Theory Gabow_SCC.Gabow_Skeleton

section ‹Skeleton for Gabow's SCC Algorithm \label{sec:skel}›
theory Gabow_Skeleton
imports CAVA_Automata.Digraph
begin

(* TODO: convenience locale, consider merging this with invariants *)
locale fr_graph =
  graph G
  for G :: "('v, 'more) graph_rec_scheme"
  +
  assumes finite_reachableE_V0[simp, intro!]: "finite (E* `` V0)"

text ‹
  In this theory, we formalize a skeleton of Gabow's SCC algorithm. 
  The skeleton serves as a starting point to develop concrete algorithms,
  like enumerating the SCCs or checking emptiness of a generalized Büchi automaton.
›

section ‹Statistics Setup›
text ‹
  We define some dummy-constants that are included into the generated code,
  and may be mapped to side-effecting ML-code that records statistics and debug information
  about the execution. In the skeleton algorithm, we count the number of visited nodes,
  and include a timing for the whole algorithm.
›

definition stat_newnode :: "unit => unit"   ― ‹Invoked if new node is visited›
  where [code]: "stat_newnode  λ_. ()"

definition stat_start :: "unit => unit"     ― ‹Invoked once if algorithm starts›
  where [code]: "stat_start  λ_. ()"

definition stat_stop :: "unit => unit"      ― ‹Invoked once if algorithm stops›
  where [code]: "stat_stop  λ_. ()"

lemma [autoref_rules]: 
  "(stat_newnode,stat_newnode)  unit_rel  unit_rel"
  "(stat_start,stat_start)  unit_rel  unit_rel"
  "(stat_stop,stat_stop)  unit_rel  unit_rel"
  by auto

abbreviation "stat_newnode_nres  RETURN (stat_newnode ())"
abbreviation "stat_start_nres  RETURN (stat_start ())"
abbreviation "stat_stop_nres  RETURN (stat_stop ())"

lemma discard_stat_refine[refine]:
  "m1m2  stat_newnode_nres  m1  m2"
  "m1m2  stat_start_nres  m1  m2"
  "m1m2  stat_stop_nres  m1  m2"
  by simp_all

section ‹Abstract Algorithm›
text ‹
  In this section, we formalize an abstract version of a path-based SCC algorithm.
  Later, this algorithm will be refined to use Gabow's data structure.
›

subsection ‹Preliminaries›
definition path_seg :: "'a set list  nat  nat  'a set"
  ― ‹Set of nodes in a segment of the path›
  where "path_seg p i j  {p!k|k. ik  k<j}"

lemma path_seg_simps[simp]: 
  "ji  path_seg p i j = {}"
  "path_seg p i (Suc i) = p!i"
  unfolding path_seg_def
  apply auto []
  apply (auto simp: le_less_Suc_eq) []
  done

lemma path_seg_drop:
  "(set (drop i p)) = path_seg p i (length p)"
  unfolding path_seg_def
  by (fastforce simp: in_set_drop_conv_nth Bex_def)

lemma path_seg_butlast: 
  "p[]  path_seg p 0 (length p - Suc 0) = (set (butlast p))"
  apply (cases p rule: rev_cases, simp)
  apply (fastforce simp: path_seg_def nth_append in_set_conv_nth)
  done

definition idx_of :: "'a set list  'a  nat"
  ― ‹Index of path segment that contains a node›
  where "idx_of p v  THE i. i<length p  vp!i"

lemma idx_of_props:
  assumes 
    p_disjoint_sym: "i j v. i<length p  j<length p  vp!i  vp!j  i=j"
  assumes ON_STACK: "v(set p)"
  shows 
    "idx_of p v < length p" and
    "v  p ! idx_of p v"
proof -
  from ON_STACK obtain i where "i<length p" "v  p ! i"
    by (auto simp add: in_set_conv_nth)
  moreover hence "j<length p. vp ! j  i=j"
    using p_disjoint_sym by auto
  ultimately show "idx_of p v < length p" 
    and "v  p ! idx_of p v" unfolding idx_of_def
    by (metis (lifting) theI')+
qed

lemma idx_of_uniq:
  assumes 
    p_disjoint_sym: "i j v. i<length p  j<length p  vp!i  vp!j  i=j"
  assumes A: "i<length p" "vp!i"
  shows "idx_of p v = i"
proof -
  from A p_disjoint_sym have "j<length p. vp ! j  i=j" by auto
  with A show ?thesis
    unfolding idx_of_def
    by (metis (lifting) the_equality)
qed


subsection ‹Invariants›
text ‹The state of the inner loop consists of the path p› of
  collapsed nodes, the set D› of finished (done) nodes, and the set
  pE› of pending edges.›
type_synonym 'v abs_state = "'v set list × 'v set × ('v×'v) set"

context fr_graph
begin
  definition touched :: "'v set list  'v set  'v set" 
    ― ‹Touched: Nodes that are done or on path›
    where "touched p D  D  (set p)"

  definition vE :: "'v set list  'v set  ('v × 'v) set  ('v × 'v) set"
    ― ‹Visited edges: No longer pending edges from touched nodes›
    where "vE p D pE  (E  (touched p D × UNIV)) - pE"

  lemma vE_ss_E: "vE p D pE  E" ― ‹Visited edges are edges›
    unfolding vE_def by auto

end

locale outer_invar_loc ― ‹Invariant of the outer loop›
  = fr_graph G for G :: "('v,'more) graph_rec_scheme" +
  fixes it :: "'v set" ― ‹Remaining nodes to iterate over›
  fixes D :: "'v set" ― ‹Finished nodes›

  assumes it_initial: "itV0"  ― ‹Only start nodes to iterate over›

  assumes it_done: "V0 - it  D"  ― ‹Nodes already iterated over are visited›
  assumes D_reachable: "DE*``V0" ― ‹Done nodes are reachable›
  assumes D_closed: "E``D  D" ― ‹Done is closed under transitions›
begin

  lemma locale_this: "outer_invar_loc G it D" by unfold_locales

  definition (in fr_graph) "outer_invar  λit D. outer_invar_loc G it D"

  lemma outer_invar_this[simp, intro!]: "outer_invar it D"
    unfolding outer_invar_def apply simp by unfold_locales 
end

locale invar_loc ― ‹Invariant of the inner loop›
  = fr_graph G
  for G :: "('v, 'more) graph_rec_scheme" +
  fixes v0 :: "'v"
  fixes D0 :: "'v set"
  fixes p :: "'v set list"
  fixes D :: "'v set"
  fixes pE :: "('v×'v) set"

  assumes v0_initial[simp, intro!]: "v0V0"
  assumes D_incr: "D0  D"

  assumes pE_E_from_p: "pE  E  ((set p)) × UNIV" 
    ― ‹Pending edges are edges from path›
  assumes E_from_p_touched: "E  ((set p) × UNIV)  pE  UNIV × touched p D" 
    ― ‹Edges from path are pending or touched›
  assumes D_reachable: "DE*``V0" ― ‹Done nodes are reachable›
  assumes p_connected: "Suc i<length p  p!i × p!Suc i  (E-pE)  {}"
    ― ‹CNodes on path are connected by non-pending edges›

  assumes p_disjoint: "i<j; j<length p  p!i  p!j = {}" 
    ― ‹CNodes on path are disjoint›
  assumes p_sc: "Uset p  U×U  (vE p D pE  U×U)*" 
    ― ‹Nodes in CNodes are mutually reachable by visited edges›

  assumes root_v0: "p[]  v0hd p" ― ‹Root CNode contains start node›
  assumes p_empty_v0: "p=[]  v0D" ― ‹Start node is done if path empty›
  
  assumes D_closed: "E``D  D" ― ‹Done is closed under transitions›
  (*assumes D_vis: "E∩D×D ⊆ vE" -- "All edges from done nodes are visited"*)

  assumes vE_no_back: "i<j; j<length p  vE p D pE  p!j × p!i = {}" 
  ― ‹Visited edges do not go back on path›
  assumes p_not_D: "(set p)  D = {}" ― ‹Path does not contain done nodes›
begin
  abbreviation ltouched where "ltouched  touched p D"
  abbreviation lvE where "lvE  vE p D pE"

  lemma locale_this: "invar_loc G v0 D0 p D pE" by unfold_locales

  definition (in fr_graph) 
    "invar  λv0 D0 (p,D,pE). invar_loc G v0 D0 p D pE"

  lemma invar_this[simp, intro!]: "invar v0 D0 (p,D,pE)"
    unfolding invar_def apply simp by unfold_locales 

  lemma finite_reachableE_v0[simp, intro!]: "finite (E*``{v0})"
    apply (rule finite_subset[OF _ finite_reachableE_V0])
    using v0_initial by auto

  lemma D_vis: "ED×UNIV  lvE" ― ‹All edges from done nodes are visited›
    unfolding vE_def touched_def using pE_E_from_p p_not_D by blast 

  lemma vE_touched: "lvE  ltouched × ltouched" 
    ― ‹Visited edges only between touched nodes›
    using E_from_p_touched D_closed unfolding vE_def touched_def by blast

  lemma lvE_ss_E: "lvE  E" ― ‹Visited edges are edges›
    unfolding vE_def by auto


  lemma path_touched: "(set p)  ltouched" by (auto simp: touched_def)
  lemma D_touched: "D  ltouched" by (auto simp: touched_def)

  lemma pE_by_vE: "pE = (E  (set p) × UNIV) - lvE"
    ― ‹Pending edges are edges from path not yet visited›
    unfolding vE_def touched_def
    using pE_E_from_p
    by auto

  lemma pick_pending: "p[]  pE  last p × UNIV = (E-lvE)  last p × UNIV"
    ― ‹Pending edges from end of path are non-visited edges from end of path›
    apply (subst pE_by_vE)
    by auto

  lemma p_connected': 
    assumes A: "Suc i<length p" 
    shows "p!i × p!Suc i  lvE  {}" 
  proof -
    from A p_not_D have "p!i  set p" "p!Suc i  set p" by auto
    with p_connected[OF A] show ?thesis unfolding vE_def touched_def
      by blast
  qed

end

subsubsection ‹Termination›

context fr_graph 
begin
  text ‹The termination argument is based on unprocessed edges: 
    Reachable edges from untouched nodes and pending edges.›
  definition "unproc_edges v0 p D pE  (E  (E*``{v0} - (D  (set p))) × UNIV)  pE"

  text ‹
    In each iteration of the loop, either the number of unprocessed edges
    decreases, or the path length decreases.›
  definition "abs_wf_rel v0  inv_image (finite_psubset <*lex*> measure length)
    (λ(p,D,pE). (unproc_edges v0 p D pE, p))"

  lemma abs_wf_rel_wf[simp, intro!]: "wf (abs_wf_rel v0)"
    unfolding abs_wf_rel_def
    by auto
end

subsection ‹Abstract Skeleton Algorithm›

context fr_graph
begin

  definition (in fr_graph) initial :: "'v  'v set  'v abs_state"
    where "initial v0 D  ([{v0}], D, (E  {v0}×UNIV))"

  definition (in -) collapse_aux :: "'a set list  nat  'a set list"
    where "collapse_aux p i  take i p @ [(set (drop i p))]"

  definition (in -) collapse :: "'a  'a abs_state  'a abs_state" 
    where "collapse v PDPE  
    let 
      (p,D,pE)=PDPE; 
      i=idx_of p v;
      p = collapse_aux p i
    in (p,D,pE)"

  definition (in -) 
    select_edge :: "'a abs_state  ('a option × 'a abs_state) nres"
    where
    "select_edge PDPE  do {
      let (p,D,pE) = PDPE;
      e  SELECT (λe. e  pE  last p × UNIV);
      case e of
        None  RETURN (None,(p,D,pE))
      | Some (u,v)  RETURN (Some v, (p,D,pE - {(u,v)}))
    }"

  definition (in fr_graph) push :: "'v  'v abs_state  'v abs_state" 
    where "push v PDPE  
    let
      (p,D,pE) = PDPE;
      p = p@[{v}];
      pE = pE  (E{v}×UNIV)
    in
      (p,D,pE)"

  definition (in -) pop :: "'v abs_state  'v abs_state"
    where "pop PDPE  let
      (p,D,pE) = PDPE;
      (p,V) = (butlast p, last p);
      D = V  D
    in
      (p,D,pE)"

  text ‹The following lemmas match the definitions presented in the paper:›
  lemma "select_edge (p,D,pE)  do {
      e  SELECT (λe. e  pE  last p × UNIV);
      case e of
        None  RETURN (None,(p,D,pE))
      | Some (u,v)  RETURN (Some v, (p,D,pE - {(u,v)}))
    }"
    unfolding select_edge_def by simp

  lemma "collapse v (p,D,pE) 
     let i=idx_of p v in (take i p @ [(set (drop i p))],D,pE)"
    unfolding collapse_def collapse_aux_def by simp

  lemma "push v (p, D, pE)  (p @ [{v}], D, pE  E  {v} × UNIV)"
    unfolding push_def by simp

  lemma "pop (p, D, pE)  (butlast p, last p  D, pE)"
    unfolding pop_def by auto

  thm pop_def[unfolded Let_def, no_vars]

  thm select_edge_def[unfolded Let_def]


  definition skeleton :: "'v set nres" 
    ― ‹Abstract Skeleton Algorithm›
    where
    "skeleton  do {
      let D = {};
      r  FOREACHi outer_invar V0 (λv0 D0. do {
        if v0D0 then do {
          let s = initial v0 D0;

          (p,D,pE)  WHILEIT (invar v0 D0)
            (λ(p,D,pE). p  []) (λ(p,D,pE). 
          do {
            ― ‹Select edge from end of path›
            (vo,(p,D,pE))  select_edge (p,D,pE);

            ASSERT (p[]);
            case vo of 
              Some v  do { ― ‹Found outgoing edge to node v›
                if v  (set p) then do {
                  ― ‹Back edge: Collapse path›
                  RETURN (collapse v (p,D,pE))
                } else if vD then do {
                  ― ‹Edge to new node. Append to path›
                  RETURN (push v (p,D,pE))
                } else do {
                  ― ‹Edge to done node. Skip›
                  RETURN (p,D,pE)
                }
              }
            | None  do {
                ASSERT (pE  last p × UNIV = {});
                ― ‹No more outgoing edges from current node on path›
                RETURN (pop (p,D,pE))
              }
          }) s;
          ASSERT (p=[]  pE={});
          RETURN D
        } else
          RETURN D0
      }) D;
      RETURN r
    }"

end

subsection ‹Invariant Preservation›

context fr_graph begin

  lemma set_collapse_aux[simp]: "(set (collapse_aux p i)) = (set p)"
    apply (subst (2) append_take_drop_id[of _ p,symmetric])
    apply (simp del: append_take_drop_id)
    unfolding collapse_aux_def by auto

  lemma touched_collapse[simp]: "touched (collapse_aux p i) D = touched p D"
    unfolding touched_def by simp

  lemma vE_collapse_aux[simp]: "vE (collapse_aux p i) D pE = vE p D pE"
    unfolding vE_def by simp

  lemma touched_push[simp]: "touched (p @ [V]) D = touched p D  V"
    unfolding touched_def by auto

end

subsubsection ‹Corollaries of the invariant›
text ‹In this section, we prove some more corollaries of the invariant,
  which are helpful to show invariant preservation›

context invar_loc
begin
  lemma cnode_connectedI: 
    "i<length p; up!i; vp!i  (u,v)(lvE  p!i×p!i)*"
    using p_sc[of "p!i"] by (auto simp: in_set_conv_nth)

  lemma cnode_connectedI': "i<length p; up!i; vp!i  (u,v)(lvE)*"
    by (metis inf.cobounded1 rtrancl_mono_mp cnode_connectedI)

  lemma p_no_empty: "{}  set p"
  proof 
    assume "{}set p"
    then obtain i where IDX: "i<length p" "p!i={}" 
      by (auto simp add: in_set_conv_nth)
    show False proof (cases i)
      case 0 with root_v0 IDX show False by (cases p) auto
    next
      case [simp]: (Suc j)
      from p_connected'[of j] IDX show False by simp
    qed
  qed

  corollary p_no_empty_idx: "i<length p  p!i{}"
    using p_no_empty by (metis nth_mem)
  
  lemma p_disjoint_sym: "i<length p; j<length p; vp!i; vp!j  i=j"
    by (metis disjoint_iff_not_equal linorder_neqE_nat p_disjoint)

  lemma pi_ss_path_seg_eq[simp]:
    assumes A: "i<length p" "ulength p"
    shows "p!ipath_seg p l u  li  i<u"
  proof
    assume B: "p!ipath_seg p l u"
    from A obtain x where "xp!i" by (blast dest: p_no_empty_idx)
    with B obtain i' where C: "xp!i'" "li'" "i'<u" 
      by (auto simp: path_seg_def)
    from p_disjoint_sym[OF i<length p _ xp!i xp!i'] i'<u ulength p
    have "i=i'" by simp
    with C show "li  i<u" by auto
  qed (auto simp: path_seg_def)

  lemma path_seg_ss_eq[simp]:
    assumes A: "l1<u1" "u1length p" "l2<u2" "u2length p"
    shows "path_seg p l1 u1  path_seg p l2 u2  l2l1  u1u2"
  proof
    assume S: "path_seg p l1 u1  path_seg p l2 u2"
    have "p!l1  path_seg p l1 u1" using A by simp
    also note S finally have 1: "l2l1" using A by simp
    have "p!(u1 - 1)  path_seg p l1 u1" using A by simp
    also note S finally have 2: "u1u2" using A by auto
    from 1 2 show "l2l1  u1u2" ..
  next
    assume "l2l1  u1u2" thus "path_seg p l1 u1  path_seg p l2 u2"
      using A
      apply (clarsimp simp: path_seg_def) []
      apply (metis dual_order.strict_trans1 dual_order.trans)
      done
  qed

  lemma pathI: 
    assumes "xp!i" "yp!j"
    assumes "ij" "j<length p"
    defines "seg  path_seg p i (Suc j)"
    shows "(x,y)(lvE  seg×seg)*"
    ― ‹We can obtain a path between cnodes on path›
    using assms(3,1,2,4) unfolding seg_def
  proof (induction arbitrary: y rule: dec_induct)
    case base thus ?case by (auto intro!: cnode_connectedI)
  next
    case (step j)

    let ?seg = "path_seg p i (Suc j)"
    let ?seg' = "path_seg p i (Suc (Suc j))"

    have SSS: "?seg  ?seg'" 
      apply (subst path_seg_ss_eq)
      using step.hyps step.prems by auto

    from p_connected'[OF Suc j < length p] obtain u v where 
      UV: "(u,v)lvE" "up!j" "vp!Suc j" by auto

    have ISS: "p!j  ?seg'" "p!Suc j  ?seg'" 
      using step.hyps step.prems by simp_all

    from p_no_empty_idx[of j] Suc j < length p obtain x' where "x'p!j" 
      by auto
    with step.IH[of x'] xp!i Suc j < length p 
    have t: "(x,x')(lvE?seg×?seg)*" by auto
    have "(x,x')(lvE?seg'×?seg')*" using SSS 
      by (auto intro: rtrancl_mono_mp[OF _ t])
    also 
    from cnode_connectedI[OF _ x'p!j up!j] Suc j < length p have
      t: "(x', u)  (lvE  p ! j × p ! j)*" by auto
    have "(x', u)  (lvE?seg'×?seg')*" using ISS
      by (auto intro: rtrancl_mono_mp[OF _ t])
    also have "(u,v)lvE?seg'×?seg'" using UV ISS by auto
    also from cnode_connectedI[OF Suc j < length p vp!Suc j yp!Suc j] 
    have t: "(v, y)  (lvE  p ! Suc j × p ! Suc j)*" by auto
    have "(v, y)  (lvE?seg'×?seg')*" using ISS
      by (auto intro: rtrancl_mono_mp[OF _ t])
    finally show "(x,y)(lvE?seg'×?seg')*" .
  qed

  lemma p_reachable: "(set p)  E*``{v0}" ― ‹Nodes on path are reachable›
  proof 
    fix v
    assume A: "v(set p)"
    then obtain i where "i<length p" and "vp!i" 
      by (metis UnionE in_set_conv_nth)
    moreover from A root_v0 have "v0p!0" by (cases p) auto
    ultimately have 
      t: "(v0,v)(lvE  path_seg p 0 (Suc i) × path_seg p 0 (Suc i))*"
      by (auto intro: pathI)
    from lvE_ss_E have "(v0,v)E*" by (auto intro: rtrancl_mono_mp[OF _ t])
    thus "vE*``{v0}" by auto
  qed

  lemma touched_reachable: "ltouched  E*``V0" ― ‹Touched nodes are reachable›
    unfolding touched_def using p_reachable D_reachable by blast

  lemma vE_reachable: "lvE  E*``V0 × E*``V0"
    apply (rule order_trans[OF vE_touched])
    using touched_reachable by blast

  lemma pE_reachable: "pE  E*``{v0} × E*``{v0}"
  proof safe
    fix u v
    assume E: "(u,v)pE"
    with pE_E_from_p p_reachable have "(v0,u)E*" "(u,v)E" by blast+
    thus "(v0,u)E*" "(v0,v)E*" by auto
  qed

  lemma D_closed_vE_rtrancl: "lvE*``D  D"
    by (metis D_closed Image_closed_trancl eq_iff reachable_mono lvE_ss_E)

  lemma D_closed_path: "path E u q w; uD  set q  D"
  proof -
    assume a1: "path E u q w"
    assume "u  D"
    hence f1: "{u}  D"
      using bot.extremum by force
    have "set q  E* `` {u}"
      using a1 by (metis insert_subset path_nodes_reachable)
    thus "set q  D"
      using f1 by (metis D_closed rtrancl_reachable_induct subset_trans)
  qed

  lemma D_closed_path_vE: "path lvE u q w; uD  set q  D"
    by (metis D_closed_path path_mono lvE_ss_E)

  lemma path_in_lastnode:
    assumes P: "path lvE u q v"
    assumes [simp]: "p[]"
    assumes ND: "ulast p" "vlast p"
    shows "set q  last p"
    ― ‹A path from the last Cnode to the last Cnode remains in the last Cnode›
    (* TODO: This can be generalized in two directions: 
      either 1) The path end anywhere. Due to vE_touched we can infer 
        that it ends in last cnode  
      or 2) We may use any cnode, not only the last one
    *)
    using P ND
  proof (induction)
    case (path_prepend u v l w) 
    from (u,v)lvE vE_touched have "vltouched" by auto
    hence "v(set p)"
      unfolding touched_def
    proof
      assume "vD"
      moreover from path lvE v l w have "(v,w)lvE*" by (rule path_is_rtrancl)
      ultimately have "wD" using D_closed_vE_rtrancl by auto
      with wlast p p_not_D have False
        by (metis IntI Misc.last_in_set Sup_inf_eq_bot_iff assms(2) 
          bex_empty path_prepend.hyps(2))
      thus ?thesis ..
    qed
    then obtain i where "i<length p" "vp!i"
      by (metis UnionE in_set_conv_nth)
    have "i=length p - 1"
    proof (rule ccontr)
      assume "ilength p - 1"
      with i<length p have "i < length p - 1" by simp
      with vE_no_back[of i "length p - 1"] i<length p 
      have "lvE  last p × p!i = {}"
        by (simp add: last_conv_nth)
      with (u,v)lvE ulast p vp!i show False by auto
    qed
    with vp!i have "vlast p" by (simp add: last_conv_nth)
    with path_prepend.IH wlast p ulast p show ?case by auto
  qed simp

  lemma loop_in_lastnode:
    assumes P: "path lvE u q u"
    assumes [simp]: "p[]"
    assumes ND: "set q  last p  {}"
    shows "ulast p" and "set q  last p"
    ― ‹A loop that touches the last node is completely inside the last node›
  proof -
    from ND obtain v where "vset q" "vlast p" by auto
    then obtain q1 q2 where [simp]: "q=q1@v#q2" 
      by (auto simp: in_set_conv_decomp)
    from P have "path lvE v (v#q2@q1) v" 
      by (auto simp: path_conc_conv path_cons_conv)
    from path_in_lastnode[OF this p[] vlast p vlast p] 
    show "set q  last p" by simp
    from P show "ulast p" 
      apply (cases q, simp)
      
      apply simp
      using set q  last p
      apply (auto simp: path_cons_conv)
      done
  qed


  lemma no_D_p_edges: "E  D × (set p) = {}"
    using D_closed p_not_D by auto

  lemma idx_of_props:
    assumes ON_STACK: "v(set p)"
    shows 
      "idx_of p v < length p" and
      "v  p ! idx_of p v"
    using idx_of_props[OF _ assms] p_disjoint_sym by blast+

end

subsubsection ‹Auxiliary Lemmas Regarding the Operations›

lemma (in fr_graph) vE_initial[simp]: "vE [{v0}] {} (E  {v0} × UNIV) = {}"
  unfolding vE_def touched_def by auto

context invar_loc
begin
  lemma vE_push: " (u,v)pE; ulast p; v(set p); vD  
     vE (p @ [{v}]) D ((pE - {(u,v)})  E{v}×UNIV) = insert (u,v) lvE"
    unfolding vE_def touched_def using pE_E_from_p
    by auto

  lemma vE_remove[simp]: 
    "p[]; (u,v)pE  vE p D (pE - {(u,v)}) = insert (u,v) lvE"
    unfolding vE_def touched_def using pE_E_from_p by blast

  lemma vE_pop[simp]: "p[]  vE (butlast p) (last p  D) pE = lvE"
    unfolding vE_def touched_def 
    by (cases p rule: rev_cases) auto


  lemma pE_fin: "p=[]  pE={}"
    using pE_by_vE by auto

  lemma (in invar_loc) lastp_un_D_closed:
    assumes NE: "p  []"
    assumes NO': "pE  (last p × UNIV) = {}"
    shows "E``(last p  D)  (last p  D)"
    ― ‹On pop, the popped CNode and D are closed under transitions›
  proof (intro subsetI, elim ImageE)
    from NO' have NO: "(E - lvE)  (last p × UNIV) = {}"
      by (simp add: pick_pending[OF NE])

    let ?i = "length p - 1"
    from NE have [simp]: "last p = p!?i" by (metis last_conv_nth) 
    
    fix u v
    assume E: "(u,v)E"
    assume UI: "ulast p  D" hence "up!?i  D" by simp
    
    {
      assume "ulast p" "vlast p" 
      moreover from E NO ulast p have "(u,v)lvE" by auto
      ultimately have "vD  v(set p)" 
        using vE_touched unfolding touched_def by auto
      moreover {
        assume "v(set p)"
        then obtain j where V: "j<length p" "vp!j" 
          by (metis UnionE in_set_conv_nth)
        with vlast p have "j<?i" by (cases "j=?i") auto
        from vE_no_back[OF j<?i _] (u,v)lvE V ulast p have False by auto
      } ultimately have "vD" by blast
    } with E UI D_closed show "vlast p  D" by auto
  qed



end


subsubsection ‹Preservation of Invariant by Operations›

context fr_graph
begin
  lemma (in outer_invar_loc) invar_initial_aux: 
    assumes "v0it - D"
    shows "invar v0 D (initial v0 D)"
    unfolding invar_def initial_def
    apply simp
    apply unfold_locales
    apply simp_all
    using assms it_initial apply auto []
    using D_reachable it_initial assms apply auto []
    using D_closed apply auto []
    using assms apply auto []
    done

  lemma invar_initial: 
    "outer_invar it D0; v0it; v0D0  invar v0 D0 (initial v0 D0)"
    unfolding outer_invar_def
    apply (drule outer_invar_loc.invar_initial_aux) 
    by auto

  lemma outer_invar_initial[simp, intro!]: "outer_invar V0 {}"
    unfolding outer_invar_def
    apply unfold_locales
    by auto

  lemma invar_pop:
    assumes INV: "invar v0 D0 (p,D,pE)"
    assumes NE[simp]: "p[]"
    assumes NO': "pE  (last p × UNIV) = {}"
    shows "invar v0 D0 (pop (p,D,pE))"
    unfolding invar_def pop_def
    apply simp
  proof -
    from INV interpret invar_loc G v0 D0 p D pE unfolding invar_def by simp

    have [simp]: "set p = insert (last p) (set (butlast p))" 
      using NE by (cases p rule: rev_cases) auto

    from p_disjoint have lp_dj_blp: "last p  (set (butlast p)) = {}"
      apply (cases p rule: rev_cases)
      apply simp
      apply (fastforce simp: in_set_conv_nth nth_append)
      done

    {
      fix i
      assume A: "Suc i < length (butlast p)"
      hence A': "Suc i < length p" by auto

      from nth_butlast[of i p] A have [simp]: "butlast p ! i = p ! i" by auto
      from nth_butlast[of "Suc i" p] A 
      have [simp]: "butlast p ! Suc i = p ! Suc i" by auto

      from p_connected[OF A'] 
      have "butlast p ! i × butlast p ! Suc i  (E - pE)  {}"
        by simp
    } note AUX_p_connected = this

    (*have [simp]: "(E ∩ (last p ∪ D ∪ ⋃set (butlast p)) × UNIV - pE) = vE"
      unfolding vE_def touched_def by auto*)

    show "invar_loc G v0 D0 (butlast p) (last p  D) pE"
      apply unfold_locales
  
      unfolding vE_pop[OF NE]

      apply simp

      using D_incr apply auto []

      using pE_E_from_p NO' apply auto []
  
      using E_from_p_touched apply (auto simp: touched_def) []
  
      using D_reachable p_reachable NE apply auto []

      apply (rule AUX_p_connected, assumption+) []

      using p_disjoint apply (simp add: nth_butlast)

      using p_sc apply simp

      using root_v0 apply (cases p rule: rev_cases) apply auto [2]

      using root_v0 p_empty_v0 apply (cases p rule: rev_cases) apply auto [2]

      apply (rule lastp_un_D_closed, insert NO', auto) []

      using vE_no_back apply (auto simp: nth_butlast) []

      using p_not_D lp_dj_blp apply auto []
      done
  qed

  thm invar_pop[of v_0 D_0, no_vars]

  lemma invar_collapse:
    assumes INV: "invar v0 D0 (p,D,pE)"
    assumes NE[simp]: "p[]"
    assumes E: "(u,v)pE" and "ulast p"
    assumes BACK: "v(set p)"
    defines "i  idx_of p v"
    defines "p'  collapse_aux p i"
    shows "invar v0 D0 (collapse v (p,D,pE - {(u,v)}))"
    unfolding invar_def collapse_def
    apply simp
    unfolding i_def[symmetric] p'_def[symmetric]
  proof -
    from INV interpret invar_loc G v0 D0 p D pE unfolding invar_def by simp

    let ?thesis="invar_loc G v0 D0 p' D (pE - {(u,v)})"

    have SETP'[simp]: "(set p') = (set p)" unfolding p'_def by simp

    have IL: "i < length p" and VMEM: "vp!i" 
      using idx_of_props[OF BACK] unfolding i_def by auto

    have [simp]: "length p' = Suc i" 
      unfolding p'_def collapse_aux_def using IL by auto

    have P'_IDX_SS: "j<Suc i. p!j  p'!j"
      unfolding p'_def collapse_aux_def using IL 
      by (auto simp add: nth_append path_seg_drop)

    from ulast p have "up!(length p - 1)" by (auto simp: last_conv_nth)

    have defs_fold: 
      "vE p' D (pE - {(u,v)}) = insert (u,v) lvE" 
      "touched p' D = ltouched"
      by (simp_all add: p'_def E)

    {
      fix j
      assume A: "Suc j < length p'" 
      hence "Suc j < length p" using IL by simp
      from p_connected[OF this] have "p!j × p!Suc j  (E-pE)  {}" .
      moreover from P'_IDX_SS A have "p!jp'!j" and "p!Suc j  p'!Suc j"
        by auto
      ultimately have "p' ! j × p' ! Suc j  (E - (pE - {(u, v)}))  {}" 
        by blast
    } note AUX_p_connected = this

    have P_IDX_EQ[simp]: "j. j < i  p'!j = p!j"
      unfolding p'_def collapse_aux_def using IL  
      by (auto simp: nth_append)

    have P'_LAST[simp]: "p'!i = path_seg p i (length p)" (is "_ = ?last_cnode")
      unfolding p'_def collapse_aux_def using IL 
      by (auto simp: nth_append path_seg_drop)

    {
      fix j k
      assume A: "j < k" "k < length p'" 
      have "p' ! j  p' ! k = {}"
      proof (safe, simp)
        fix v
        assume "vp'!j" and "vp'!k"
        with A have "vp!j" by simp
        show False proof (cases)
          assume "k=i"
          with vp'!k obtain k' where "vp!k'" "ik'" "k'<length p" 
            by (auto simp: path_seg_def)
          hence "p ! j  p ! k' = {}"
            using A by (auto intro!: p_disjoint)
          with vp!j vp!k' show False by auto
        next
          assume "ki" with A have "k<i" by simp
          hence "k<length p" using IL by simp
          note p_disjoint[OF j<k this] 
          also have "p!j = p'!j" using j<k k<i by simp
          also have "p!k = p'!k" using k<i by simp
          finally show False using vp'!j vp'!k by auto
        qed
      qed
    } note AUX_p_disjoint = this

    {
      fix U
      assume A: "Uset p'"
      then obtain j where "j<Suc i" and [simp]: "U=p'!j"
        by (auto simp: in_set_conv_nth)
      hence "U × U  (insert (u, v) lvE  U × U)*" 
      proof cases
        assume [simp]: "j=i"
        show ?thesis proof (clarsimp)
          fix x y
          assume "xpath_seg p i (length p)" "ypath_seg p i (length p)"
          then obtain ix iy where 
            IX: "xp!ix" "iix" "ix<length p" and
            IY: "yp!iy" "iiy" "iy<length p"
            by (auto simp: path_seg_def)
            

          from IX have SS1: "path_seg p ix (length p)  ?last_cnode"
            by (subst path_seg_ss_eq) auto

          from IY have SS2: "path_seg p i (Suc iy)  ?last_cnode"
            by (subst path_seg_ss_eq) auto

          let ?rE = "λR. (lvE  R×R)"
          let ?E = "(insert (u,v) lvE  ?last_cnode × ?last_cnode)"

          from pathI[OF xp!ix up!(length p - 1)] have
            "(x,u)(?rE (path_seg p ix (Suc (length p - 1))))*" using IX by auto
          hence "(x,u)?E*" 
            apply (rule rtrancl_mono_mp[rotated]) 
            using SS1
            by auto

          also have "(u,v)?E" using i<length p
            apply (clarsimp)
            apply (intro conjI)
            apply (rule rev_subsetD[OF up!(length p - 1)])
            apply (simp)
            apply (rule rev_subsetD[OF VMEM])
            apply (simp)
            done
          also 
          from pathI[OF vp!i yp!iy] have
            "(v,y)(?rE (path_seg p i (Suc iy)))*" using IY by auto
          hence "(v,y)?E*"
            apply (rule rtrancl_mono_mp[rotated]) 
            using SS2
            by auto
          finally show "(x,y)?E*" .
        qed
      next
        assume "ji"
        with j<Suc i have [simp]: "j<i" by simp
        with i<length p have "p!jset p"
          by (metis Suc_lessD in_set_conv_nth less_trans_Suc) 

        thus ?thesis using p_sc[of U] p!jset p
          apply (clarsimp)
          apply (subgoal_tac "(a,b)(lvE  p ! j × p ! j)*")
          apply (erule rtrancl_mono_mp[rotated])
          apply auto
          done
      qed
    } note AUX_p_sc = this

    { fix j k
      assume A: "j<k" "k<length p'"
      hence "j<i" by simp
      have "insert (u, v) lvE  p' ! k × p' ! j = {}"
      proof -
        have "{(u,v)}  p' ! k × p' ! j = {}" 
          apply auto
          by (metis IL P_IDX_EQ Suc_lessD VMEM j < i 
            less_irrefl_nat less_trans_Suc p_disjoint_sym)
        moreover have "lvE  p' ! k × p' ! j = {}" 
        proof (cases "k<i")
          case True thus ?thesis
            using vE_no_back[of j k] A i<length p by auto
        next
          case False with A have [simp]: "k=i" by simp
          show ?thesis proof (rule disjointI, clarsimp simp: j<i)
            fix x y
            assume B: "(x,y)lvE" "xpath_seg p i (length p)" "yp!j"
            then obtain ix where "xp!ix" "iix" "ix<length p" 
              by (auto simp: path_seg_def)
            moreover with A have "j<ix" by simp
            ultimately show False using vE_no_back[of j ix] B by auto
          qed
        qed
        ultimately show ?thesis by blast
      qed
    } note AUX_vE_no_back = this

    show ?thesis
      apply unfold_locales
      unfolding defs_fold

      apply simp

      using D_incr apply auto []

      using pE_E_from_p apply auto []

      using E_from_p_touched BACK apply (simp add: touched_def) apply blast

      apply (rule D_reachable)

      apply (rule AUX_p_connected, assumption+) []

      apply (rule AUX_p_disjoint, assumption+) []

      apply (rule AUX_p_sc, assumption+) []

      using root_v0 
      apply (cases i) 
      apply (simp add: p'_def collapse_aux_def)
      apply (metis NE hd_in_set)
      apply (cases p, simp_all add: p'_def collapse_aux_def) []

      apply (simp add: p'_def collapse_aux_def)

      apply (rule D_closed)

      apply (drule (1) AUX_vE_no_back, auto) []

      using p_not_D apply simp
      done
  qed
  
  lemma invar_push:
    assumes INV: "invar v0 D0 (p,D,pE)"
    assumes NE[simp]: "p[]"
    assumes E: "(u,v)pE" and UIL: "ulast p"
    assumes VNE: "v(set p)" "vD"
    shows "invar v0 D0 (push v (p,D,pE - {(u,v)}))"
    unfolding invar_def push_def
    apply simp
  proof -
    from INV interpret invar_loc G v0 D0 p D pE unfolding invar_def by simp

    let ?thesis 
      = "invar_loc G v0 D0 (p @ [{v}]) D (pE - {(u, v)}  E  {v} × UNIV)"

    note defs_fold = vE_push[OF E UIL VNE] touched_push

    {
      fix i
      assume SILL: "Suc i < length (p @ [{v}])"
      have "(p @ [{v}]) ! i × (p @ [{v}]) ! Suc i 
              (E - (pE - {(u, v)}  E  {v} × UNIV))  {}"
      proof (cases "i = length p - 1")
        case True thus ?thesis using SILL E pE_E_from_p UIL VNE
          by (simp add: nth_append last_conv_nth) fast
      next
        case False
        with SILL have SILL': "Suc i < length p" by simp
            
        with SILL' VNE have X1: "vp!i" "vp!Suc i" by auto
            
        from p_connected[OF SILL'] obtain a b where 
          "ap!i" "bp!Suc i" "(a,b)E" "(a,b)pE" 
          by auto
        with X1 have "av" "bv" by auto
        with (a,b)E (a,b)pE have "(a,b)(E - (pE - {(u, v)}  E  {v} × UNIV))"
          by auto
        with ap!i bp!Suc i
        show ?thesis using  SILL'
          by (simp add: nth_append; blast) 
      qed
    } note AUX_p_connected = this

    {
      fix U
      assume A: "U  set (p @ [{v}])"
      have "U × U  (insert (u, v) lvE  U × U)*"
      proof cases
        assume "Uset p"
        with p_sc have "U×U  (lvE  U×U)*" .
        thus ?thesis
          by (metis (lifting, no_types) Int_insert_left_if0 Int_insert_left_if1 
            in_mono insert_subset rtrancl_mono_mp subsetI)
      next
        assume "Uset p" with A have "U={v}" by simp
        thus ?thesis by auto
      qed
    } note AUX_p_sc = this

    {
      fix i j
      assume A: "i < j" "j < length (p @ [{v}])"
      have "insert (u, v) lvE  (p @ [{v}]) ! j × (p @ [{v}]) ! i = {}"
      proof (cases "j=length p")
        case False with A have "j<length p" by simp
        from vE_no_back i<j this VNE show ?thesis 
          by (auto simp add: nth_append)
      next
        from p_not_D A have PDDJ: "p!i  D = {}" 
          by (auto simp: Sup_inf_eq_bot_iff)
        case True thus ?thesis
          using A apply (simp add: nth_append)
          apply (rule conjI)
          using UIL A p_disjoint_sym
          apply (metis Misc.last_in_set NE UnionI VNE(1))

          using vE_touched VNE PDDJ apply (auto simp: touched_def) []
          done
      qed
    } note AUX_vE_no_back = this
        
    show ?thesis
      apply unfold_locales
      unfolding defs_fold

      apply simp

      using D_incr apply auto []

      using pE_E_from_p apply auto []

      using E_from_p_touched VNE apply (auto simp: touched_def) []

      apply (rule D_reachable)

      apply (rule AUX_p_connected, assumption+) []

      using p_disjoint v(set p) apply (auto simp: nth_append) []

      apply (rule AUX_p_sc, assumption+) []

      using root_v0 apply simp

      apply simp

      apply (rule D_closed)

      apply (rule AUX_vE_no_back, assumption+) []

      using p_not_D VNE apply auto []
      done
  qed

  lemma invar_skip:
    assumes INV: "invar v0 D0 (p,D,pE)"
    assumes NE[simp]: "p[]"
    assumes E: "(u,v)pE" and UIL: "ulast p"
    assumes VNP: "v(set p)" and VD: "vD"
    shows "invar v0 D0 (p,D,pE - {(u, v)})"
    unfolding invar_def
    apply simp
  proof -
    from INV interpret invar_loc G v0 D0 p D pE unfolding invar_def by simp
    let ?thesis = "invar_loc G v0 D0 p D (pE - {(u, v)})"
    note defs_fold = vE_remove[OF NE E]

    show ?thesis
      apply unfold_locales
      unfolding defs_fold
      
      apply simp

      using D_incr apply auto []

      using pE_E_from_p apply auto []

      using E_from_p_touched VD apply (auto simp: touched_def) []

      apply (rule D_reachable)

      using p_connected apply auto []

      apply (rule p_disjoint, assumption+) []

      apply (drule p_sc)
      apply (erule order_trans)
      apply (rule rtrancl_mono)
      apply blast []

      apply (rule root_v0, assumption+) []

      apply (rule p_empty_v0, assumption+) []

      apply (rule D_closed)

      using vE_no_back VD p_not_D 
      apply clarsimp
      apply (metis Suc_lessD UnionI VNP less_trans_Suc nth_mem)

      apply (rule p_not_D)
      done
  qed


  lemma fin_D_is_reachable: 
    ― ‹When inner loop terminates, all nodes reachable from start node are
      finished›
    assumes INV: "invar v0 D0 ([], D, pE)"
    shows "D  E*``{v0}"
  proof -
    from INV interpret invar_loc G v0 D0 "[]" D pE unfolding invar_def by auto

    from p_empty_v0 rtrancl_reachable_induct[OF order_refl D_closed] D_reachable
    show ?thesis by auto
  qed

  lemma fin_reachable_path: 
    ― ‹When inner loop terminates, nodes reachable from start node are
      reachable over visited edges›
    assumes INV: "invar v0 D0 ([], D, pE)"
    assumes UR: "uE*``{v0}"
    shows "path (vE [] D pE) u q v  path E u q v"
  proof -
    from INV interpret invar_loc G v0 D0 "[]" D pE unfolding invar_def by auto
    
    show ?thesis
    proof
      assume "path lvE u q v"
      thus "path E u q v" using path_mono[OF lvE_ss_E] by blast
    next
      assume "path E u q v"
      thus "path lvE u q v" using UR
      proof induction
        case (path_prepend u v p w)
        with fin_D_is_reachable[OF INV] have "uD" by auto
        with D_closed (u,v)E have "vD" by auto
        from path_prepend.prems path_prepend.hyps have "vE*``{v0}" by auto
        with path_prepend.IH fin_D_is_reachable[OF INV] have "path lvE v p w" 
          by simp
        moreover from uD vD (u,v)E D_vis have "(u,v)lvE" by auto
        ultimately show ?case by (auto simp: path_cons_conv)
      qed simp
    qed
  qed

  lemma invar_outer_newnode: 
    assumes A: "v0D0" "v0it" 
    assumes OINV: "outer_invar it D0"
    assumes INV: "invar v0 D0 ([],D',pE)"
    shows "outer_invar (it-{v0}) D'"
  proof -
    from OINV interpret outer_invar_loc G it D0 unfolding outer_invar_def .
    from INV interpret inv: invar_loc G v0 D0 "[]" D' pE 
      unfolding invar_def by simp
    
    from fin_D_is_reachable[OF INV] have [simp]: "v0D'" by auto

    show ?thesis
      unfolding outer_invar_def
      apply unfold_locales
      using it_initial apply auto []
      using it_done inv.D_incr apply auto []
      using inv.D_reachable apply assumption
      using inv.D_closed apply assumption
      done
  qed

  lemma invar_outer_Dnode:
    assumes A: "v0D0" "v0it" 
    assumes OINV: "outer_invar it D0"
    shows "outer_invar (it-{v0}) D0"
  proof -
    from OINV interpret outer_invar_loc G it D0 unfolding outer_invar_def .
    
    show ?thesis
      unfolding outer_invar_def
      apply unfold_locales
      using it_initial apply auto []
      using it_done A apply auto []
      using D_reachable apply assumption
      using D_closed apply assumption
      done
  qed

  lemma pE_fin': "invar x σ ([], D, pE)  pE={}"
    unfolding invar_def by (simp add: invar_loc.pE_fin)

end

subsubsection ‹Termination›

context invar_loc 
begin
  lemma unproc_finite[simp, intro!]: "finite (unproc_edges v0 p D pE)"
    ― ‹The set of unprocessed edges is finite›
  proof -
    have "unproc_edges v0 p D pE  E*``{v0} × E*``{v0}"
      unfolding unproc_edges_def 
      using pE_reachable
      by auto
    thus ?thesis
      by (rule finite_subset) simp
  qed

  lemma unproc_decreasing: 
    ― ‹As effect of selecting a pending edge, the set of unprocessed edges
      decreases›
    assumes [simp]: "p[]" and A: "(u,v)pE" "ulast p"
    shows "unproc_edges v0 p D (pE-{(u,v)})  unproc_edges v0 p D pE"
    using A unfolding unproc_edges_def
    by fastforce
end

context fr_graph 
begin

  lemma abs_wf_pop:
    assumes INV: "invar v0 D0 (p,D,pE)"
    assumes NE[simp]: "p[]"
    assumes NO: "pE  last aba × UNIV = {}"
    shows "(pop (p,D,pE), (p, D, pE))  abs_wf_rel v0"
    unfolding pop_def
    apply simp
  proof -
    from INV interpret invar_loc G v0 D0 p D pE unfolding invar_def by simp 
    let ?thesis = "((butlast p, last p  D, pE), p, D, pE)  abs_wf_rel v0"
    have "unproc_edges v0 (butlast p) (last p  D) pE = unproc_edges v0 p D pE"
      unfolding unproc_edges_def
      apply (cases p rule: rev_cases, simp)
      apply auto
      done
    thus ?thesis
      by (auto simp: abs_wf_rel_def)
  qed

  lemma abs_wf_collapse:
    assumes INV: "invar v0 D0 (p,D,pE)"
    assumes NE[simp]: "p[]"
    assumes E: "(u,v)pE" "ulast p"
    shows "(collapse v (p,D,pE-{(u,v)}), (p, D, pE)) abs_wf_rel v0"
    unfolding collapse_def
    apply simp
  proof -
    from INV interpret invar_loc G v0 D0 p D pE unfolding invar_def by simp 
    define i where "i = idx_of p v"
    let ?thesis 
      = "((collapse_aux p i, D, pE-{(u,v)}), (p, D, pE))  abs_wf_rel v0"

    have "unproc_edges v0 (collapse_aux p i) D (pE-{(u,v)}) 
      = unproc_edges v0 p D (pE-{(u,v)})"
      unfolding unproc_edges_def by (auto)
    also note unproc_decreasing[OF NE E]
    finally show ?thesis
      by (auto simp: abs_wf_rel_def)
  qed
    
  lemma abs_wf_push:
    assumes INV: "invar v0 D0 (p,D,pE)"
    assumes NE[simp]: "p[]"
    assumes E: "(u,v)pE" "ulast p" and A: "vD" "v(set p)"
    shows "(push v (p,D,pE-{(u,v)}), (p, D, pE))  abs_wf_rel v0"
    unfolding push_def
    apply simp
  proof -
    from INV interpret invar_loc G v0 D0 p D pE unfolding invar_def by simp 
    let ?thesis 
      = "((p@[{v}], D, pE-{(u,v)}  E{v}×UNIV), (p, D, pE))  abs_wf_rel v0"

    have "unproc_edges v0 (p@[{v}]) D (pE-{(u,v)}  E{v}×UNIV) 
      = unproc_edges v0 p D (pE-{(u,v)})"
      unfolding unproc_edges_def
      using E A pE_reachable
      by auto
    also note unproc_decreasing[OF NE E]
    finally show ?thesis
      by (auto simp: abs_wf_rel_def)
  qed

  lemma abs_wf_skip:
    assumes INV: "invar v0 D0 (p,D,pE)"
    assumes NE[simp]: "p[]"
    assumes E: "(u,v)pE" "ulast p"
    shows "((p, D, pE-{(u,v)}), (p, D, pE))  abs_wf_rel v0"
  proof -
    from INV interpret invar_loc G v0 D0 p D pE unfolding invar_def by simp 
    from unproc_decreasing[OF NE E] show ?thesis
      by (auto simp: abs_wf_rel_def)
  qed
end

subsubsection ‹Main Correctness Theorem›

context fr_graph 
begin
  lemmas invar_preserve = 
    invar_initial
    invar_pop invar_push invar_skip invar_collapse 
    abs_wf_pop abs_wf_collapse abs_wf_push abs_wf_skip 
    outer_invar_initial invar_outer_newnode invar_outer_Dnode

  text ‹The main correctness theorem for the dummy-algorithm just states that
    it satisfies the invariant when finished, and the path is empty.
›
  theorem skeleton_spec: "skeleton  SPEC (λD. outer_invar {} D)"
  proof -
    note [simp del] = Union_iff
    note [[goals_limit = 4]]

    show ?thesis
      unfolding skeleton_def select_edge_def select_def
      apply (refine_vcg WHILEIT_rule[OF abs_wf_rel_wf])
      apply (vc_solve solve: invar_preserve simp: pE_fin' finite_V0)
      apply auto
      done
  qed

  text ‹Short proof, as presented in the paper›
  context 
    notes [refine] = refine_vcg 
  begin
    theorem "skeleton  SPEC (λD. outer_invar {} D)"
      unfolding skeleton_def select_edge_def select_def
      by (refine_vcg WHILEIT_rule[OF abs_wf_rel_wf])
         (auto intro: invar_preserve simp: pE_fin' finite_V0)
  end

end

subsection "Consequences of Invariant when Finished"
context fr_graph
begin
  lemma fin_outer_D_is_reachable:
    ― ‹When outer loop terminates, exactly the reachable nodes are finished›
    assumes INV: "outer_invar {} D"
    shows "D = E*``V0"
  proof -
    from INV interpret outer_invar_loc G "{}" D unfolding outer_invar_def by auto

    from it_done rtrancl_reachable_induct[OF order_refl D_closed] D_reachable
    show ?thesis by auto
  qed

end


section ‹Refinement to Gabow's Data Structure›text_raw‹\label{sec:algo-ds}›

text ‹
  The implementation due to Gabow cite"Gabow00" represents a path as
  a stack S› of single nodes, and a stack B› that contains the
  boundaries of the collapsed segments. Moreover, a map I› maps nodes
  to their stack indices.

  As we use a tail-recursive formulation, we use another stack 
  P :: (nat × 'v set) list› to represent the pending edges. The
  entries in P› are sorted by ascending first component,
  and P› only contains entries with non-empty second component. 
  An entry (i,l)› means that the edges from the node at 
  S[i]› to the nodes stored in l› are pending.
›

subsection ‹Preliminaries›
primrec find_max_nat :: "nat  (natbool)  nat" 
  ― ‹Find the maximum number below an upper bound for which a predicate holds›
  where
  "find_max_nat 0 _ = 0"
| "find_max_nat (Suc n) P = (if (P n) then n else find_max_nat n P)"

lemma find_max_nat_correct: 
  "P 0; 0<u  find_max_nat u P = Max {i. i<u  P i}"
  apply (induction u)
  apply auto

  apply (rule Max_eqI[THEN sym])
  apply auto [3]
  
  apply (case_tac u)
  apply simp
  apply clarsimp
  by (metis less_SucI less_antisym)

lemma find_max_nat_param[param]:
  assumes "(n,n')nat_rel"
  assumes "j j'. (j,j')nat_rel; j'<n'  (P j,P' j')bool_rel"
  shows "(find_max_nat n P,find_max_nat n' P')  nat_rel"
  using assms
  by (induction n arbitrary: n') auto

context begin interpretation autoref_syn .
  lemma find_max_nat_autoref[autoref_rules]:
    assumes "(n,n')nat_rel"
    assumes "j j'. (j,j')nat_rel; j'<n'  (P j,P'$j')bool_rel"
    shows "(find_max_nat n P,
        (OP find_max_nat ::: nat_rel  (nat_relbool_rel)  nat_rel) $n'$P'
      )  nat_rel"
    using find_max_nat_param[OF assms]
    by simp

end

subsection ‹Gabow's Datastructure›

subsubsection ‹Definition and Invariant›
datatype node_state = STACK nat | DONE

type_synonym 'v oGS = "'v  node_state"

definition oGS_α :: "'v oGS  'v set" where "oGS_α I  {v. I v = Some DONE}"

locale oGS_invar = 
  fixes I :: "'v oGS"
  assumes I_no_stack: "I v  Some (STACK j)"


type_synonym 'a GS 
  = "'a list × nat list × ('a  node_state) × (nat × 'a set) list"
locale GS =  
  fixes SBIP :: "'a GS"
begin
  definition "S  (λ(S,B,I,P). S) SBIP"
  definition "B  (λ(S,B,I,P). B) SBIP"
  definition "I  (λ(S,B,I,P). I) SBIP"
  definition "P  (λ(S,B,I,P). P) SBIP"

  definition seg_start :: "nat  nat" ― ‹Start index of segment, inclusive›
    where "seg_start i  B!i" 

  definition seg_end :: "nat  nat"  ― ‹End index of segment, exclusive›
    where "seg_end i  if i+1 = length B then length S else B!(i+1)"

  definition seg :: "nat  'a set" ― ‹Collapsed set at index›
    where "seg i  {S!j | j. seg_start i  j  j < seg_end i }"

  definition "p_α  map seg [0..<length B]" ― ‹Collapsed path›

  definition "D_α  {v. I v = Some DONE}" ― ‹Done nodes›
  
  definition "pE_α  { (u,v) . j I. (j,I)set P  u = S!j  vI }" 
    ― ‹Pending edges›

  definition "α  (p_α,D_α,pE_α)" ― ‹Abstract state›

end

lemma GS_sel_simps[simp]:
  "GS.S (S,B,I,P) = S"
  "GS.B (S,B,I,P) = B"
  "GS.I (S,B,I,P) = I"
  "GS.P (S,B,I,P) = P"
  unfolding GS.S_def GS.B_def GS.I_def GS.P_def
  by auto

context GS begin
  lemma seg_start_indep[simp]: "GS.seg_start (S',B,I',P') = seg_start"  
    unfolding GS.seg_start_def[abs_def] by (auto)
  lemma seg_end_indep[simp]: "GS.seg_end (S,B,I',P') = seg_end"  
    unfolding GS.seg_end_def[abs_def] by auto
  lemma seg_indep[simp]: "GS.seg (S,B,I',P') = seg"  
    unfolding GS.seg_def[abs_def] by auto
  lemma p_α_indep[simp]: "GS.p_α (S,B,I',P') = p_α"
    unfolding GS.p_α_def by auto

  lemma D_α_indep[simp]: "GS.D_α (S',B',I,P') = D_α"
    unfolding GS.D_α_def by auto

  lemma pE_α_indep[simp]: "GS.pE_α (S,B',I',P) = pE_α" 
    unfolding GS.pE_α_def by auto

  definition find_seg ― ‹Abs-path index for stack index›
    where "find_seg j  Max {i. i<length B  B!ij}"

  definition S_idx_of ― ‹Stack index for node›
    where "S_idx_of v  case I v of Some (STACK i)  i"

end

locale GS_invar = GS +
  assumes B_in_bound: "set B  {0..<length S}"
  assumes B_sorted: "sorted B"
  assumes B_distinct: "distinct B"
  assumes B0: "S[]  B[]  B!0=0"
  assumes S_distinct: "distinct S"

  assumes I_consistent: "(I v = Some (STACK j))  (j<length S  v = S!j)"
  
  assumes P_sorted: "sorted (map fst P)"
  assumes P_distinct: "distinct (map fst P)"
  assumes P_bound: "set P  {0..<length S}×Collect ((≠) {})"
begin
  lemma locale_this: "GS_invar SBIP" by unfold_locales

end

definition "oGS_rel  br oGS_α oGS_invar"
lemma oGS_rel_sv[intro!,simp,relator_props]: "single_valued oGS_rel"
  unfolding oGS_rel_def by auto

definition "GS_rel  br GS.α GS_invar"
lemma GS_rel_sv[intro!,simp,relator_props]: "single_valued GS_rel"
  unfolding GS_rel_def by auto

context GS_invar
begin
  lemma empty_eq: "S=[]  B=[]"
    using B_in_bound B0 by auto

  lemma B_in_bound': "i<length B  B!i < length S"
    using B_in_bound nth_mem by fastforce

  lemma seg_start_bound:
    assumes A: "i<length B" shows "seg_start i < length S"
    using B_in_bound nth_mem[OF A] unfolding seg_start_def by auto

  lemma seg_end_bound:
    assumes A: "i<length B" shows "seg_end i  length S"
  proof (cases "i+1=length B")
    case True thus ?thesis by (simp add: seg_end_def)
  next
    case False with A have "i+1<length B" by simp
    from nth_mem[OF this] B_in_bound have " B ! (i + 1) < length S" by auto
    thus ?thesis using False by (simp add: seg_end_def)
  qed

  lemma seg_start_less_end: "i<length B  seg_start i < seg_end i"
    unfolding seg_start_def seg_end_def
    using B_in_bound' distinct_sorted_mono[OF B_sorted B_distinct]
    by auto

  lemma seg_end_less_start: "i<j; j<length B  seg_end i  seg_start j"
    unfolding seg_start_def seg_end_def
    by (auto simp: distinct_sorted_mono_iff[OF B_distinct B_sorted])

  lemma find_seg_bounds:
    assumes A: "j<length S"
    shows "seg_start (find_seg j)  j" 
    and "j < seg_end (find_seg j)" 
    and "find_seg j < length B"
  proof -
    let ?M = "{i. i<length B  B!ij}"
    from A have [simp]: "B[]" using empty_eq by (cases S) auto
    have NE: "?M{}" using A B0 by (cases B) auto

    have F: "finite ?M" by auto
    
    from Max_in[OF F NE]
    have LEN: "find_seg j < length B" and LB: "B!find_seg j  j"
      unfolding find_seg_def
      by auto

    thus "find_seg j < length B" by -
    
    from LB show LB': "seg_start (find_seg j)  j"
      unfolding seg_start_def by simp

    moreover show UB': "j < seg_end (find_seg j)"
      unfolding seg_end_def 
    proof (split if_split, intro impI conjI)
      show "j<length S" using A .
      
      assume "find_seg j + 1  length B" 
      with LEN have P1: "find_seg j + 1 < length B" by simp

      show "j < B ! (find_seg j + 1)"
      proof (rule ccontr, simp only: linorder_not_less)
        assume P2: "B ! (find_seg j + 1)  j"
        with P1 Max_ge[OF F, of "find_seg j + 1", folded find_seg_def]
        show False by simp
      qed
    qed
  qed
    
  lemma find_seg_correct:
    assumes A: "j<length S"
    shows "S!j  seg (find_seg j)" and "find_seg j < length B"
    using find_seg_bounds[OF A]
      unfolding seg_def by auto

  lemma set_p_α_is_set_S:
    "(set p_α) = set S"
    apply rule
    unfolding p_α_def seg_def[abs_def]
    using seg_end_bound apply fastforce []

    apply (auto simp: in_set_conv_nth)

    using find_seg_bounds
    apply (fastforce simp: in_set_conv_nth)
    done

  lemma S_idx_uniq: 
    "i<length S; j<length S  S!i=S!j  i=j"
    using S_distinct
    by (simp add: nth_eq_iff_index_eq)

  lemma S_idx_of_correct: 
    assumes A: "v(set p_α)"
    shows "S_idx_of v < length S" and "S!S_idx_of v = v"
  proof -
    from A have "vset S" by (simp add: set_p_α_is_set_S)
    then obtain j where G1: "j<length S" "v=S!j" by (auto simp: in_set_conv_nth)
    with I_consistent have "I v = Some (STACK j)" by simp
    hence "S_idx_of v = j" by (simp add: S_idx_of_def)
    with G1 show "S_idx_of v < length S" and "S!S_idx_of v = v" by simp_all
  qed

  lemma p_α_disjoint_sym: 
    shows "i j v. i<length p_α  j<length p_α  vp_α!i  vp_α!j  i=j"
  proof (intro allI impI, elim conjE)
    fix i j v
    assume A: "i < length p_α" "j < length p_α" "v  p_α ! i" "v  p_α ! j"
    from A have LI: "i<length B" and LJ: "j<length B" by (simp_all add: p_α_def)

    from A have B1: "seg_start j < seg_end i" and B2: "seg_start i < seg_end j"
      unfolding p_α_def seg_def[abs_def]
      apply clarsimp_all
      apply (subst (asm) S_idx_uniq)
      apply (metis dual_order.strict_trans1 seg_end_bound)
      apply (metis dual_order.strict_trans1 seg_end_bound)
      apply simp
      apply (subst (asm) S_idx_uniq)
      apply (metis dual_order.strict_trans1 seg_end_bound)
      apply (metis dual_order.strict_trans1 seg_end_bound)
      apply simp
      done

    from B1 have B1: "(B!j < B!Suc i  Suc i < length B)  i=length B - 1"
      using LI unfolding seg_start_def seg_end_def by (auto split: if_split_asm)

    from B2 have B2: "(B!i < B!Suc j  Suc j < length B)  j=length B - 1"
      using LJ unfolding seg_start_def seg_end_def by (auto split: if_split_asm)

    from B1 have B1: "j<Suc i  i=length B - 1"
      using LI LJ distinct_sorted_strict_mono_iff[OF B_distinct B_sorted]
      by auto

    from B2 have B2: "i<Suc j  j=length B - 1"
      using LI LJ distinct_sorted_strict_mono_iff[OF B_distinct B_sorted]
      by auto

    from B1 B2 show "i=j"
      using LI LJ
      by auto
  qed

end


subsection ‹Refinement of the Operations›

definition GS_initial_impl :: "'a oGS  'a  'a set  'a GS" where
  "GS_initial_impl I v0 succs  (
    [v0],
    [0],
    I(v0(STACK 0)),
    if succs={} then [] else [(0,succs)])"

context GS
begin
  definition "push_impl v succs  
    let
      _ = stat_newnode ();
      j = length S;
      S = S@[v];
      B = B@[j];
      I = I(v  STACK j);
      P = if succs={} then P else P@[(j,succs)]
    in
      (S,B,I,P)"

  
  definition mark_as_done 
    where "l u I. mark_as_done l u I  do {
    (_,I)WHILET 
      (λ(l,I). l<u) 
      (λ(l,I). do { ASSERT (l<length S); RETURN (Suc l,I(S!l  DONE))}) 
      (l,I);
    RETURN I
  }"

  definition mark_as_done_abs where
    "l u I. mark_as_done_abs l u I 
     (λv. if v{S!j | j. lj  j<u} then Some DONE else I v)"

  lemma mark_as_done_aux:
    fixes l u I
    shows "l<u; ulength S  mark_as_done l u I 
     SPEC (λr. r = mark_as_done_abs l u I)"
    unfolding mark_as_done_def mark_as_done_abs_def
    apply (refine_rcg 
      WHILET_rule[where 
        I="λ(l',I'). 
          I' = (λv. if v{S!j | j. lj  j<l'} then Some DONE else I v)
           ll'  l'u"
        and R="measure (λ(l',_). u-l')" 
      ]
      refine_vcg)
    
    apply (auto intro!: ext simp: less_Suc_eq)
    done    

  definition "pop_impl  
    do {
      let lsi = length B - 1;
      ASSERT (lsi<length B);
      I  mark_as_done (seg_start lsi) (seg_end lsi) I;
      ASSERT (B[]);
      let S = take (last B) S;
      ASSERT (B[]);
      let B = butlast B;
      RETURN (S,B,I,P)
    }"

  definition "sel_rem_last  
    if P=[] then 
      RETURN (None,(S,B,I,P))
    else do {
      let (j,succs) = last P;
      ASSERT (length B - 1 < length B);
      if j  seg_start (length B - 1) then do {
        ASSERT (succs{});
        v  SPEC (λx. xsuccs);
        let succs = succs - {v};
        ASSERT (P[]  length P - 1 < length P);
        let P = (if succs={} then butlast P else P[length P - 1 := (j,succs)]);
        RETURN (Some v,(S,B,I,P))
      } else RETURN (None,(S,B,I,P))
    }" 


  definition "find_seg_impl j  find_max_nat (length B) (λi. B!ij)"

  lemma (in GS_invar) find_seg_impl:
    "j<length S  find_seg_impl j = find_seg j"
    unfolding find_seg_impl_def
    thm find_max_nat_correct
    apply (subst find_max_nat_correct)
    apply (simp add: B0)
    apply (simp add: B0)
    apply (simp add: find_seg_def)
    done


  definition "idx_of_impl v  do {
      ASSERT (i. I v = Some (STACK i));
      let j = S_idx_of v;
      ASSERT (j<length S);
      let i = find_seg_impl j;
      RETURN i
    }"

  definition "collapse_impl v  
    do { 
      iidx_of_impl v;
      ASSERT (i+1  length B);
      let B = take (i+1) B;
      RETURN (S,B,I,P)
    }"

end

lemma (in -) GS_initial_correct: 
  assumes REL: "(I,D)oGS_rel"
  assumes A: "v0D"
  shows "GS.α (GS_initial_impl I v0 succs) = ([{v0}],D,{v0}×succs)" (is ?G1)
  and "GS_invar (GS_initial_impl I v0 succs)" (is ?G2)
proof -
  from REL have [simp]: "D = oGS_α I" and I: "oGS_invar I"
    by (simp_all add: oGS_rel_def br_def)

  from I have [simp]: "j v. I v  Some (STACK j)"
    by (simp add: oGS_invar_def)

  show ?G1
    unfolding GS.α_def GS_initial_impl_def
    apply (simp split del: if_split) apply (intro conjI)

    unfolding GS.p_α_def GS.seg_def[abs_def] GS.seg_start_def GS.seg_end_def
    apply (auto) []

    using A unfolding GS.D_α_def apply (auto simp: oGS_α_def) []

    unfolding GS.pE_α_def apply auto []
    done

  show ?G2
    unfolding GS_initial_impl_def
    apply unfold_locales
    apply auto
    done
qed

context GS_invar
begin
  lemma push_correct:
    assumes A: "v(set p_α)" and B: "vD_α"
    shows "GS.α (push_impl v succs) = (p_α@[{v}],D_α,pE_α  {v}×succs)" 
      (is ?G1)
    and "GS_invar (push_impl v succs)" (is ?G2)
  proof -

    note [simp] = Let_def

    have A1: "GS.D_α (push_impl v succs) = D_α"
      using B
      by (auto simp: push_impl_def GS.D_α_def)

    have iexI: "a b j P. a!j = b!j; P j  j'. a!j = b!j'  P j'"
      by blast

    have A2: "GS.p_α (push_impl v succs) = p_α @ [{v}]"
      unfolding push_impl_def GS.p_α_def GS.seg_def[abs_def] 
        GS.seg_start_def GS.seg_end_def
      apply (clarsimp split del: if_split)

      apply clarsimp
      apply safe
      apply (((rule iexI)?, 
        (auto  
          simp: nth_append nat_in_between_eq 
          dest: order.strict_trans[OF _ B_in_bound']
        )) []
      ) +
      done

    have iexI2: "j I Q. (j,I)set P; (j,I)set P  Q j  j. Q j"
      by blast

    have A3: "GS.pE_α (push_impl v succs) = pE_α  {v} × succs"
      unfolding push_impl_def GS.pE_α_def
      using P_bound
      apply (force simp: nth_append elim!: iexI2)
      done

    show ?G1
      unfolding GS.α_def
      by (simp add: A1 A2 A3)

    show ?G2
      apply unfold_locales
      unfolding push_impl_def
      apply simp_all

      using B_in_bound B_sorted B_distinct apply (auto simp: sorted_append) [3]
      using B_in_bound B0 apply (cases S) apply (auto simp: nth_append) [2]

      using S_distinct A apply (simp add: set_p_α_is_set_S)

      using A I_consistent 
      apply (auto simp: nth_append set_p_α_is_set_S split: if_split_asm) []
      
      using P_sorted P_distinct P_bound apply (auto simp: sorted_append) [3]
      done
  qed

  lemma no_last_out_P_aux:
    assumes NE: "p_α[]" and NS: "pE_α  last p_α × UNIV = {}"
    shows "set P  {0..<last B} × UNIV"
  proof -
    {
      fix j I
      assume jII: "(j,I)set P"
        and JL: "last Bj"
      with P_bound have JU: "j<length S" and INE: "I{}" by auto
      with JL JU have "S!j  last p_α"
        using NE
        unfolding p_α_def 
        apply (auto 
          simp: last_map seg_def seg_start_def seg_end_def last_conv_nth) 
        done
      moreover from jII have "{S!j} × I  pE_α" unfolding pE_α_def
        by auto
      moreover note INE NS
      ultimately have False by blast
    } thus ?thesis by fastforce
  qed

  lemma pop_correct:
    assumes NE: "p_α[]" and NS: "pE_α  last p_α × UNIV = {}"
    shows "pop_impl 
       GS_rel (SPEC (λr. r=(butlast p_α, D_α  last p_α, pE_α)))"
  proof -
    have iexI: "a b j P. a!j = b!j; P j  j'. a!j = b!j'  P j'"
      by blast
    
    have [simp]: "n. n - Suc 0  n  n0" by auto

    from NE have BNE: "B[]"
      unfolding p_α_def by auto

    {
      fix i j
      assume B: "j<B!i" and A: "i<length B"
      note B
      also from sorted_nth_mono[OF B_sorted, of i "length B - 1"] A 
      have "B!i  last B"
        by (simp add: last_conv_nth)
      finally have "j < last B" .
      hence "take (last B) S ! j = S ! j" 
        and "take (B!(length B - Suc 0)) S !j = S!j"
        by (simp_all add: last_conv_nth BNE)
    } note AUX1=this

    {
      fix v j
      have "(mark_as_done_abs 
              (seg_start (length B - Suc 0))
              (seg_end (length B - Suc 0)) I v = Some (STACK j)) 
         (j < length S  j < last B  v = take (last B) S ! j)"
        apply (simp add: mark_as_done_abs_def)
        apply safe []
        using I_consistent
        apply (clarsimp_all
          simp: seg_start_def seg_end_def last_conv_nth BNE
          simp: S_idx_uniq)

        apply (force)
        apply (subst nth_take)
        apply force
        apply force
        done
    } note AUX2 = this

    define ci where "ci = ( 
      take (last B) S, 
      butlast B,
      mark_as_done_abs 
        (seg_start (length B - Suc 0)) (seg_end (length B - Suc 0)) I,
      P)"

    have ABS: "GS.α ci = (butlast p_α, D_α  last p_α, pE_α)"
      apply (simp add: GS.α_def ci_def)
      apply (intro conjI)
      apply (auto  
        simp del: map_butlast
        simp add: map_butlast[symmetric] butlast_upt
        simp add: GS.p_α_def GS.seg_def[abs_def] GS.seg_start_def GS.seg_end_def
        simp: nth_butlast last_conv_nth nth_take AUX1
        cong: if_cong
        intro!: iexI
        dest: order.strict_trans[OF _ B_in_bound']
      ) []

      apply (auto 
        simp: GS.D_α_def p_α_def last_map BNE seg_def mark_as_done_abs_def) []

      using AUX1 no_last_out_P_aux[OF NE NS]
      apply (auto simp: GS.pE_α_def mark_as_done_abs_def elim!: bex2I) []
      done

    have INV: "GS_invar ci"
      apply unfold_locales
      apply (simp_all add: ci_def)

      using B_in_bound B_sorted B_distinct 
      apply (cases B rule: rev_cases, simp) 
      apply (auto simp: sorted_append order.strict_iff_order) [] 

      using B_sorted BNE apply (auto simp: sorted_butlast) []

      using B_distinct BNE apply (auto simp: distinct_butlast) []

      using B0 apply (cases B rule: rev_cases, simp add: BNE) 
      apply (auto simp: nth_append split: if_split_asm) []
   
      using S_distinct apply (auto) []

      apply (rule AUX2)

      using P_sorted P_distinct 
      apply (auto) [2]

      using P_bound no_last_out_P_aux[OF NE NS]
      apply (auto simp: in_set_conv_decomp)
      done
      

    show ?thesis
      unfolding pop_impl_def
      apply (refine_rcg 
        SPEC_refine refine_vcg order_trans[OF mark_as_done_aux])
      apply (simp_all add: BNE seg_start_less_end seg_end_bound)
      apply (fold ci_def)
      unfolding GS_rel_def
      apply (rule brI)
      apply (simp_all add: ABS INV)
      done
  qed


  lemma sel_rem_last_correct:
    assumes NE: "p_α[]"
    shows
    "sel_rem_last  (Id ×r GS_rel) (select_edge (p_α,D_α,pE_α))"
  proof -
    {
      fix l i a b b'
      have "i<length l; l!i=(a,b)  map fst (l[i:=(a,b')]) = map fst l"
        by (induct l arbitrary: i) (auto split: nat.split)
    } note map_fst_upd_snd_eq = this

    from NE have BNE[simp]: "B[]" unfolding p_α_def by simp

    have INVAR: "sel_rem_last  SPEC (GS_invar o snd)"
      unfolding sel_rem_last_def
      apply (refine_rcg refine_vcg)
      using locale_this apply (cases SBIP) apply simp

      apply simp

      using P_bound apply (cases P rule: rev_cases, auto) []

      apply simp

      apply simp apply (intro impI conjI)

      apply (unfold_locales, simp_all) []
      using B_in_bound B_sorted B_distinct B0 S_distinct I_consistent 
      apply auto [6]

      using P_sorted P_distinct 
      apply (auto simp: map_butlast sorted_butlast distinct_butlast) [2]

      using P_bound apply (auto dest: in_set_butlastD) []

      apply (unfold_locales, simp_all) []
      using B_in_bound B_sorted B_distinct B0 S_distinct I_consistent 
      apply auto [6]

      using P_sorted P_distinct 
      apply (auto simp: last_conv_nth map_fst_upd_snd_eq) [2]

      using P_bound 
      apply (cases P rule: rev_cases, simp)
      apply (auto) []

      using locale_this apply (cases SBIP) apply simp
      done


    {
      assume NS: "pE_αlast p_α×UNIV = {}"
      hence "sel_rem_last 
         SPEC (λr. case r of (None,SBIP')  SBIP'=SBIP | _  False)"
        unfolding sel_rem_last_def
        apply (refine_rcg refine_vcg)
        apply (cases SBIP)
        apply simp

        apply simp
        using P_bound apply (cases P rule: rev_cases, auto) []
        apply simp

        using no_last_out_P_aux[OF NE NS]
        apply (auto simp: seg_start_def last_conv_nth) []

        apply (cases SBIP)
        apply simp
        done
    } note SPEC_E = this

    {
      assume NON_EMPTY: "pE_αlast p_α×UNIV  {}"

      then obtain j succs P' where 
        EFMT: "P = P'@[(j,succs)]"
        unfolding pE_α_def
        by (cases P rule: rev_cases) auto
        
      with P_bound have J_UPPER: "j<length S" and SNE: "succs{}" 
        by auto

      have J_LOWER: "seg_start (length B - Suc 0)  j"
      proof (rule ccontr)
        assume "¬(seg_start (length B - Suc 0)  j)"
        hence "j < seg_start (length B - 1)" by simp
        with P_sorted EFMT 
        have P_bound': "set P  {0..<seg_start (length B - 1)} × UNIV"
          by (auto simp: sorted_append)
        hence "pE_α  last p_α×UNIV = {}"
          by (auto 
            simp: p_α_def last_conv_nth seg_def pE_α_def S_idx_uniq seg_end_def)
        thus False using NON_EMPTY by simp
      qed

      from J_UPPER J_LOWER have SJL: "S!jlast p_α" 
        unfolding p_α_def seg_def[abs_def] seg_end_def
        by (auto simp: last_map)

      from EFMT have SSS: "{S!j}×succspE_α"
        unfolding pE_α_def
        by auto


      {
        fix v
        assume "vsuccs"
        with SJL SSS have G: "(S!j,v)pE_α  last p_α×UNIV" by auto
        
        {
          fix j' succs'
          assume "S ! j' = S ! j" "(j', succs')  set P'"
          with J_UPPER P_bound S_idx_uniq EFMT have "j'=j" by auto
          with P_distinct (j', succs')  set P' EFMT have False by auto
        } note AUX3=this

        have G1: "GS.pE_α (S,B,I,P' @ [(j, succs - {v})]) = pE_α - {(S!j, v)}"
          unfolding GS.pE_α_def using AUX3
          by (auto simp: EFMT)
        
        {
          assume "succs{v}"
          hence "GS.pE_α (S,B,I,P' @ [(j, succs - {v})]) = GS.pE_α (S,B,I,P')"
            unfolding GS.pE_α_def by auto

          with G1 have "GS.pE_α (S,B,I,P') = pE_α - {(S!j, v)}" by simp
        } note G2 = this

        note G G1 G2
      } note AUX3 = this

      have "sel_rem_last  SPEC (λr. case r of 
        (Some v,SBIP')  u. 
            (u,v)(pE_αlast p_α×UNIV) 
           GS.α SBIP' = (p_α,D_α,pE_α-{(u,v)})
      | _  False)"
        unfolding sel_rem_last_def
        apply (refine_rcg refine_vcg)

        using SNE apply (vc_solve simp: J_LOWER EFMT)

        apply (frule AUX3(1))

        apply safe

        apply (drule (1) AUX3(3)) apply (auto simp: EFMT GS.α_def) []
        apply (drule AUX3(2)) apply (auto simp: GS.α_def) []
        done
    } note SPEC_NE=this

    have SPEC: "sel_rem_last  SPEC (λr. case r of 
        (None, SBIP')  SBIP' = SBIP  pE_α  last p_α × UNIV = {}  GS_invar SBIP
      | (Some v, SBIP')  u. (u, v)  pE_α  last p_α × UNIV 
                         GS.α SBIP' = (p_α, D_α, pE_α - {(u, v)})
                         GS_invar SBIP'
    )"  
      using INVAR
      apply (cases "pE_α  last p_α × UNIV = {}") 
      apply (frule SPEC_E)
      apply (auto split: option.splits simp: pw_le_iff; blast; fail)
      apply (frule SPEC_NE)
      apply (auto split: option.splits simp: pw_le_iff; blast; fail)
      done    
      
      
    have X1: "(y. (y=None  Φ y)  (a b. y=Some (a,b)  Ψ y a b)) 
      (Φ None  (a b. Ψ (Some (a,b)) a b))" for Φ Ψ
      by auto
      

    show ?thesis
      apply (rule order_trans[OF SPEC])
      unfolding select_edge_def select_def 
      apply (simp 
        add: pw_le_iff refine_pw_simps prod_rel_sv 
        del: SELECT_pw
        split: option.splits prod.splits)
      apply (fastforce simp: br_def GS_rel_def GS.α_def)
      done  
  qed

  lemma find_seg_idx_of_correct:
    assumes A: "v(set p_α)"
    shows "(find_seg (S_idx_of v)) = idx_of p_α v"
  proof -
    note S_idx_of_correct[OF A] idx_of_props[OF p_α_disjoint_sym A]
    from find_seg_correct[OF S_idx_of v < length S] have 
      "find_seg (S_idx_of v) < length p_α" 
      and "S!S_idx_of v  p_α!find_seg (S_idx_of v)"
      unfolding p_α_def by auto
    from idx_of_uniq[OF p_α_disjoint_sym this] S ! S_idx_of v = v 
    show ?thesis by auto
  qed


  lemma idx_of_correct:
    assumes A: "v(set p_α)"
    shows "idx_of_impl v  SPEC (λx. x=idx_of p_α v  x<length B)"
    using assms
    unfolding idx_of_impl_def
    apply (refine_rcg refine_vcg)
    apply (metis I_consistent in_set_conv_nth set_p_α_is_set_S)
    apply (erule S_idx_of_correct)
    apply (simp add: find_seg_impl find_seg_idx_of_correct)
    by (metis find_seg_correct(2) find_seg_impl)

  lemma collapse_correct:
    assumes A: "v(set p_α)"
    shows "collapse_impl v GS_rel (SPEC (λr. r=collapse v α))"
  proof -
    {
      fix i
      assume "i<length p_α"
      hence ILEN: "i<length B" by (simp add: p_α_def)

      let ?SBIP' = "(S, take (Suc i) B, I, P)"

      {
        have [simp]: "GS.seg_start ?SBIP' i = seg_start i"
          by (simp add: GS.seg_start_def)

        have [simp]: "GS.seg_end ?SBIP' i = seg_end (length B - 1)"
          using ILEN by (simp add: GS.seg_end_def min_absorb2)

        {
          fix j
          assume B: "seg_start i  j" "j < seg_end (length B - Suc 0)"
          hence "j<length S" using ILEN seg_end_bound 
          proof -
            note B(2)
            also from i<length B have "(length B - Suc 0) < length B" by auto
            from seg_end_bound[OF this] 
            have "seg_end (length B - Suc 0)  length S" .
            finally show ?thesis .
          qed

          have "i  find_seg j  find_seg j < length B 
             seg_start (find_seg j)  j  j < seg_end (find_seg j)" 
          proof (intro conjI)
            show "ifind_seg j"
              by (metis le_trans not_less B(1) find_seg_bounds(2) 
                seg_end_less_start ILEN j < length S)
          qed (simp_all add: find_seg_bounds[OF j<length S])
        } note AUX1 = this

        {
          fix Q and j::nat
          assume "Q j"
          hence "i. S!j = S!i  Q i"
            by blast
        } note AUX_ex_conj_SeqSI = this

        have "GS.seg ?SBIP' i =  (seg ` {i..<length B})"
          unfolding GS.seg_def[abs_def]
          apply simp
          apply (rule)
          apply (auto dest!: AUX1) []

          (* The following three lines complete the proof. AUX_ex_conj_SeqSI
            and all stuff 
            below would be unnecessary, if smt would be allowed for AFP.
          apply (auto simp: seg_start_def seg_end_def split: if_split_asm)
          apply (smt distinct_sorted_mono[OF B_sorted B_distinct])
          apply (smt distinct_sorted_mono[OF B_sorted B_distinct] B_in_bound')
          *)

          apply (auto 
            simp: seg_start_def seg_end_def 
            split: if_split_asm
            intro!: AUX_ex_conj_SeqSI
          )

         apply (metis diff_diff_cancel le_diff_conv le_eq_less_or_eq 
           lessI trans_le_add1 
           distinct_sorted_mono[OF B_sorted B_distinct, of i])

         apply (metis diff_diff_cancel le_diff_conv le_eq_less_or_eq 
           trans_le_add1 distinct_sorted_mono[OF B_sorted B_distinct, of i])
         
         apply (metis (opaque_lifting, no_types) Suc_lessD Suc_lessI less_trans_Suc
           B_in_bound')
         done
      } note AUX2 = this
      
      from ILEN have "GS.p_α (S, take (Suc i) B, I, P) = collapse_aux p_α i"
        unfolding GS.p_α_def collapse_aux_def
        apply (simp add: min_absorb2 drop_map)
        apply (rule conjI)
        apply (auto 
          simp: GS.seg_def[abs_def] GS.seg_start_def GS.seg_end_def take_map) []

        apply (simp add: AUX2)
        done
    } note AUX1 = this

    from A obtain i where [simp]: "I v = Some (STACK i)"
      using I_consistent set_p_α_is_set_S
      by (auto simp: in_set_conv_nth)

    {
      have "(collapse_aux p_α (idx_of p_α v), D_α, pE_α) =
        GS.α (S, take (Suc (idx_of p_α v)) B, I, P)"
      unfolding GS.α_def
      using idx_of_props[OF p_α_disjoint_sym A]
      by (simp add: AUX1)
    } note ABS=this

    {
      have "GS_invar (S, take (Suc (idx_of p_α v)) B, I, P)"
        apply unfold_locales
        apply simp_all

        using B_in_bound B_sorted B_distinct
        apply (auto simp: sorted_wrt_take dest: in_set_takeD) [3]

        using B0 S_distinct apply auto [2]

        using I_consistent apply simp

        using P_sorted P_distinct P_bound apply auto [3]
        done
    } note INV=this

    show ?thesis
      unfolding collapse_impl_def
      apply (refine_rcg SPEC_refine refine_vcg order_trans[OF idx_of_correct])
        apply fact
       apply simp
      apply (simp add: collapse_def α_def find_seg_impl GS_rel_def)
      apply (rule brI)
        apply (rule ABS)
        apply (rule INV)
      done
  qed

end

text ‹Technical adjustment for avoiding case-splits for definitions
  extracted from GS-locale›
lemma opt_GSdef: "f  g  f s  case s of (S,B,I,P)  g (S,B,I,P)" by auto

lemma ext_def: "fg  f x  g x" by auto

context fr_graph begin
  definition "push_impl v s  GS.push_impl s v (E``{v})" 
  lemmas push_impl_def_opt = 
    push_impl_def[abs_def, 
    THEN ext_def, THEN opt_GSdef, unfolded GS.push_impl_def GS_sel_simps]

  text ‹Definition for presentation›
  lemma "push_impl v (S,B,I,P)  (S@[v], B@[length S], I(vSTACK (length S)),
    if E``{v}={} then P else P@[(length S,E``{v})])"
    unfolding push_impl_def GS.push_impl_def GS.P_def GS.S_def
    by (auto simp: Let_def)

  lemma GS_α_split: 
    "GS.α s = (p,D,pE)  (p=GS.p_α s  D=GS.D_α s  pE=GS.pE_α s)"
    "(p,D,pE) = GS.α s  (p=GS.p_α s  D=GS.D_α s  pE=GS.pE_α s)"
    by (auto simp add: GS.α_def)

  lemma push_refine:
    assumes A: "(s,(p,D,pE))GS_rel" "(v,v')Id"
    assumes B: "v(set p)" "vD"
    shows "(push_impl v s, push v' (p,D,pE))GS_rel"
  proof -
    from A have [simp]: "p=GS.p_α s  D=GS.D_α s  pE=GS.pE_α s" "v'=v" 
      and INV: "GS_invar s"
      by (auto simp add: GS_rel_def br_def GS_α_split)

    from INV B show ?thesis
      by (auto 
        simp: GS_rel_def br_def GS_invar.push_correct push_impl_def push_def)
  qed

  definition "pop_impl s  GS.pop_impl s"
  lemmas pop_impl_def_opt = 
    pop_impl_def[abs_def, THEN opt_GSdef, unfolded GS.pop_impl_def
    GS.mark_as_done_def GS.seg_start_def GS.seg_end_def 
    GS_sel_simps]

  lemma pop_refine:
    assumes A: "(s,(p,D,pE))GS_rel"
    assumes B: "p  []" "pE  last p × UNIV = {}"
    shows "pop_impl s  GS_rel (RETURN (pop (p,D,pE)))"
  proof -
    from A have [simp]: "p=GS.p_α s  D=GS.D_α s  pE=GS.pE_α s" 
      and INV: "GS_invar s"
      by (auto simp add: GS_rel_def br_def GS_α_split)

    show ?thesis
      unfolding pop_impl_def[abs_def] pop_def
      apply (rule order_trans[OF GS_invar.pop_correct])
      using INV B
      apply (simp_all add: Un_commute RETURN_def) 
      done
  qed

  thm pop_refine[no_vars]

  definition "collapse_impl v s  GS.collapse_impl s v"
  lemmas collapse_impl_def_opt = 
    collapse_impl_def[abs_def, 
    THEN ext_def, THEN opt_GSdef, unfolded GS.collapse_impl_def GS_sel_simps]

  lemma collapse_refine:
    assumes A: "(s,(p,D,pE))GS_rel" "(v,v')Id"
    assumes B: "v'(set p)"
    shows "collapse_impl v s GS_rel (RETURN (collapse v' (p,D,pE)))"
  proof -
    from A have [simp]: "p=GS.p_α s  D=GS.D_α s  pE=GS.pE_α s" "v'=v" 
      and INV: "GS_invar s"
      by (auto simp add: GS_rel_def br_def GS_α_split)

    show ?thesis
      unfolding collapse_impl_def[abs_def]
      apply (rule order_trans[OF GS_invar.collapse_correct])
      using INV B by (simp_all add: GS.α_def RETURN_def)
  qed

  definition "select_edge_impl s  GS.sel_rem_last s"
  lemmas select_edge_impl_def_opt = 
    select_edge_impl_def[abs_def, 
      THEN opt_GSdef, 
      unfolded GS.sel_rem_last_def GS.seg_start_def GS_sel_simps]

  lemma select_edge_refine: 
    assumes A: "(s,(p,D,pE))GS_rel"
    assumes NE: "p  []"
    shows "select_edge_impl s  (Id ×r GS_rel) (select_edge (p,D,pE))"
  proof -
    from A have [simp]: "p=GS.p_α s  D=GS.D_α s  pE=GS.pE_α s" 
      and INV: "GS_invar s"
      by (auto simp add: GS_rel_def br_def GS_α_split)

    from INV NE show ?thesis
      unfolding select_edge_impl_def
      using GS_invar.sel_rem_last_correct[OF INV] NE
      by (simp)
  qed

  definition "initial_impl v0 I  GS_initial_impl I v0 (E``{v0})"

  lemma initial_refine:
    "v0D0; (I,D0)oGS_rel; (v0i,v0)Id 
     (initial_impl v0i I,initial v0 D0)GS_rel"
    unfolding initial_impl_def GS_rel_def br_def
    apply (simp_all add: GS_initial_correct)
    apply (auto simp: initial_def)
    done


  definition "path_is_empty_impl s  GS.S s = []"
  lemma path_is_empty_refine: 
    "GS_invar s  path_is_empty_impl s  GS.p_α s=[]"
    unfolding path_is_empty_impl_def GS.p_α_def GS_invar.empty_eq
    by auto

  definition (in GS) "is_on_stack_impl v 
     case I v of Some (STACK _)  True | _  False"

  lemma (in GS_invar) is_on_stack_impl_correct:
    shows "is_on_stack_impl v  v(set p_α)"
    unfolding is_on_stack_impl_def
    using I_consistent[of v]
    apply (force 
      simp: set_p_α_is_set_S in_set_conv_nth 
      split: option.split node_state.split)
    done

  definition "is_on_stack_impl v s  GS.is_on_stack_impl s v"
  lemmas is_on_stack_impl_def_opt = 
    is_on_stack_impl_def[abs_def, THEN ext_def, THEN opt_GSdef, 
      unfolded GS.is_on_stack_impl_def GS_sel_simps]

  lemma is_on_stack_refine:
    " GS_invar s   is_on_stack_impl v s  v(set (GS.p_α s))"
    unfolding is_on_stack_impl_def GS_rel_def br_def
    by (simp add: GS_invar.is_on_stack_impl_correct)


  definition (in GS) "is_done_impl v 
     case I v of Some DONE  True | _  False"

  lemma (in GS_invar) is_done_impl_correct:
    shows "is_done_impl v  vD_α"
    unfolding is_done_impl_def D_α_def
    apply (auto split: option.split node_state.split)
    done

  definition "is_done_oimpl v I  case I v of Some DONE  True | _  False"

  definition "is_done_impl v s  GS.is_done_impl s v"

  lemma is_done_orefine:
    " oGS_invar s   is_done_oimpl v s  voGS_α s"
    unfolding is_done_oimpl_def oGS_rel_def br_def
    by (auto 
      simp: oGS_invar_def oGS_α_def 
      split: option.splits node_state.split)

  lemma is_done_refine:
    " GS_invar s   is_done_impl v s  vGS.D_α s"
    unfolding is_done_impl_def GS_rel_def br_def
    by (simp add: GS_invar.is_done_impl_correct)

  lemma oinitial_refine: "(Map.empty, {})  oGS_rel"
    by (auto simp: oGS_rel_def br_def oGS_α_def oGS_invar_def)

end

subsection ‹Refined Skeleton Algorithm›

context fr_graph begin

  lemma I_to_outer:
    assumes "((S, B, I, P), ([], D, {}))  GS_rel"
    shows "(I,D)oGS_rel"
    using assms
    unfolding GS_rel_def oGS_rel_def br_def oGS_α_def GS.α_def GS.D_α_def GS_invar_def oGS_invar_def
    apply (auto simp: GS.p_α_def)
    done
  
  
  definition skeleton_impl :: "'v oGS nres" where
    "skeleton_impl  do {
      stat_start_nres;
      let I=Map.empty;
      r  FOREACHi (λit I. outer_invar it (oGS_α I)) V0 (λv0 I0. do {
        if ¬is_done_oimpl v0 I0 then do {
          let s = initial_impl v0 I0;

          (S,B,I,P)WHILEIT (invar v0 (oGS_α I0) o GS.α)
            (λs. ¬path_is_empty_impl s) (λs.
          do {
            ― ‹Select edge from end of path›
            (vo,s)  select_edge_impl s;

            case vo of 
              Some v  do {
                if is_on_stack_impl v s then do {
                  collapse_impl v s
                } else if ¬is_done_impl v s then do {
                  ― ‹Edge to new node. Append to path›
                  RETURN (push_impl v s)
                } else do {
                  ― ‹Edge to done node. Skip›
                  RETURN s
                }
              }
            | None  do {
                ― ‹No more outgoing edges from current node on path›
                pop_impl s
              }
          }) s;
          RETURN I
        } else
          RETURN I0
      }) I;
      stat_stop_nres;
      RETURN r
    }"

  subsubsection ‹Correctness Theorem›

  lemma "skeleton_impl  oGS_rel skeleton"
    using [[goals_limit = 1]]
    unfolding skeleton_impl_def skeleton_def
    apply (refine_rcg
      bind_refine'
      select_edge_refine push_refine 
      pop_refine
      collapse_refine 
      initial_refine
      oinitial_refine
      inj_on_id
    )
    using [[goals_limit = 5]]
    apply refine_dref_type  

    apply (vc_solve (nopre) solve: asm_rl I_to_outer
      simp: GS_rel_def br_def GS.α_def oGS_rel_def oGS_α_def 
      is_on_stack_refine path_is_empty_refine is_done_refine is_done_orefine
    )

    done

  lemmas skeleton_refines 
    = select_edge_refine push_refine pop_refine collapse_refine 
      initial_refine oinitial_refine
  lemmas skeleton_refine_simps 
    = GS_rel_def br_def GS.α_def oGS_rel_def oGS_α_def 
      is_on_stack_refine path_is_empty_refine is_done_refine is_done_orefine

  text ‹Short proof, for presentation›
  context
    notes [[goals_limit = 1]]
    notes [refine] = inj_on_id bind_refine'
  begin
  lemma "skeleton_impl  oGS_rel skeleton"
    unfolding skeleton_impl_def skeleton_def
    by (refine_rcg skeleton_refines, refine_dref_type)
       (vc_solve (nopre) solve: asm_rl I_to_outer simp: skeleton_refine_simps)  

  end

end

end