_writes_separate_forwards)
    using  ‹heap_is_wellformed h2› apply simp
    using ancestors_h2 apply simp
    apply(auto simp add: get_ancestors_si_locs_def get_parent_locs_def)[1]
     apply (simp add: local.get_ancestors_si_locs_def local.get_parent_reads_pointers
        local.set_disconnected_nodes_get_ancestors_si)
    using local.get_ancestors_si_locs_def local.set_disconnected_nodes_get_ancestors_si by blast
  have node_not_in_ancestors_h2: "cast node ∉ set ancestors_h2"
    using ‹heap_is_wellformed h› ‹heap_is_wellformed h2› ancestors ancestors_h2
    apply(rule get_ancestors_si_remains_not_in_ancestors)
    using assms(2) assms(3) h2 local.adopt_node_children_subset apply blast
    using shadow_root_eq_h2 node_not_in_ancestors object_ptr_kinds_M_eq2_h assms(2) assms(3)
      ‹type_wf h2›
    by(auto dest: returns_result_eq)

  moreover
  have "parent_child_rel h2 = parent_child_rel h3"
    by(auto simp add: CD.parent_child_rel_def object_ptr_kinds_M_eq3_h2 children_eq2_h2)
  have "parent_child_rel h' = insert (ptr, cast node) ((parent_child_rel h3))"
    using children_h3 children_h' ptr_in_heap
    apply(auto simp add: CD.parent_child_rel_def object_ptr_kinds_M_eq3_h' children_eq2_h3
        insert_before_list_node_in_set)[1]
     apply (metis (no_types, lifting) children_eq2_h3 insert_before_list_in_set select_result_I2)
    by (metis (no_types, lifting) children_eq2_h3 imageI insert_before_list_in_set select_result_I2)
  have "CD.a_acyclic_heap h'"
  proof -
    have "acyclic (parent_child_rel h2)"
      using wellformed_h2
      by (simp add: heap_is_wellformed_def CD.heap_is_wellformed_def CD.acyclic_heap_def)
    then have "acyclic (parent_child_rel h3)"
      by(auto simp add: CD.parent_child_rel_def object_ptr_kinds_M_eq3_h2 children_eq2_h2)
    moreover have "cast node ∉ {x. (x, ptr) ∈ (parent_child_rel h2)^*}"
      using get_ancestors_si_parent_child_rel
      using ‹known_ptrs h2› ‹type_wf h2› ancestors_h2 node_not_in_ancestors_h2 wellformed_h2
      by blast
    then have "cast node ∉ {x. (x, ptr) ∈ (parent_child_rel h3)^*}"
      by(auto simp add: CD.parent_child_rel_def object_ptr_kinds_M_eq3_h2 children_eq2_h2)
    ultimately show ?thesis
      using ‹parent_child_rel h' = insert (ptr, cast node) ((parent_child_rel h3))›
      by(auto simp add: CD.acyclic_heap_def)
  qed

  moreover have "CD.a_all_ptrs_in_heap h2"
    using wellformed_h2  by (simp add: heap_is_wellformed_def CD.heap_is_wellformed_def)
  have "CD.a_all_ptrs_in_heap h'"
  proof -
    have "CD.a_all_ptrs_in_heap h3"
      using ‹CD.a_all_ptrs_in_heap h2›
      apply(auto simp add: CD.a_all_ptrs_in_heap_def object_ptr_kinds_M_eq2_h2
          node_ptr_kinds_eq2_h2 children_eq_h2)[1]
      using disconnected_nodes_eq2_h2 disconnected_nodes_h2 disconnected_nodes_h3
      using node_ptr_kinds_eq2_h2 apply auto[1]
       apply (metis (no_types, opaque_lifting) children_eq2_h2 in_mono object_ptr_kinds_M_eq3_h2)
      by (metis (no_types, opaque_lifting) NodeMonad.ptr_kinds_ptr_kinds_M disconnected_nodes_eq2_h2
          disconnected_nodes_h2 disconnected_nodes_h3 document_ptr_kinds_commutes
          node_ptr_kinds_eq2_h2 object_ptr_kinds_M_eq3_h2 select_result_I2 set_remove1_subset subsetD)
    have "set children_h3 ⊆ set |h' ⊨ node_ptr_kinds_M|_r"
      using children_h3 ‹CD.a_all_ptrs_in_heap h3›
      apply(auto simp add: CD.a_all_ptrs_in_heap_def node_ptr_kinds_eq2_h3)[1]
      using CD.parent_child_rel_child_nodes2 ‹known_ptr ptr›
        ‹parent_child_rel h2 = parent_child_rel h3› ‹type_wf h2›
        local.parent_child_rel_child_in_heap node_ptr_kinds_commutes object_ptr_kinds_M_eq3_h'
        object_ptr_kinds_M_eq3_h2 wellformed_h2 by blast
    then have "set (insert_before_list node reference_child children_h3) ⊆ set |h' ⊨ node_ptr_kinds_M|_r"
      using node_in_heap
      apply(auto simp add: node_ptr_kinds_eq2_h node_ptr_kinds_eq2_h2 node_ptr_kinds_eq2_h3)[1]
      by (metis (no_types, opaque_lifting) contra_subsetD insert_before_list_in_set
          node_ptr_kinds_commutes object_ptr_kinds_M_eq3_h object_ptr_kinds_M_eq3_h' object_ptr_kinds_M_eq3_h2)
    then show ?thesis
      using ‹CD.a_all_ptrs_in_heap h3›
      apply(auto simp add: object_ptr_kinds_M_eq3_h' CD.a_all_ptrs_in_heap_def node_ptr_kinds_def
          node_ptr_kinds_eq2_h3 disconnected_nodes_eq_h3)[1]
      using children_eq_h3 children_h'
       apply (metis (no_types, lifting) children_eq2_h3 select_result_I2 subsetD)
      by (metis (no_types, lifting) DocumentMonad.ptr_kinds_ptr_kinds_M disconnected_nodes_eq2_h3
          document_ptr_kinds_eq2_h3 subsetD)
  qed

  moreover have "CD.a_distinct_lists h2"
    using wellformed_h2  by (simp add: heap_is_wellformed_def CD.heap_is_wellformed_def )
  then have "CD.a_distinct_lists h3"
  proof(auto simp add: CD.a_distinct_lists_def object_ptr_kinds_M_eq2_h2 document_ptr_kinds_eq2_h2
      children_eq2_h2 intro!: distinct_concat_map_I)
    fix x
    assume 1: "x |∈| document_ptr_kinds h3"
      and 2: "distinct (concat (map (λdocument_ptr. |h2 ⊨ get_disconnected_nodes document_ptr|_r)
(sorted_list_of_set (fset (document_ptr_kinds h3)))))"
    show "distinct |h3 ⊨ get_disconnected_nodes x|_r"
      using distinct_concat_map_E(2)[OF 2] select_result_I2[OF disconnected_nodes_h3]
        disconnected_nodes_eq2_h2 select_result_I2[OF disconnected_nodes_h2] 1
      by (metis (full_types) distinct_remove1 finite_fset set_sorted_list_of_set)
  next
    fix x y xa
    assume 1: "distinct (concat (map (λdocument_ptr. |h2 ⊨ get_disconnected_nodes document_ptr|_r)
(sorted_list_of_set (fset (document_ptr_kinds h3)))))"
      and 2: "x |∈| document_ptr_kinds h3"
      and 3: "y |∈| document_ptr_kinds h3"
      and 4: "x ≠ y"
      and 5: "xa ∈ set |h3 ⊨ get_disconnected_nodes x|_r"
      and 6: "xa ∈ set |h3 ⊨ get_disconnected_nodes y|_r"
    show False
    proof (cases "x = owner_document")
      case True
      then have "y ≠ owner_document"
        using 4 by simp
      show ?thesis
        using distinct_concat_map_E(1)[OF 1]
        using 2 3 4 5 6 select_result_I2[OF disconnected_nodes_h3]
          select_result_I2[OF disconnected_nodes_h2]
        apply(auto simp add: True disconnected_nodes_eq2_h2[OF ‹y ≠ owner_document›])[1]
        by (metis (no_types, opaque_lifting) disconnected_nodes_eq2_h2 disjoint_iff_not_equal
            notin_set_remove1)
    next
      case False
      then show ?thesis
      proof (cases "y = owner_document")
        case True
        then show ?thesis
          using distinct_concat_map_E(1)[OF 1]
          using 2 3 4 5 6 select_result_I2[OF disconnected_nodes_h3]
            select_result_I2[OF disconnected_nodes_h2]
          apply(auto simp add: True disconnected_nodes_eq2_h2[OF ‹x ≠ owner_document›])[1]
          by (metis (no_types, opaque_lifting) disconnected_nodes_eq2_h2 disjoint_iff_not_equal
              notin_set_remove1)
      next
        case False
        then show ?thesis
          using distinct_concat_map_E(1)[OF 1, simplified, OF 2 3 4] 5 6
          using disconnected_nodes_eq2_h2 disconnected_nodes_h2 disconnected_nodes_h3
            disjoint_iff_not_equal notin_set_remove1 select_result_I2
          by (metis (no_types, opaque_lifting))
      qed
    qed
  next
    fix x xa xb
    assume 1: "(∪x∈fset (object_ptr_kinds h3). set |h3 ⊨ get_child_nodes x|_r) ∩
(∪x∈fset (document_ptr_kinds h3). set |h2 ⊨ get_disconnected_nodes x|_r) = {}"
      and 2: "xa |∈| object_ptr_kinds h3"
      and 3: "x ∈ set |h3 ⊨ get_child_nodes xa|_r"
      and 4: "xb |∈| document_ptr_kinds h3"
      and 5: "x ∈ set |h3 ⊨ get_disconnected_nodes xb|_r"
    have 6: "set |h3 ⊨ get_child_nodes xa|_r ∩ set |h2 ⊨ get_disconnected_nodes xb|_r = {}"
      using 1 2 4
      by (metis ‹type_wf h2› children_eq2_h2 document_ptr_kinds_commutes ‹known_ptrs h›
          local.get_child_nodes_ok local.get_disconnected_nodes_ok
          local.heap_is_wellformed_children_disc_nodes_different local.known_ptrs_known_ptr
          object_ptr_kinds_M_eq3_h object_ptr_kinds_M_eq3_h2 returns_result_select_result wellformed_h2)
    show False
    proof (cases "xb = owner_document")
      case True
      then show ?thesis
        using select_result_I2[OF disconnected_nodes_h3,folded select_result_I2[OF disconnected_nodes_h2]]
        by (metis (no_types, lifting) "3" "5" "6" disjoint_iff_not_equal notin_set_remove1)
    next
      case False
      show ?thesis
        using 2 3 4 5 6 unfolding disconnected_nodes_eq2_h2[OF False] by auto
    qed
  qed
  then have "CD.a_distinct_lists h'"
  proof(auto simp add: CD.a_distinct_lists_def document_ptr_kinds_eq2_h3 object_ptr_kinds_M_eq2_h3
      disconnected_nodes_eq2_h3 intro!: distinct_concat_map_I)
    fix x
    assume 1: "distinct (concat (map (λptr. |h3 ⊨ get_child_nodes ptr|_r)
(sorted_list_of_set (fset (object_ptr_kinds h')))))" and
      2: "x |∈| object_ptr_kinds h'"
    have 3: "∧p. p |∈| object_ptr_kinds h' ==> distinct |h3 ⊨ get_child_nodes p|_r"
      using 1  by (auto elim: distinct_concat_map_E)
    show "distinct |h' ⊨ get_child_nodes x|_r"
    proof(cases "ptr = x")
      case True
      show ?thesis
        using 3[OF 2] children_h3 children_h'
        by(auto simp add: True  insert_before_list_distinct
            dest: child_not_in_any_children[unfolded children_eq_h2])
    next
      case False
      show ?thesis
        using children_eq2_h3[OF False] 3[OF 2] by auto
    qed
  next
    fix x y xa
    assume 1:"distinct (concat (map (λptr. |h3 ⊨ get_child_nodes ptr|_r)
(sorted_list_of_set (fset (object_ptr_kinds h')))))"
      and 2: "x |∈| object_ptr_kinds h'"
      and 3: "y |∈| object_ptr_kinds h'"
      and 4: "x ≠ y"
      and 5: "xa ∈ set |h' ⊨ get_child_nodes x|_r"
      and 6: "xa ∈ set |h' ⊨ get_child_nodes y|_r"
    have 7:"set |h3 ⊨ get_child_nodes x|_r ∩ set |h3 ⊨ get_child_nodes y|_r = {}"
      using distinct_concat_map_E(1)[OF 1] 2 3 4 by auto
    show False
    proof (cases "ptr = x")
      case True
      then have "ptr ≠ y"
        using 4 by simp
      then show ?thesis
        using  children_h3 children_h' child_not_in_any_children[unfolded children_eq_h2] 5 6
        apply(auto simp add:  True children_eq2_h3[OF ‹ptr ≠ y›])[1]
        by (metis (no_types, opaque_lifting) "3" "7" ‹type_wf h3› children_eq2_h3 disjoint_iff_not_equal
            get_child_nodes_ok insert_before_list_in_set ‹known_ptrs h› local.known_ptrs_known_ptr
            object_ptr_kinds_M_eq3_h object_ptr_kinds_M_eq3_h' object_ptr_kinds_M_eq3_h2
            returns_result_select_result select_result_I2)
    next
      case False
      then show ?thesis
      proof (cases "ptr = y")
        case True
        then show ?thesis
          using  children_h3 children_h' child_not_in_any_children[unfolded children_eq_h2] 5 6
          apply(auto simp add:  True children_eq2_h3[OF ‹ptr ≠ x›])[1]
          by (metis (no_types, opaque_lifting) "2" "4" "7" IntI ‹known_ptrs h3› ‹type_wf h'›
              children_eq_h3 empty_iff insert_before_list_in_set local.get_child_nodes_ok
              local.known_ptrs_known_ptr object_ptr_kinds_M_eq3_h' returns_result_select_result select_result_I2)
      next
        case False
        then show ?thesis
          using children_eq2_h3[OF ‹ptr ≠ x›] children_eq2_h3[OF ‹ptr ≠ y›] 5 6 7 by auto
      qed
    qed
  next
    fix x xa xb
    assume 1: " (∪x∈fset (object_ptr_kinds h'). set |h3 ⊨ get_child_nodes x|_r) ∩
(∪x∈fset (document_ptr_kinds h'). set |h' ⊨ get_disconnected_nodes x|_r) = {} "
      and 2: "xa |∈| object_ptr_kinds h'"
      and 3: "x ∈ set |h' ⊨ get_child_nodes xa|_r"
      and 4: "xb |∈| document_ptr_kinds h'"
      and 5: "x ∈ set |h' ⊨ get_disconnected_nodes xb|_r"
    have 6: "set |h3 ⊨ get_child_nodes xa|_r ∩ set |h' ⊨ get_disconnected_nodes xb|_r = {}"
      using 1 2 3 4 5
    proof -
      have "∀h d. ¬ type_wf h ∨ d |∉| document_ptr_kinds h ∨ h ⊨ ok get_disconnected_nodes d"
        using local.get_disconnected_nodes_ok by satx
      then have "h' ⊨ ok get_disconnected_nodes xb"
        using "4" ‹type_wf h'› by fastforce
      then have f1: "h3 ⊨ get_disconnected_nodes xb →_r |h' ⊨ get_disconnected_nodes xb|_r"
        by (simp add: disconnected_nodes_eq_h3)
      have "xa |∈| object_ptr_kinds h3"
        using "2" object_ptr_kinds_M_eq3_h' by blast
      then show ?thesis
        using f1 ‹local.CD.a_distinct_lists h3› CD.distinct_lists_no_parent by fastforce
    qed
    show False
    proof (cases "ptr = xa")
      case True
      show ?thesis
        using 6 node_not_in_disconnected_nodes  3 4 5  select_result_I2[OF children_h']
          select_result_I2[OF children_h3] True disconnected_nodes_eq2_h3
        by (metis (no_types, lifting) "2" DocumentMonad.ptr_kinds_ptr_kinds_M
            ‹CD.a_distinct_lists h3› ‹type_wf h'› disconnected_nodes_eq_h3 CD.distinct_lists_no_parent
            document_ptr_kinds_eq2_h3 get_disconnected_nodes_ok insert_before_list_in_set
            object_ptr_kinds_M_eq3_h' returns_result_select_result)

    next
      case False
      then show ?thesis
        using 1 2 3 4 5 children_eq2_h3[OF False] by fastforce
    qed
  qed

  moreover have "CD.a_owner_document_valid h2"
    using wellformed_h2  by (simp add: heap_is_wellformed_def CD.heap_is_wellformed_def)
  then have "CD.a_owner_document_valid h'"
    apply(auto simp add: CD.a_owner_document_valid_def object_ptr_kinds_M_eq2_h2
        object_ptr_kinds_M_eq2_h3 node_ptr_kinds_eq2_h2 node_ptr_kinds_eq2_h3 document_ptr_kinds_eq2_h2
        document_ptr_kinds_eq2_h3 children_eq2_h2 )[1] thm children_eq2_h3
    apply(auto simp add: document_ptr_kinds_eq2_h2[simplified]  document_ptr_kinds_eq2_h3[simplified]
        object_ptr_kinds_M_eq2_h2[simplified] object_ptr_kinds_M_eq2_h3[simplified]
        node_ptr_kinds_eq2_h2[simplified] node_ptr_kinds_eq2_h3[simplified])[1]
    apply(auto simp add: disconnected_nodes_eq2_h3[symmetric])[1]
    by (metis (no_types, opaque_lifting) Core_DOM_Functions.i_insert_before.insert_before_list_in_set
        children_eq2_h3 children_h' children_h3 disconnected_nodes_eq2_h2 disconnected_nodes_h2
        disconnected_nodes_h3 in_set_remove1 is_OK_returns_result_I object_ptr_kinds_M_eq3_h'
        ptr_in_heap returns_result_eq returns_result_select_result)
  ultimately have "heap_is_wellformed_C_o_r_e_{_}_D_O_M h'"
    by (simp add: CD.heap_is_wellformed_def)

  have shadow_root_eq_h2:
    "∧ptr' shadow_root_ptr_opt. h2 ⊨ get_shadow_root ptr' →_r shadow_root_ptr_opt =
h3 ⊨ get_shadow_root ptr' →_r shadow_root_ptr_opt"
    using get_shadow_root_reads set_disconnected_nodes_writes h3
    apply(rule reads_writes_preserved)
    by(auto simp add: adopt_node_locs_def remove_child_locs_def set_child_nodes_get_shadow_root
        set_disconnected_nodes_get_shadow_root)
  then
  have shadow_root_eq2_h2: "∧ptr'. |h2 ⊨ get_shadow_root ptr'|_r = |h3 ⊨ get_shadow_root ptr'|_r"
    by (meson select_result_eq)

  have shadow_root_eq_h3:
    "∧ptr' shadow_root_ptr_opt. h3 ⊨ get_shadow_root ptr' →_r shadow_root_ptr_opt =
h' ⊨ get_shadow_root ptr' →_r shadow_root_ptr_opt"
    using get_shadow_root_reads insert_node_writes h'
    apply(rule reads_writes_preserved)
    by(auto simp add: adopt_node_locs_def remove_child_locs_def set_child_nodes_get_shadow_root
        set_disconnected_nodes_get_shadow_root)
  then
  have shadow_root_eq2_h3: "∧ptr'. |h3 ⊨ get_shadow_root ptr'|_r = |h' ⊨ get_shadow_root ptr'|_r"
    by (meson select_result_eq)

  have tag_name_eq_h2: "∧ptr' tag. h2 ⊨ get_tag_name ptr' →_r tag = h3 ⊨ get_tag_name ptr' →_r tag"
    using get_tag_name_reads set_disconnected_nodes_writes h3
    apply(rule reads_writes_preserved)
    by(auto simp add: adopt_node_locs_def remove_child_locs_def set_child_nodes_get_tag_name
        set_disconnected_nodes_get_tag_name)
  then
  have tag_name_eq2_h2: "∧ptr'. |h2 ⊨ get_tag_name ptr'|_r = |h3 ⊨ get_tag_name ptr'|_r"
    by (meson select_result_eq)

  have tag_name_eq_h3: "∧ptr' tag. h3 ⊨ get_tag_name ptr' →_r tag = h' ⊨ get_tag_name ptr' →_r tag"
    using get_tag_name_reads insert_node_writes h'
    apply(rule reads_writes_preserved)
    by(auto simp add: adopt_node_locs_def remove_child_locs_def set_child_nodes_get_tag_name
        set_disconnected_nodes_get_tag_name)
  then
  have tag_name_eq2_h3: "∧ptr'. |h3 ⊨ get_tag_name ptr'|_r = |h' ⊨ get_tag_name ptr'|_r"
    by (meson select_result_eq)

  have object_ptr_kinds_eq_h2: "object_ptr_kinds h2 = 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 h3])
    unfolding adopt_node_locs_def remove_child_locs_def
    using set_disconnected_nodes_pointers_preserved set_child_nodes_pointers_preserved
    by (auto simp add: reflp_def transp_def  split: if_splits)

  have object_ptr_kinds_eq_h3: "object_ptr_kinds h3 = object_ptr_kinds h'"
    apply(rule writes_small_big[where P="λh h'. object_ptr_kinds h = object_ptr_kinds h'",
          OF insert_node_writes h'])
    using set_disconnected_nodes_pointers_preserved set_child_nodes_pointers_preserved
    by (auto simp add: reflp_def transp_def  split: if_splits)

  have shadow_root_ptr_kinds_eq_h2: "shadow_root_ptr_kinds h2 = shadow_root_ptr_kinds h3"
    using object_ptr_kinds_eq_h2
    by(auto simp add: shadow_root_ptr_kinds_def)
  have element_ptr_kinds_eq_h2: "element_ptr_kinds h2 = element_ptr_kinds h3"
    using object_ptr_kinds_eq_h2
    by(auto simp add: element_ptr_kinds_def node_ptr_kinds_def)

  have shadow_root_ptr_kinds_eq_h3: "shadow_root_ptr_kinds h3 = shadow_root_ptr_kinds h'"
    using object_ptr_kinds_eq_h3
    by(auto simp add: shadow_root_ptr_kinds_def)
  have element_ptr_kinds_eq_h3: "element_ptr_kinds h3 = element_ptr_kinds h'"
    using object_ptr_kinds_eq_h3
    by(auto simp add: element_ptr_kinds_def node_ptr_kinds_def)

  have "a_host_shadow_root_rel h2 = a_host_shadow_root_rel h3"
    by(auto simp add: a_host_shadow_root_rel_def element_ptr_kinds_eq_h2 shadow_root_eq2_h2)
  have "a_host_shadow_root_rel h3 = a_host_shadow_root_rel h'"
    by(auto simp add: a_host_shadow_root_rel_def element_ptr_kinds_eq_h3 shadow_root_eq2_h3)

  have "cast node ∉ {x. (x, ptr) ∈ (parent_child_rel h3 ∪ a_host_shadow_root_rel h3)^*}"
    using get_ancestors_si_parent_child_host_shadow_root_rel
    using ‹known_ptrs h2› ‹local.a_host_shadow_root_rel h2 = local.a_host_shadow_root_rel h3›
      ‹parent_child_rel h2 = parent_child_rel h3› ‹type_wf h2› ancestors_h2 node_not_in_ancestors_h2
      wellformed_h2
    by auto

  have "acyclic (parent_child_rel h3 ∪ a_host_shadow_root_rel h3)"
    using ‹heap_is_wellformed h2›
    by(auto simp add: heap_is_wellformed_def ‹parent_child_rel h2 = parent_child_rel h3›
        ‹a_host_shadow_root_rel h2 = a_host_shadow_root_rel h3›)
  then
  have "acyclic (parent_child_rel h' ∪ a_host_shadow_root_rel h')"
    apply(auto simp add: ‹a_host_shadow_root_rel h3 = a_host_shadow_root_rel h'›
        ‹parent_child_rel h' = insert (ptr, cast node) ((parent_child_rel h3))›)[1]
    using ‹cast node ∉ {x. (x, ptr) ∈ (parent_child_rel h3 ∪ a_host_shadow_root_rel h3)^*}›
    by (simp add: ‹local.a_host_shadow_root_rel h3 = local.a_host_shadow_root_rel h'›)
  then
  show "heap_is_wellformed h'"
    using ‹heap_is_wellformed h2›
    using ‹heap_is_wellformed_C_o_r_e_{_}_D_O_M h'›
    apply(auto simp add: heap_is_wellformed_def CD.heap_is_wellformed_def CD.acyclic_heap_def
        a_all_ptrs_in_heap_def a_distinct_lists_def a_shadow_root_valid_def)[1]
    by(auto simp add: object_ptr_kinds_eq_h2 object_ptr_kinds_eq_h3 element_ptr_kinds_eq_h2
        element_ptr_kinds_eq_h3 shadow_root_ptr_kinds_eq_h2 shadow_root_ptr_kinds_eq_h3  shadow_root_eq_h2
        shadow_root_eq_h3 shadow_root_eq2_h2 shadow_root_eq2_h3 tag_name_eq_h2 tag_name_eq_h3
        tag_name_eq2_h2 tag_name_eq2_h3)

qed
end

interpretation i_insert_before_wf?: l_insert_before_wf_C_o_r_e_{_}_D_O_M get_parent get_parent_locs
  get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs get_ancestors_si
  get_ancestors_si_locs adopt_node adopt_node_locs set_disconnected_nodes set_disconnected_nodes_locs
  get_disconnected_nodes get_disconnected_nodes_locs get_owner_document insert_before
  insert_before_locs append_child type_wf known_ptr known_ptrs heap_is_wellformed parent_child_rel
  by(simp add: l_insert_before_wf_C_o_r_e_{_}_D_O_{M_def} instances)
declare l_insert_before_wf_C_o_r_e_{_}_D_O_{M_axioms} [instances]

lemma insert_before_wf_is_l_insert_before_wf [instances]:
  "l_insert_before_wf Shadow_DOM.heap_is_wellformed ShadowRootClass.type_wf
ShadowRootClass.known_ptr ShadowRootClass.known_ptrs
     Shadow_DOM.insert_before Shadow_DOM.get_child_nodes"
  apply(auto simp add: l_insert_before_wf_def l_insert_before_wf_axioms_def instances)[1]
  using insert_before_removes_child apply fast
  done

lemma l_set_disconnected_nodes_get_disconnected_nodes_wf [instances]: "l_set_disconnected_nodes_get_disconnected_nodes_wf ShadowRootClass.type_wf
ShadowRootClass.known_ptr Shadow_DOM.heap_is_wellformed Shadow_DOM.parent_child_rel Shadow_DOM.get_child_nodes
     get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs"
  apply(auto simp add: l_set_disconnected_nodes_get_disconnected_nodes_wf_def
      l_set_disconnected_nodes_get_disconnected_nodes_wf_axioms_def instances)[1]
  by (metis Diff_iff Shadow_DOM.i_heap_is_wellformed.heap_is_wellformed_disconnected_nodes_distinct
      Shadow_DOM.i_remove_child.set_disconnected_nodes_get_disconnected_nodes insert_iff
      returns_result_eq set_remove1_eq)

interpretation l_get_root_node_si_wf_S_h_a_d_o_w_{_}_D_O_M
  type_wf known_ptr known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs
  get_host get_host_locs get_ancestors_si get_ancestors_si_locs get_root_node_si get_root_node_si_locs
  get_disconnected_nodes get_disconnected_nodes_locs get_shadow_root get_shadow_root_locs
  get_tag_name get_tag_name_locs heap_is_wellformed parent_child_rel heap_is_wellformed_C_o_r_e_{_}_D_O_M
  get_disconnected_document get_disconnected_document_locs
  by(auto simp add: l_get_root_node_si_wf_S_h_a_d_o_w_{_}_D_O_{M_def} instances)
declare l_get_root_node_si_wf_S_h_a_d_o_w_{_}_D_O_{M_axioms} [instances]

interpretation i_insert_before_wf2?: l_insert_before_wf2_S_h_a_d_o_w_{_}_D_O_M type_wf known_ptr get_child_nodes
  get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs set_child_nodes set_child_nodes_locs get_shadow_root get_shadow_root_locs set_disconnected_nodes set_disconnected_nodes_locs get_tag_name get_tag_name_locs heap_is_wellformed parent_child_rel get_ancestors_si get_ancestors_si_locs get_parent get_parent_locs adopt_node adopt_node_locs get_owner_document insert_before insert_before_locs append_child known_ptrs get_host get_host_locs get_root_node_si get_root_node_si_locs heap_is_wellformed_C_o_r_e_{_}_D_O_M get_disconnected_document get_disconnected_document_locs
  by(auto simp add: l_insert_before_wf2_S_h_a_d_o_w_{_}_D_O_{M_def} instances)
declare l_insert_before_wf2_S_h_a_d_o_w_{_}_D_O_{M_axioms} [instances]

lemma insert_before_wf2_is_l_insert_before_wf2 [instances]:
  "l_insert_before_wf2 ShadowRootClass.type_wf ShadowRootClass.known_ptr ShadowRootClass.known_ptrs
Shadow_DOM.insert_before
     Shadow_DOM.heap_is_wellformed"
  apply(auto simp add: l_insert_before_wf2_def l_insert_before_wf2_axioms_def instances)[1]
  using insert_before_child_preserves apply(fast, fast, fast)
  done

subsubsection ‹append\_child›

interpretation i_append_child_wf?: l_append_child_wf_C_o_r_e_{_}_D_O_M get_owner_document get_parent
  get_parent_locs remove_child remove_child_locs
  get_disconnected_nodes get_disconnected_nodes_locs
  set_disconnected_nodes set_disconnected_nodes_locs
  adopt_node adopt_node_locs known_ptr type_wf get_child_nodes
  get_child_nodes_locs known_ptrs set_child_nodes
  set_child_nodes_locs remove get_ancestors_si get_ancestors_si_locs
  insert_before insert_before_locs append_child heap_is_wellformed
  parent_child_rel
  by(auto simp add: l_append_child_wf_C_o_r_e_{_}_D_O_{M_def} instances)
declare l_append_child_wf_C_o_r_e_{_}_D_O_{M_axioms} [instances]

lemma append_child_wf_is_l_append_child_wf [instances]:
  "l_append_child_wf type_wf known_ptr known_ptrs append_child heap_is_wellformed"
  apply(auto simp add: l_append_child_wf_def l_append_child_wf_axioms_def instances)[1]
  using append_child_heap_is_wellformed_preserved by fast+

subsubsection ‹to\_tree\_order›

interpretation i_to_tree_order_wf?: l_to_tree_order_wf_C_o_r_e_{_}_D_O_M known_ptr type_wf get_child_nodes
  get_child_nodes_locs to_tree_order known_ptrs get_parent get_parent_locs heap_is_wellformed
  parent_child_rel get_disconnected_nodes get_disconnected_nodes_locs
  apply(auto simp add: l_to_tree_order_wf_C_o_r_e_{_}_D_O_{M_def} instances)[1]
  done
declare l_to_tree_order_wf_C_o_r_e_{_}_D_O_{M_axioms} [instances]

lemma to_tree_order_wf_is_l_to_tree_order_wf [instances]:
  "l_to_tree_order_wf heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs
to_tree_order get_parent get_child_nodes"
  apply(auto simp add: l_to_tree_order_wf_def l_to_tree_order_wf_axioms_def instances)[1]
  using to_tree_order_ok apply fast
  using to_tree_order_ptrs_in_heap apply fast
  using to_tree_order_parent_child_rel apply(fast, fast)
  using to_tree_order_child2 apply blast
  using to_tree_order_node_ptrs apply fast
  using to_tree_order_child apply fast
  using to_tree_order_ptr_in_result apply fast
  using to_tree_order_parent apply fast
  using to_tree_order_subset apply fast
  done

paragraph ‹get\_root\_node›

interpretation i_to_tree_order_wf_get_root_node_wf?: l_to_tree_order_wf_get_root_node_wf_C_o_r_e_{_}_D_O_M
  known_ptr type_wf known_ptrs heap_is_wellformed parent_child_rel
  get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs
  get_parent get_parent_locs get_ancestors get_ancestors_locs get_root_node get_root_node_locs
  to_tree_order
  by(auto simp add: l_to_tree_order_wf_get_root_node_wf_C_o_r_e_{_}_D_O_{M_def} instances)
declare l_to_tree_order_wf_get_root_node_wf_C_o_r_e_{_}_D_O_{M_axioms} [instances]

lemma to_tree_order_wf_get_root_node_wf_is_l_to_tree_order_wf_get_root_node_wf [instances]:
  "l_to_tree_order_wf_get_root_node_wf ShadowRootClass.type_wf ShadowRootClass.known_ptr
ShadowRootClass.known_ptrs to_tree_order Shadow_DOM.get_root_node
     Shadow_DOM.heap_is_wellformed"
  apply(auto simp add: l_to_tree_order_wf_get_root_node_wf_def
      l_to_tree_order_wf_get_root_node_wf_axioms_def instances)[1]
  using to_tree_order_get_root_node apply fast
  using to_tree_order_same_root apply fast
  done

subsubsection ‹to\_tree\_order\_si›

locale l_to_tree_order_si_wf_S_h_a_d_o_w_{_}_D_O_M =
  l_get_child_nodes +
  l_get_parent_get_host_wf_S_h_a_d_o_w_{_}_D_O_M +
  l_to_tree_order_si_S_h_a_d_o_w_{_}_D_O_M
begin
lemma to_tree_order_si_ok:
  assumes "heap_is_wellformed h" and "known_ptrs h" and "type_wf h"
    and "ptr |∈| object_ptr_kinds h"
  shows "h ⊨ ok (to_tree_order_si ptr)"
proof(insert assms(1) assms(4), induct rule: heap_wellformed_induct_si)
  case (step parent)
  have "known_ptr parent"
    using assms(2) local.known_ptrs_known_ptr step.prems
    by blast
  then show ?case
    using step
    using assms(1) assms(2) assms(3)
    using local.heap_is_wellformed_children_in_heap local.get_shadow_root_shadow_root_ptr_in_heap
    by(auto simp add: to_tree_order_si_def[of parent] intro: get_child_nodes_ok get_shadow_root_ok
        intro!: bind_is_OK_pure_I map_M_pure_I bind_pure_I map_M_ok_I split: option.splits)
qed
end

interpretation i_to_tree_order_si_wf?: l_to_tree_order_si_wf_S_h_a_d_o_w_{_}_D_O_M
  type_wf known_ptr get_child_nodes get_child_nodes_locs get_disconnected_nodes
  get_disconnected_nodes_locs get_shadow_root get_shadow_root_locs get_tag_name
  get_tag_name_locs heap_is_wellformed parent_child_rel heap_is_wellformed_C_o_r_e_{_}_D_O_M
  get_host get_host_locs get_disconnected_document get_disconnected_document_locs
  known_ptrs get_parent get_parent_locs to_tree_order_si
  by(auto simp add: l_to_tree_order_si_wf_S_h_a_d_o_w_{_}_D_O_{M_def} instances)
declare l_to_tree_order_si_wf_S_h_a_d_o_w_{_}_D_O_{M_axioms} [instances]

subsubsection ‹get\_assigned\_nodes›

lemma forall_M_small_big: "h ⊨ forall_M f xs →_h h' ==> P h ==>
(∧h h' x. x ∈ set xs ==> h ⊨ f x →_h h' ==> P h ==> P h') ==> P h'"
  by(induct xs arbitrary: h) (auto elim!: bind_returns_heap_E)

locale l_assigned_nodes_wf_S_h_a_d_o_w_{_}_D_O_M =
  l_assigned_nodes_S_h_a_d_o_w_{_}_D_O_M +
  l_heap_is_wellformed +
  l_remove_child_wf2 +
  l_append_child_wf +
  l_remove_shadow_root_wf_S_h_a_d_o_w_{_}_D_O_M
begin

lemma assigned_nodes_distinct:
  assumes "heap_is_wellformed h"
  assumes "h ⊨ assigned_nodes slot →_r nodes"
  shows "distinct nodes"
proof -
  have "∧ptr children. h ⊨ get_child_nodes ptr →_r children ==> distinct children"
    using assms(1) local.heap_is_wellformed_children_distinct by blast
  then show ?thesis
    using assms
    apply(auto simp add: assigned_nodes_def elim!: bind_returns_result_E2 split: if_splits)[1]
    by (simp add: filter_M_distinct)
qed

lemma flatten_dom_preserves:
  assumes "heap_is_wellformed h" and "known_ptrs h" and "type_wf h"
  assumes "h ⊨ flatten_dom →_h h'"
  shows "heap_is_wellformed h'" and "known_ptrs h'" and "type_wf h'"
proof -
  obtain tups h2 element_ptrs shadow_root_ptrs where
    "h ⊨ element_ptr_kinds_M →_r element_ptrs" and
    tups: "h ⊨ map_filter_M2 (λelement_ptr. do {
      tag ← get_tag_name element_ptr;
      assigned_nodes ← assigned_nodes element_ptr;
      (if tag = ''slot'' ∧ assigned_nodes ≠ [] then
return (Some (element_ptr, assigned_nodes)) else return None)}) element_ptrs →_r tups"
    (is "h ⊨ map_filter_M2 ?f element_ptrs →_r tups") and
    h2: "h ⊨ forall_M (λ(slot, assigned_nodes). do {
      get_child_nodes (cast slot) 🍋 forall_M remove;
      forall_M (append_child (cast slot)) assigned_nodes
    }) tups →_h h2" and
    "h2 ⊨ shadow_root_ptr_kinds_M →_r shadow_root_ptrs" and
    h': "h2 ⊨ forall_M (λshadow_root_ptr. do {
      host ← get_host shadow_root_ptr;
      get_child_nodes (cast host) 🍋 forall_M remove;
      get_child_nodes (cast shadow_root_ptr) 🍋 forall_M (append_child (cast host));
      remove_shadow_root host
    }) shadow_root_ptrs →_h h'"
    using ‹h ⊨ flatten_dom →_h h'›
    apply(auto simp add: flatten_dom_def elim!: bind_returns_heap_E
        bind_returns_heap_E2[rotated, OF ElementMonad.ptr_kinds_M_pure, rotated]
        bind_returns_heap_E2[rotated, OF ShadowRootMonad.ptr_kinds_M_pure, rotated])[1]
    apply(drule pure_returns_heap_eq)
    by(auto intro!: map_filter_M2_pure bind_pure_I)
  have "heap_is_wellformed h2 ∧ known_ptrs h2 ∧ type_wf h2"
    using h2 ‹heap_is_wellformed h› ‹known_ptrs h› ‹type_wf h›
    by(auto elim!: bind_returns_heap_E bind_returns_heap_E2[rotated,
          OF get_child_nodes_pure, rotated]
        elim!: forall_M_small_big[where P = "λh. heap_is_wellformed h ∧ known_ptrs h ∧ type_wf h",
          simplified]
        intro: remove_preserves_known_ptrs remove_heap_is_wellformed_preserved
        remove_preserves_type_wf
        append_child_preserves_known_ptrs append_child_heap_is_wellformed_preserved
        append_child_preserves_type_wf)
  then
  show "heap_is_wellformed h'" and "known_ptrs h'" and "type_wf h'"
    using h'
    by(auto elim!: bind_returns_heap_E bind_returns_heap_E2[rotated, OF get_host_pure, rotated]
        bind_returns_heap_E2[rotated, OF get_child_nodes_pure, rotated]
        dest!: forall_M_small_big[where P = "λh. heap_is_wellformed h ∧ known_ptrs h ∧ type_wf h",
          simplified]
        intro: remove_preserves_known_ptrs remove_heap_is_wellformed_preserved
        remove_preserves_type_wf
        append_child_preserves_known_ptrs append_child_heap_is_wellformed_preserved
        append_child_preserves_type_wf
        remove_shadow_root_preserves
        )
qed
end

interpretation i_assigned_nodes_wf?: l_assigned_nodes_wf_S_h_a_d_o_w_{_}_D_O_M
  known_ptr assigned_nodes assigned_nodes_flatten flatten_dom get_child_nodes get_child_nodes_locs
  get_tag_name get_tag_name_locs get_root_node get_root_node_locs get_host get_host_locs find_slot
  assigned_slot remove insert_before insert_before_locs append_child remove_shadow_root
  remove_shadow_root_locs type_wf get_shadow_root get_shadow_root_locs set_shadow_root
  set_shadow_root_locs get_parent get_parent_locs to_tree_order heap_is_wellformed parent_child_rel
  get_disconnected_nodes get_disconnected_nodes_locs known_ptrs remove_child remove_child_locs
  heap_is_wellformed_C_o_r_e_{_}_D_O_M get_disconnected_document get_disconnected_document_locs
  by(auto simp add: l_assigned_nodes_wf_S_h_a_d_o_w_{_}_D_O_{M_def} instances)
declare l_assigned_nodes_wf_S_h_a_d_o_w_{_}_D_O_{M_axioms} [instances]

subsubsection ‹get\_shadow\_root\_safe›

locale l_get_shadow_root_safe_wf_S_h_a_d_o_w_{_}_D_O_M =
  l_heap_is_wellformed_S_h_a_d_o_w_{_}_D_O_M  get_child_nodes get_child_nodes_locs get_disconnected_nodes
  get_disconnected_nodes_locs get_shadow_root get_shadow_root_locs get_tag_name get_tag_name_locs
  known_ptr type_wf heap_is_wellformed parent_child_rel heap_is_wellformed_C_o_r_e_{_}_D_O_M get_host
  get_host_locs +
  l_type_wf type_wf +
  l_get_shadow_root_safe_S_h_a_d_o_w_{_}_D_O_M type_wf get_shadow_root_safe get_shadow_root_safe_locs
  get_shadow_root get_shadow_root_locs get_mode get_mode_locs
  for known_ptr :: "(_::linorder) object_ptr ==> bool"
    and known_ptrs :: "(_) heap ==> bool"
    and type_wf :: "(_) heap ==> bool"
    and get_host :: "(_) shadow_root_ptr ==> ((_) heap, exception, (_) element_ptr) prog"
    and get_host_locs :: "((_) heap ==> (_) heap ==> bool) set"
    and get_shadow_root ::
    "(_) element_ptr ==> ((_) heap, exception, (_) shadow_root_ptr option) prog"
    and get_shadow_root_locs :: "(_) element_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
    and get_shadow_root_safe ::
    "(_) element_ptr ==> ((_) heap, exception, (_) shadow_root_ptr option) prog"
    and get_shadow_root_safe_locs ::
    "(_) element_ptr ==> (_) shadow_root_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
    and get_child_nodes :: "(_::linorder) object_ptr ==> ((_) heap, exception, (_) node_ptr list) prog"
    and get_child_nodes_locs :: "(_) object_ptr ==> ((_) heap ==> (_) heap ==> bool) 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 get_tag_name :: "(_) element_ptr ==> ((_) heap, exception, char list) prog"
    and get_tag_name_locs :: "(_) element_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
    and heap_is_wellformed :: "(_) heap ==> bool"
    and parent_child_rel :: "(_) heap ==> ((_) object_ptr × (_) object_ptr) set"
    and heap_is_wellformed_C_o_r_e_{_}_D_O_M :: "(_) heap ==> bool"
    and get_mode :: "(_) shadow_root_ptr ==> ((_) heap, exception, shadow_root_mode) prog"
    and get_mode_locs :: "(_) shadow_root_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
begin

end

subsubsection ‹create\_element›

locale l_create_element_wf_S_h_a_d_o_w_{_}_D_O_M =
  l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs +
  l_set_tag_name type_wf set_tag_name set_tag_name_locs +
  l_create_element_defs create_element +
  l_heap_is_wellformed_S_h_a_d_o_w_{_}_D_O_M  get_child_nodes get_child_nodes_locs get_disconnected_nodes
  get_disconnected_nodes_locs
  get_shadow_root get_shadow_root_locs get_tag_name get_tag_name_locs known_ptr type_wf
  heap_is_wellformed parent_child_rel
  heap_is_wellformed_C_o_r_e_{_}_D_O_M get_host get_host_locs get_disconnected_document
  get_disconnected_document_locs +
  l_new_element_get_disconnected_nodes get_disconnected_nodes get_disconnected_nodes_locs +
  l_set_tag_name_get_disconnected_nodes type_wf set_tag_name set_tag_name_locs
  get_disconnected_nodes get_disconnected_nodes_locs +
  l_create_element_S_h_a_d_o_w_{_}_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
  type_wf_C_o_r_e_{_}_D_O_M known_ptr_C_o_r_e_{_}_D_O_M +
  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_set_tag_name_get_tag_name type_wf get_tag_name get_tag_name_locs set_tag_name set_tag_name_locs +
  l_new_element_get_tag_name type_wf get_tag_name get_tag_name_locs +
  l_set_disconnected_nodes_get_child_nodes set_disconnected_nodes set_disconnected_nodes_locs
  get_child_nodes get_child_nodes_locs +
  l_set_disconnected_nodes_get_shadow_root set_disconnected_nodes set_disconnected_nodes_locs
  get_shadow_root get_shadow_root_locs +
  l_set_disconnected_nodes_get_tag_name type_wf set_disconnected_nodes set_disconnected_nodes_locs
  get_tag_name get_tag_name_locs +
  l_set_disconnected_nodes type_wf set_disconnected_nodes set_disconnected_nodes_locs +
  l_set_disconnected_nodes_get_disconnected_nodes  type_wf get_disconnected_nodes
  get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs +
  l_new_element_get_shadow_root type_wf get_shadow_root get_shadow_root_locs +
  l_set_tag_name_get_shadow_root type_wf set_tag_name set_tag_name_locs get_shadow_root
  get_shadow_root_locs +
  l_new_element type_wf +
  l_known_ptrs known_ptr known_ptrs
  for known_ptr :: "(_::linorder) object_ptr ==> bool"
    and known_ptrs :: "(_) heap ==> bool"
    and type_wf :: "(_) heap ==> bool"
    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_disconnected_nodes :: "(_) document_ptr ==> ((_) heap, exception, (_) node_ptr list) prog"
    and get_disconnected_nodes_locs :: "(_) document_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
    and heap_is_wellformed :: "(_) heap ==> bool"
    and parent_child_rel :: "(_) heap ==> ((_) object_ptr × (_) object_ptr) 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 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_element ::
    "(_) document_ptr ==> char list ==> ((_) heap, exception, (_) element_ptr) prog"
    and get_shadow_root :: "(_) element_ptr ==> ((_) heap, exception, (_) shadow_root_ptr option) prog"
    and get_shadow_root_locs :: "(_) element_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
    and get_tag_name :: "(_) element_ptr ==> ((_) heap, exception, char list) prog"
    and get_tag_name_locs :: "(_) element_ptr ==> ((_) heap ==> (_) heap ==> bool) set"
    and heap_is_wellformed_C_o_r_e_{_}_D_O_M :: "(_) heap ==> bool"
    and get_host :: "(_) shadow_root_ptr ==> ((_) heap, exception, (_) element_ptr) prog"
    and get_host_locs :: "((_) heap ==> (_) heap ==> bool) set"
    and get_disconnected_document :: "(_) node_ptr ==> ((_) heap, exception, (_) document_ptr) prog"
    and get_disconnected_document_locs :: "((_) heap ==> (_) heap ==> bool) set"
    and known_ptr_C_o_r_e_{_}_D_O_M :: "(_::linorder) object_ptr ==> bool"
    and type_wf_C_o_r_e_{_}_D_O_M :: "(_) heap ==> bool"
begin
lemma create_element_preserves_wellformedness:
  assumes "heap_is_wellformed h"
    and "h ⊨ create_element document_ptr tag →_h h'"
    and "type_wf h"
    and "known_ptrs h"
  shows "heap_is_wellformed h'" and "type_wf h'" and "known_ptrs 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(2)
    by(auto simp add: create_element_def
        elim!: bind_returns_heap_E
        bind_returns_heap_E2[rotated, OF CD.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 CD.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)
  then have element_ptr_kinds_eq_h2: "element_ptr_kinds h3 = element_ptr_kinds h2"
    by(simp add: element_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)
  then have element_ptr_kinds_eq_h3: "element_ptr_kinds h' = element_ptr_kinds h3"
    by(simp add: element_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
  show "known_ptrs h'"
    using known_ptrs_preserved object_ptr_kinds_eq_h3 by blast

  have "document_ptr |\<in>| document_ptr_kinds h"
    using disc_nodes_h3 document_ptr_kinds_eq_h object_ptr_kinds_eq_h2
      CD.get_disconnected_nodes_ptr_in_heap \<open>type_wf h\<close> document_ptr_kinds_def
    by (metis is_OK_returns_result_I)

  have children_eq_h: "\<And>(ptr'::(_) object_ptr) children. ptr' \<noteq> cast new_element_ptr
              \<Longrightarrow> h \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children = h2 \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children"
    using CD.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: "\<And>ptr'. ptr' \<noteq> cast new_element_ptr
                                    \<Longrightarrow> |h \<turnstile> get_child_nodes ptr'|\<^sub>r = |h2 \<turnstile> get_child_nodes ptr'|\<^sub>r"
    using select_result_eq by force
  have "h2 \<turnstile> get_child_nodes (cast new_element_ptr) \<rightarrow>\<^sub>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 tag_name_eq_h:
    "\<And>ptr' disc_nodes. ptr' \<noteq>  new_element_ptr
              \<Longrightarrow> h \<turnstile> get_tag_name ptr' \<rightarrow>\<^sub>r disc_nodes
                                             = h2 \<turnstile> get_tag_name ptr' \<rightarrow>\<^sub>r disc_nodes"
    using get_tag_name_reads h2 get_tag_name_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 tag_name_eq2_h: "\<And>ptr'. ptr' \<noteq> new_element_ptr
                                    \<Longrightarrow> |h \<turnstile> get_tag_name ptr'|\<^sub>r = |h2 \<turnstile> get_tag_name ptr'|\<^sub>r"
    using select_result_eq by force
  have "h2 \<turnstile> get_tag_name new_element_ptr \<rightarrow>\<^sub>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_empty_tag_name
    by blast

  have disconnected_nodes_eq_h:
    "\<And>doc_ptr disc_nodes. h \<turnstile> get_disconnected_nodes doc_ptr \<rightarrow>\<^sub>r disc_nodes
                                             = h2 \<turnstile> get_disconnected_nodes doc_ptr \<rightarrow>\<^sub>r disc_nodes"
    using CD.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:
    "\<And>doc_ptr. |h \<turnstile> get_disconnected_nodes doc_ptr|\<^sub>r = |h2 \<turnstile> get_disconnected_nodes doc_ptr|\<^sub>r"
    using select_result_eq by force

  have children_eq_h2:
    "\<And>ptr' children. h2 \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children = h3 \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children"
    using CD.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: "\<And>ptr'. |h2 \<turnstile> get_child_nodes ptr'|\<^sub>r = |h3 \<turnstile> get_child_nodes ptr'|\<^sub>r"
    using select_result_eq by force
  have disconnected_nodes_eq_h2:
    "\<And>doc_ptr disc_nodes. h2 \<turnstile> get_disconnected_nodes doc_ptr \<rightarrow>\<^sub>r disc_nodes
                                               = h3 \<turnstile> get_disconnected_nodes doc_ptr \<rightarrow>\<^sub>r disc_nodes"
    using CD.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:
    "\<And>doc_ptr. |h2 \<turnstile> get_disconnected_nodes doc_ptr|\<^sub>r = |h3 \<turnstile> get_disconnected_nodes doc_ptr|\<^sub>r"
    using select_result_eq by force
  have tag_name_eq_h2:
    "\<And>ptr' disc_nodes. ptr' \<noteq>  new_element_ptr
              \<Longrightarrow> h2 \<turnstile> get_tag_name ptr' \<rightarrow>\<^sub>r disc_nodes
                                             = h3 \<turnstile> get_tag_name ptr' \<rightarrow>\<^sub>r disc_nodes"
    apply(rule reads_writes_preserved[OF get_tag_name_reads set_tag_name_writes h3])
    by (metis local.set_tag_name_get_tag_name_different_pointers)
  then have tag_name_eq2_h2: "\<And>ptr'. ptr' \<noteq> new_element_ptr
                                    \<Longrightarrow> |h2 \<turnstile> get_tag_name ptr'|\<^sub>r = |h3 \<turnstile> get_tag_name ptr'|\<^sub>r"
    using select_result_eq by force
  have "h2 \<turnstile> get_tag_name new_element_ptr \<rightarrow>\<^sub>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_empty_tag_name
    by blast

  have "type_wf h2"
    using \<open>type_wf h\<close> new_element_types_preserved h2 by blast
  then have "type_wf h3"
    using writes_small_big[where P="\<lambda>h h'. type_wf h \<longrightarrow> type_wf h'", OF set_tag_name_writes h3]
    using  set_tag_name_types_preserved
    by(auto simp add: reflp_def transp_def)
  then show "type_wf h'"
    using writes_small_big[where P="\<lambda>h h'. type_wf h \<longrightarrow> 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:
    "\<And>ptr' children. h3 \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children = h' \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children"
    using CD.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: "\<And>ptr'. |h3 \<turnstile> get_child_nodes ptr'|\<^sub>r = |h' \<turnstile> get_child_nodes ptr'|\<^sub>r"
    using select_result_eq by force
  have disconnected_nodes_eq_h3:
    "\<And>doc_ptr disc_nodes. document_ptr \<noteq> doc_ptr
       \<Longrightarrow> h3 \<turnstile> get_disconnected_nodes doc_ptr \<rightarrow>\<^sub>r disc_nodes
                                               = h' \<turnstile> get_disconnected_nodes doc_ptr \<rightarrow>\<^sub>r disc_nodes"
    using CD.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:
    "\<And>doc_ptr. document_ptr \<noteq> doc_ptr
                \<Longrightarrow> |h3 \<turnstile> get_disconnected_nodes doc_ptr|\<^sub>r = |h' \<turnstile> get_disconnected_nodes doc_ptr|\<^sub>r"
    using select_result_eq by force
  have tag_name_eq_h2:
    "\<And>ptr' disc_nodes. ptr' \<noteq>  new_element_ptr
              \<Longrightarrow> h2 \<turnstile> get_tag_name ptr' \<rightarrow>\<^sub>r disc_nodes
                                             = h3 \<turnstile> get_tag_name ptr' \<rightarrow>\<^sub>r disc_nodes"
    apply(rule reads_writes_preserved[OF get_tag_name_reads set_tag_name_writes h3])
    by (metis local.set_tag_name_get_tag_name_different_pointers)
  then have tag_name_eq2_h2: "\<And>ptr'. ptr' \<noteq> new_element_ptr
                                    \<Longrightarrow> |h2 \<turnstile> get_tag_name ptr'|\<^sub>r = |h3 \<turnstile> get_tag_name ptr'|\<^sub>r"
    using select_result_eq by force

  have disc_nodes_document_ptr_h2: "h2 \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes_h3"
    using disconnected_nodes_eq_h2 disc_nodes_h3 by auto
  then have disc_nodes_document_ptr_h: "h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes_h3"
    using disconnected_nodes_eq_h by auto
  then have "cast new_element_ptr \<notin> set disc_nodes_h3"
    using \<open>heap_is_wellformed h\<close>
    using \<open>cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr \<notin> set |h \<turnstile> node_ptr_kinds_M|\<^sub>r\<close>
      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 tag_name_eq_h3:
    "\<And>ptr' disc_nodes. h3 \<turnstile> get_tag_name ptr' \<rightarrow>\<^sub>r disc_nodes
                                             = h' \<turnstile> get_tag_name ptr' \<rightarrow>\<^sub>r disc_nodes"
    apply(rule reads_writes_preserved[OF get_tag_name_reads set_disconnected_nodes_writes h'])
    using set_disconnected_nodes_get_tag_name
    by blast
  then have tag_name_eq2_h3: "\<And>ptr'. |h3 \<turnstile> get_tag_name ptr'|\<^sub>r = |h' \<turnstile> get_tag_name ptr'|\<^sub>r"
    using select_result_eq by force

  have "acyclic (parent_child_rel h)"
    using \<open>heap_is_wellformed h\<close>
    by (simp add: heap_is_wellformed_def CD.heap_is_wellformed_def CD.acyclic_heap_def)
  also have "parent_child_rel h = parent_child_rel h2"
  proof(auto simp add: CD.parent_child_rel_def)[1]
    fix a x
    assume 0: "a |\<in>| object_ptr_kinds h"
      and 1: "x \<in> set |h \<turnstile> get_child_nodes a|\<^sub>r"
    then show "a |\<in>| object_ptr_kinds h2"
      by (simp add: object_ptr_kinds_eq_h)
  next
    fix a x
    assume 0: "a |\<in>| object_ptr_kinds h"
      and 1: "x \<in> set |h \<turnstile> get_child_nodes a|\<^sub>r"
    then show "x \<in> set |h2 \<turnstile> get_child_nodes a|\<^sub>r"
      by (metis ObjectMonad.ptr_kinds_ptr_kinds_M
          \<open>cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr \<notin> set |h \<turnstile> object_ptr_kinds_M|\<^sub>r\<close> children_eq2_h)
  next
    fix a x
    assume 0: "a |\<in>| object_ptr_kinds h2"
      and 1: "x \<in> set |h2 \<turnstile> get_child_nodes a|\<^sub>r"
    then show "a |\<in>| object_ptr_kinds h"
      using object_ptr_kinds_eq_h \<open>h2 \<turnstile> get_child_nodes (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr) \<rightarrow>\<^sub>r []\<close>
      by(auto)
  next
    fix a x
    assume 0: "a |\<in>| object_ptr_kinds h2"
      and 1: "x \<in> set |h2 \<turnstile> get_child_nodes a|\<^sub>r"
    then show "x \<in> set |h \<turnstile> get_child_nodes a|\<^sub>r"
      by (metis (no_types, lifting)
          \<open>h2 \<turnstile> get_child_nodes (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr) \<rightarrow>\<^sub>r []\<close>
          children_eq2_h empty_iff empty_set image_eqI select_result_I2)
  qed
  also have "\<dots> = parent_child_rel h3"
    by(auto simp add: CD.parent_child_rel_def object_ptr_kinds_eq_h2 children_eq2_h2)
  also have "\<dots> = parent_child_rel h'"
    by(auto simp add: CD.parent_child_rel_def object_ptr_kinds_eq_h3 children_eq2_h3)
  finally have "CD.a_acyclic_heap h'"
    by (simp add: CD.acyclic_heap_def)

  have "CD.a_all_ptrs_in_heap h"
    using \<open>heap_is_wellformed h\<close>  by (simp add: heap_is_wellformed_def CD.heap_is_wellformed_def)
  then have "CD.a_all_ptrs_in_heap h2"
    apply(auto simp add: CD.a_all_ptrs_in_heap_def)[1]
     apply (metis \<open>known_ptrs h2\<close> \<open>parent_child_rel h = parent_child_rel h2\<close> \<open>type_wf h2\<close> assms(1)
        assms(3) funion_iff CD.get_child_nodes_ok local.known_ptrs_known_ptr
        local.parent_child_rel_child_in_heap CD.parent_child_rel_child_nodes2 node_ptr_kinds_commutes
        node_ptr_kinds_eq_h returns_result_select_result)
    by (metis (no_types, opaque_lifting) CD.get_child_nodes_ok CD.get_child_nodes_ptr_in_heap
        \<open>h2 \<turnstile> get_child_nodes (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr) \<rightarrow>\<^sub>r []\<close> assms(3) assms(4)
        children_eq_h disconnected_nodes_eq2_h document_ptr_kinds_eq_h is_OK_returns_result_I
        local.known_ptrs_known_ptr node_ptr_kinds_commutes returns_result_select_result subsetD)
  then have "CD.a_all_ptrs_in_heap h3"
    by (simp add: children_eq2_h2 disconnected_nodes_eq2_h2 document_ptr_kinds_eq_h2
        CD.a_all_ptrs_in_heap_def node_ptr_kinds_eq_h2 object_ptr_kinds_eq_h2)
  then have "CD.a_all_ptrs_in_heap h'"
    by (smt (verit) children_eq2_h3 disc_nodes_document_ptr_h2 disconnected_nodes_eq2_h2
        disconnected_nodes_eq2_h3 document_ptr_kinds_eq_h3 finsertCI funion_finsert_right
        h' insert_iff list.simps(15) local.CD.a_all_ptrs_in_heap_def
        local.set_disconnected_nodes_get_disconnected_nodes node_ptr_kinds_eq_h node_ptr_kinds_eq_h2
        node_ptr_kinds_eq_h3 object_ptr_kinds_eq_h3 select_result_I2 subsetD subsetI)
  have "\<And>p. p |\<in>| object_ptr_kinds h \<Longrightarrow> cast new_element_ptr \<notin> set |h \<turnstile> get_child_nodes p|\<^sub>r"
    using \<open>heap_is_wellformed h\<close> \<open>cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr \<notin> set |h \<turnstile> node_ptr_kinds_M|\<^sub>r\<close>
      heap_is_wellformed_children_in_heap
    by (meson NodeMonad.ptr_kinds_ptr_kinds_M CD.a_all_ptrs_in_heap_def assms(3) assms(4) fset_mp
        fset_of_list_elem CD.get_child_nodes_ok known_ptrs_known_ptr returns_result_select_result)
  then have "\<And>p. p |\<in>| object_ptr_kinds h2 \<Longrightarrow> cast new_element_ptr \<notin> set |h2 \<turnstile> get_child_nodes p|\<^sub>r"
    using children_eq2_h
    apply(auto simp add: object_ptr_kinds_eq_h)[1]
    using \<open>h2 \<turnstile> get_child_nodes (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr) \<rightarrow>\<^sub>r []\<close> apply auto[1]
    by (metis ObjectMonad.ptr_kinds_ptr_kinds_M
        \<open>cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr \<notin> set |h \<turnstile> object_ptr_kinds_M|\<^sub>r\<close>)
  then have "\<And>p. p |\<in>| object_ptr_kinds h3 \<Longrightarrow> cast new_element_ptr \<notin> set |h3 \<turnstile> get_child_nodes p|\<^sub>r"
    using object_ptr_kinds_eq_h2 children_eq2_h2 by auto
  then have new_element_ptr_not_in_any_children:
    "\<And>p. p |\<in>| object_ptr_kinds h' \<Longrightarrow> cast new_element_ptr \<notin> set |h' \<turnstile> get_child_nodes p|\<^sub>r"
    using object_ptr_kinds_eq_h3 children_eq2_h3 by auto

  have "CD.a_distinct_lists h"
    using \<open>heap_is_wellformed h\<close>
    by (simp add: CD.heap_is_wellformed_def heap_is_wellformed_def)
  then have "CD.a_distinct_lists h2"
    using \<open>h2 \<turnstile> get_child_nodes (cast new_element_ptr) \<rightarrow>\<^sub>r []\<close>
    apply(auto simp add: CD.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_remove)
      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(1) assms(3) assms(4) empty_iff  CD.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 \<open> CD.a_distinct_lists h\<close> \<open>type_wf h2\<close> disconnected_nodes_eq_h document_ptr_kinds_eq_h
--> --------------------

--> maximum size reached

--> --------------------

Messung V0.5 in Prozent

¤ Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.1.23Bemerkung: (vorverarbeitet am 2026-06-10) ¤

*Bot Zugriff

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.