# Theory 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 ())"

"m1≤m2 ⟹ stat_newnode_nres ⪢ m1 ≤ m2"
"m1≤m2 ⟹ stat_start_nres ⪢ m1 ≤ m2"
"m1≤m2 ⟹ 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. i≤k ∧ k<j}"

lemma path_seg_simps[simp]:
"j≤i ⟹ 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 ∧ v∈p!i"

lemma idx_of_props:
assumes
p_disjoint_sym: "∀i j v. i<length p ∧ j<length p ∧ v∈p!i ∧ v∈p!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"
moreover hence "∀j<length p. v∈p ! 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 ∧ v∈p!i ∧ v∈p!j ⟶ i=j"
assumes A: "i<length p" "v∈p!i"
shows "idx_of p v = i"
proof -
from A p_disjoint_sym have "∀j<length p. v∈p ! 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: "it⊆V0"  ― ‹Only start nodes to iterate over›

assumes it_done: "V0 - it ⊆ D"  ― ‹Nodes already iterated over are visited›
assumes D_reachable: "D⊆E⇧*V0" ― ‹Done nodes are reachable›
assumes D_closed: "ED ⊆ 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!]: "v0∈V0"
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: "D⊆E⇧*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: "U∈set p ⟹ U×U ⊆ (vE p D pE ∩ U×U)⇧*"
― ‹Nodes in CNodes are mutually reachable by visited edges›

assumes root_v0: "p≠[] ⟹ v0∈hd p" ― ‹Root CNode contains start node›
assumes p_empty_v0: "p=[] ⟹ v0∈D" ― ‹Start node is done if path empty›

assumes D_closed: "ED ⊆ 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: "E∩D×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 v0∉D0 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 v∉D 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; u∈p!i; v∈p!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; u∈p!i; v∈p!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={}"
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; v∈p!i; v∈p!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" "u≤length p"
shows "p!i⊆path_seg p l u ⟷ l≤i ∧ i<u"
proof
assume B: "p!i⊆path_seg p l u"
from A obtain x where "x∈p!i" by (blast dest: p_no_empty_idx)
with B obtain i' where C: "x∈p!i'" "l≤i'" "i'<u"
by (auto simp: path_seg_def)
from p_disjoint_sym[OF ‹i<length p› _ ‹x∈p!i› ‹x∈p!i'›] ‹i'<u› ‹u≤length p›
have "i=i'" by simp
with C show "l≤i ∧ i<u" by auto
qed (auto simp: path_seg_def)

lemma path_seg_ss_eq[simp]:
assumes A: "l1<u1" "u1≤length p" "l2<u2" "u2≤length p"
shows "path_seg p l1 u1 ⊆ path_seg p l2 u2 ⟷ l2≤l1 ∧ u1≤u2"
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: "l2≤l1" using A by simp
have "p!(u1 - 1) ⊆ path_seg p l1 u1" using A by simp
also note S finally have 2: "u1≤u2" using A by auto
from 1 2 show "l2≤l1 ∧ u1≤u2" ..
next
assume "l2≤l1 ∧ u1≤u2" 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 "x∈p!i" "y∈p!j"
assumes "i≤j" "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" "u∈p!j" "v∈p!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'] ‹x∈p!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› ‹u∈p!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› ‹v∈p!Suc j› ‹y∈p!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 "v∈p!i"
by (metis UnionE in_set_conv_nth)
moreover from A root_v0 have "v0∈p!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 "v∈E⇧*{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; u∈D⟧ ⟹ 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; u∈D⟧ ⟹ 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: "u∈last p" "v∈last 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 "v∈ltouched" by auto
hence "v∈⋃(set p)"
unfolding touched_def
proof
assume "v∈D"
moreover from ‹path lvE v l w› have "(v,w)∈lvE⇧*" by (rule path_is_rtrancl)
ultimately have "w∈D" using D_closed_vE_rtrancl by auto
with ‹w∈last 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" "v∈p!i"
by (metis UnionE in_set_conv_nth)
have "i=length p - 1"
proof (rule ccontr)
assume "i≠length 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 = {}"
with ‹(u,v)∈lvE› ‹u∈last p› ‹v∈p!i› show False by auto
qed
with ‹v∈p!i› have "v∈last p" by (simp add: last_conv_nth)
with path_prepend.IH ‹w∈last p› ‹u∈last 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 "u∈last 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 "v∈set q" "v∈last 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≠[]› ‹v∈last p› ‹v∈last p›]
show "set q ⊆ last p" by simp
from P show "u∈last 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; u∈last p; v∉⋃(set p); v∉D ⟧
⟹ 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) = {}"

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: "u∈last p ∪ D" hence "u∈p!?i ∪ D" by simp

{
assume "u∈last p" "v∉last p"
moreover from E NO ‹u∈last p› have "(u,v)∈lvE" by auto
ultimately have "v∈D ∨ v∈⋃(set p)"
using vE_touched unfolding touched_def by auto
moreover {
assume "v∈⋃(set p)"
then obtain j where V: "j<length p" "v∈p!j"
by (metis UnionE in_set_conv_nth)
with ‹v∉last p› have "j<?i" by (cases "j=?i") auto
from vE_no_back[OF ‹j<?i› _] ‹(u,v)∈lvE› V ‹u∈last p› have False by auto
} ultimately have "v∈D" by blast
} with E UI D_closed show "v∈last 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 "v0∈it - 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; v0∈it; v0∉D0⟧ ⟹ 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 "u∈last 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: "v∈p!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 ‹u∈last p› have "u∈p!(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"

{
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!j⊆p'!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 "v∈p'!j" and "v∈p'!k"
with A have "v∈p!j" by simp
show False proof (cases)
assume "k=i"
with ‹v∈p'!k› obtain k' where "v∈p!k'" "i≤k'" "k'<length p"
by (auto simp: path_seg_def)
hence "p ! j ∩ p ! k' = {}"
using A by (auto intro!: p_disjoint)
with ‹v∈p!j› ‹v∈p!k'› show False by auto
next
assume "k≠i" 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 ‹v∈p'!j› ‹v∈p'!k› by auto
qed
qed
} note AUX_p_disjoint = this

{
fix U
assume A: "U∈set 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 "x∈path_seg p i (length p)" "y∈path_seg p i (length p)"
then obtain ix iy where
IX: "x∈p!ix" "i≤ix" "ix<length p" and
IY: "y∈p!iy" "i≤iy" "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 ‹x∈p!ix› ‹u∈p!(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 ‹u∈p!(length p - 1)›])
apply (simp)
apply (rule rev_subsetD[OF VMEM])
apply (simp)
done
also
from pathI[OF ‹v∈p!i› ‹y∈p!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 "j≠i"
with ‹j<Suc i› have [simp]: "j<i" by simp
with ‹i<length p› have "p!j∈set p"
by (metis Suc_lessD in_set_conv_nth less_trans_Suc)

thus ?thesis using p_sc[of U] ‹p!j∈set 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" "x∈path_seg p i (length p)" "y∈p!j"
then obtain ix where "x∈p!ix" "i≤ix" "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 (metis NE hd_in_set)
apply (cases p, simp_all 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: "u∈last p"
assumes VNE: "v∉⋃(set p)" "v∉D"
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: "v∉p!i" "v∉p!Suc i" by auto

from p_connected[OF SILL'] obtain a b where
"a∈p!i" "b∈p!Suc i" "(a,b)∈E" "(a,b)∉pE"
by auto
with X1 have "a≠v" "b≠v" by auto
with ‹(a,b)∈E› ‹(a,b)∉pE› have "(a,b)∈(E - (pE - {(u, v)} ∪ E ∩ {v} × UNIV))"
by auto
with ‹a∈p!i› ‹b∈p!Suc i›
show ?thesis using  SILL'
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 "U∈set 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 "U∉set 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
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: "u∈last p"
assumes VNP: "v∉⋃(set p)" and VD: "v∈D"
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: "u∈E⇧*{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 "u∈D" by auto
with D_closed ‹(u,v)∈E› have "v∈D" by auto
from path_prepend.prems path_prepend.hyps have "v∈E⇧*{v0}" by auto
with path_prepend.IH fin_D_is_reachable[OF INV] have "path lvE v p w"
by simp
moreover from ‹u∈D› ‹v∈D› ‹(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: "v0∉D0" "v0∈it"
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]: "v0∈D'" 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: "v0∈D0" "v0∈it"
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" "u∈last 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" "u∈last 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" "u∈last p" and A: "v∉D" "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" "u∈last 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{Gabow2000} 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 ⇒ (nat⇒bool) ⇒ 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'$j')∈bool_rel" shows "(find_max_nat n P, (OP find_max_nat ::: nat_rel → (nat_rel→bool_rel) → nat_rel)$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 ∧ v∈I }"
― ‹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!i≤j}"

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!i≤j}"
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

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 "v∈set 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 ∧ v∈p_α!i ∧ v∈p_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. l≤j ∧ j<u} then Some DONE else I v)"

lemma mark_as_done_aux:
fixes l u I
shows "⟦l<u; u≤length 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. l≤j ∧ j<l'} then Some DONE else I v)
∧ l≤l' ∧ 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. x∈succs);
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!i≤j)"

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)
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 {
i←idx_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: "v0∉D"
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"

from I have [simp]: "⋀j v. I v ≠ Some (STACK j)"

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: "v∉D_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 B≤j"
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 ⟷ n≠0" 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"
finally have "j < last B" .
hence "take (last B) S ! j = S ! j"
and "take (B!(length B - Suc 0)) S !j = S!j"
} 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 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 (intro conjI)
apply (auto
simp del: map_butlast
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

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)
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_α∩last 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_α∩last 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!j∈last p_α"
unfolding p_α_def seg_def[abs_def] seg_end_def
by (auto simp: last_map)

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

{
fix v
assume "v∈succs"
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_α∩last 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
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)
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"

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 "i≤find_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
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 (hide_lams, 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 (rule conjI)
apply (auto
simp: GS.seg_def[abs_def] GS.seg_start_def GS.seg_end_def take_map) []

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]
} 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_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 (metis discrete)

apply (simp add: collapse_def α_def find_seg_impl)
unfolding 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: "f≡g ⟹ 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(v↦STACK (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)"

lemma push_refine:
assumes A: "(s,(p,D,pE))∈GS_rel" "(v,v')∈Id"
assumes B: "v∉⋃(set p)" "v∉D"
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
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:
"⟦v0∉D0; (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 (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

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 ⟷ v∈D_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 ⟷ v∈oGS_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 ⟷ v∈GS.D_α s"
unfolding is_done_impl_def GS_rel_def br_def

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


# Theory Gabow_SCC

section ‹Enumerating the SCCs of a Graph \label{sec:scc}›
theory Gabow_SCC
imports Gabow_Skeleton
begin

text ‹
As a first variant, we implement an algorithm that computes a list of SCCs
of a graph, in topological order. This is the standard variant described by
Gabow~\cite{Gabow2000}.
›

section ‹Specification›
context fr_graph
begin
text ‹We specify a distinct list that covers all reachable nodes and
contains SCCs in topological order›

definition "compute_SCC_spec ≡ SPEC (λl.
distinct l ∧ ⋃(set l) = E⇧*V0 ∧ (∀U∈set l. is_scc E U)
∧ (∀i j. i<j ∧ j<length l ⟶ l!j × l!i ∩ E⇧* = {}) )"
end

section ‹Extended Invariant›

locale cscc_invar_ext = fr_graph G
for G :: "('v,'more) graph_rec_scheme" +
fixes l :: "'v set list" and D :: "'v set"
assumes l_is_D: "⋃(set l) = D" ― ‹The output contains all done CNodes›
assumes l_scc: "set l ⊆ Collect (is_scc E)" ― ‹The output contains only SCCs›
assumes l_no_fwd: "⋀i j. ⟦i<j; j<length l⟧ ⟹ l!j × l!i ∩ E⇧* = {}"
― ‹The output contains no forward edges›
begin
lemma l_no_empty: "{}∉set l" using l_scc by (auto simp: in_set_conv_decomp)
end

locale cscc_outer_invar_loc = outer_invar_loc G it D + cscc_invar_ext G l D
for G :: "('v,'more) graph_rec_scheme" and it l D
begin
lemma locale_this: "cscc_outer_invar_loc G it l D" by unfold_locales
lemma abs_outer_this: "outer_invar_loc G it D" by unfold_locales
end

locale cscc_invar_loc = invar_loc G v0 D0 p D pE + cscc_invar_ext G l D
for G :: "('v,'more) graph_rec_scheme" and v0 D0 and l :: "'v set list"
and p D pE
begin
lemma locale_this: "cscc_invar_loc G v0 D0 l p D pE" by unfold_locales
lemma invar_this: "invar_loc G v0 D0 p D pE" by unfold_locales
end

context fr_graph
begin
definition "cscc_outer_invar ≡ λit (l,D). cscc_outer_invar_loc G it l D"
definition "cscc_invar ≡ λv0 D0 (l,p,D,pE). cscc_invar_loc G v0 D0 l p D pE"
end

section ‹Definition of the SCC-Algorithm›

context fr_graph
begin
definition compute_SCC :: "'v set list nres" where
"compute_SCC ≡ do {
let so = ([],{});
(l,D) ← FOREACHi cscc_outer_invar V0 (λv0 (l,D0). do {
if v0∉D0 then do {
let s = (l,initial v0 D0);

(l,p,D,pE) ←
WHILEIT (cscc_invar v0 D0)
(λ(l,p,D,pE). p ≠ []) (λ(l,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 {
if v ∈ ⋃(set p) then do {
― ‹Collapse›
RETURN (l,collapse v (p,D,pE))
} else if v∉D then do {
― ‹Edge to new node. Append to path›
RETURN (l,push v (p,D,pE))
} else RETURN (l,p,D,pE)
}
| None ⇒ do {
― ‹No more outgoing edges from current node on path›
ASSERT (pE ∩ last p × UNIV = {});
let V = last p;
let (p,D,pE) = pop (p,D,pE);
let l = V#l;
RETURN (l,p,D,pE)
}
}) s;
ASSERT (p=[] ∧ pE={});
RETURN (l,D)
} else
RETURN (l,D0)
}) so;
RETURN l
}"
end

section ‹Preservation of Invariant Extension›
context cscc_invar_ext
begin
lemma l_disjoint:
assumes A: "i<j" "j<length l"
shows "l!i ∩ l!j = {}"
proof (rule disjointI)
fix u
assume "u∈l!i" "u∈l!j"
with l_no_fwd A show False by auto
qed

corollary l_distinct: "distinct l"
using l_disjoint l_no_empty
by (metis distinct_conv_nth inf_idem linorder_cases nth_mem)
end

context fr_graph
begin
definition "cscc_invar_part ≡ λ(l,p,D,pE). cscc_invar_ext G l D"

lemma cscc_invarI[intro?]:
assumes "invar v0 D0 PDPE"
assumes "invar v0 D0 PDPE ⟹ cscc_invar_part (l,PDPE)"
shows "cscc_invar v0 D0 (l,PDPE)"
using assms
unfolding initial_def cscc_invar_def invar_def
apply (simp split: prod.split_asm)
apply intro_locales
done

thm cscc_invarI[of v_0 D_0 s l]

lemma cscc_outer_invarI[intro?]:
assumes "outer_invar it D"
assumes "outer_invar it D ⟹ cscc_invar_ext G l D"
shows "cscc_outer_invar it (l,D)"
using assms
unfolding initial_def cscc_outer_invar_def outer_invar_def
apply (simp split: prod.split_asm)
apply intro_locales
done

lemma cscc_invar_initial[simp, intro!]:
assumes A: "v0∈it" "v0∉D0"
assumes INV: "cscc_outer_invar it (l,D0)"
shows "cscc_invar_part (l,initial v0 D0)"
proof -
from INV interpret cscc_outer_invar_loc G it l D0
unfolding cscc_outer_invar_def by simp

show ?thesis
unfolding cscc_invar_part_def initial_def
apply simp
by unfold_locales
qed

lemma cscc_invar_pop:
assumes INV: "cscc_invar v0 D0 (l,p,D,pE)"
assumes "invar v0 D0 (pop (p,D,pE))"
assumes NE[simp]: "p≠[]"
assumes NO': "pE ∩ (last p × UNIV) = {}"
shows "cscc_invar_part (last p # l, pop (p,D,pE))"
proof -
from INV interpret cscc_invar_loc G v0 D0 l p D pE
unfolding cscc_invar_def by simp

have AUX_l_scc: "is_scc E (last p)"
unfolding is_scc_pointwise
proof safe
{
assume "last p = {}" thus False
using p_no_empty by (cases p rule: rev_cases) auto
}

fix u v
assume "u∈last p" "v∈last p"
with p_sc[of "last p"] have "(u,v) ∈ (lvE ∩ last p × last p)⇧*" by auto
with lvE_ss_E show "(u,v)∈(E ∩ last p × last p)⇧*"
by (metis Int_mono equalityE rtrancl_mono_mp)

fix u'
assume "u'∉last p" "(u,u')∈E⇧*" "(u',v)∈E⇧*"

from ‹u'∉last p› ‹u∈last p› ‹(u,u')∈E⇧*›
and rtrancl_reachable_induct[OF order_refl lastp_un_D_closed[OF NE NO']]
have "u'∈D" by auto
with ‹(u',v)∈E⇧*› and rtrancl_reachable_induct[OF order_refl D_closed]
have "v∈D" by auto
with ‹v∈last p› p_not_D show False by (cases p rule: rev_cases) auto
qed

{
fix i j
assume A: "i<j" "j<Suc (length l)"
have "l ! (j - Suc 0) × (last p # l) ! i ∩ E⇧* = {}"
proof (rule disjointI, safe)
fix u v
assume "(u, v) ∈ E⇧*" "u ∈ l ! (j - Suc 0)" "v ∈ (last p # l) ! i"
from ‹u ∈ l ! (j - Suc 0)› A have "u∈⋃(set l)"
by (metis Ex_list_of_length Suc_pred UnionI length_greater_0_conv
less_nat_zero_code not_less_eq nth_mem)
with l_is_D have "u∈D" by simp
with rtrancl_reachable_induct[OF order_refl D_closed] ‹(u,v)∈E⇧*›
have "v∈D" by auto

show False proof cases
assume "i=0" hence "v∈last p" using ‹v ∈ (last p # l) ! i› by simp
with p_not_D ‹v∈D› show False by (cases p rule: rev_cases) auto
next
assume "i≠0" with ‹v ∈ (last p # l) ! i› have "v∈l!(i - 1)" by auto
with l_no_fwd[of "i - 1" "j - 1"]
and ‹u ∈ l ! (j - Suc 0)› ‹(u, v) ∈ E⇧*› ‹i≠0› A
show False by fastforce
qed
qed
} note AUX_l_no_fwd = this

show ?thesis
unfolding cscc_invar_part_def pop_def apply simp
apply unfold_locales
apply clarsimp_all
using l_is_D apply auto []

using l_scc AUX_l_scc apply auto []

apply (rule AUX_l_no_fwd, assumption+) []
done
qed

thm cscc_invar_pop[of v_0 D_0 l p D pE]

lemma cscc_invar_unchanged:
assumes INV: "cscc_invar v0 D0 (l,p,D,pE)"
shows "cscc_invar_part (l,p',D,pE')"
using INV unfolding cscc_invar_def cscc_invar_part_def cscc_invar_loc_def
by simp

corollary cscc_invar_collapse:
assumes INV: "cscc_invar v0 D0 (l,p,D,pE)"
shows "cscc_invar_part (l,collapse v (p',D,pE'))"
unfolding collapse_def

corollary cscc_invar_push:
assumes INV: "cscc_invar v0 D0 (l,p,D,pE)"
shows "cscc_invar_part (l,push v (p',D,pE'))"
unfolding push_def

lemma cscc_outer_invar_initial: "cscc_invar_ext G [] {}"
by unfold_locales auto

lemma cscc_invar_outer_newnode:
assumes A: "v0∉D0" "v0∈it"
assumes OINV: "cscc_outer_invar it (l,D0)"
assumes INV: "cscc_invar v0 D0 (l',[],D',pE)"
shows "cscc_invar_ext G l' D'"
proof -
from OINV interpret cscc_outer_invar_loc G it l D0
unfolding cscc_outer_invar_def by simp
from INV interpret inv: cscc_invar_loc G v0 D0 l' "[]" D' pE
unfolding cscc_invar_def by simp

show ?thesis
by unfold_locales

qed

lemma cscc_invar_outer_Dnode:
assumes "cscc_outer_invar it (l, D)"
shows "cscc_invar_ext G l D"
using assms

lemmas cscc_invar_preserve = invar_preserve
cscc_invar_initial
cscc_invar_pop cscc_invar_collapse cscc_invar_push cscc_invar_unchanged
cscc_outer_invar_initial cscc_invar_outer_newnode cscc_invar_outer_Dnode

text ‹On termination, the invariant implies the specification›
lemma cscc_finI:
assumes INV: "cscc_outer_invar {} (l,D)"
shows fin_l_is_scc: "⟦U∈set l⟧ ⟹ is_scc E U"
and fin_l_distinct: "distinct l"
and fin_l_is_reachable: "⋃(set l) = E⇧*  V0"
and fin_l_no_fwd: "⟦i<j; j<length l⟧ ⟹ l!j ×l!i ∩ E⇧* = {}"
proof -
from INV interpret cscc_outer_invar_loc G "{}" l D
unfolding cscc_outer_invar_def by simp

show "⟦U∈set l⟧ ⟹ is_scc E U" using l_scc by auto

show "distinct l" by (rule l_distinct)

show "⋃(set l) = E⇧*  V0"
using fin_outer_D_is_reachable[OF outer_invar_this] l_is_D
by auto

show "⟦i<j; j<length l⟧ ⟹ l!j ×l!i ∩ E⇧* = {}"
by (rule l_no_fwd)

qed

end

section ‹Main Correctness Proof›

context fr_graph
begin
lemma invar_from_cscc_invarI: "cscc_invar v0 D0 (L,PDPE) ⟹ invar v0 D0 PDPE"
unfolding cscc_invar_def invar_def
apply (simp split: prod.splits)
unfolding cscc_invar_loc_def by simp

lemma outer_invar_from_cscc_invarI:
"cscc_outer_invar it (L,D) ⟹outer_invar it D"
unfolding cscc_outer_invar_def outer_invar_def
apply (simp split: prod.splits)
unfolding cscc_outer_invar_loc_def by simp

text ‹With the extended invariant and the auxiliary lemmas, the actual
correctness proof is straightforward:›
theorem compute_SCC_correct: "compute_SCC ≤ compute_SCC_spec"
proof -
note [[goals_limit = 2]]
note [simp del] = Union_iff

show ?thesis
unfolding compute_SCC_def compute_SCC_spec_def select_edge_def select_def
apply (refine_rcg
WHILEIT_rule[where R="inv_image (abs_wf_rel v0) snd" for v0]
refine_vcg
)

apply (vc_solve
rec: cscc_invarI cscc_outer_invarI
solve: cscc_invar_preserve cscc_finI
intro: invar_from_cscc_invarI outer_invar_from_cscc_invarI
dest!: sym[of "pop A" for A]
simp: pE_fin'[OF invar_from_cscc_invarI] finite_V0
)
apply auto
done
qed

text ‹Simple proof, for presentation›
context
notes [refine]=refine_vcg
notes [[goals_limit = 1]]
begin
theorem "compute_SCC ≤ compute_SCC_spec"
unfolding compute_SCC_def compute_SCC_spec_def select_edge_def select_def
by (refine_rcg
WHILEIT_rule[where R="inv_image (abs_wf_rel v0) snd" for v0])
(vc_solve
rec: cscc_invarI cscc_outer_invarI solve: cscc_invar_preserve cscc_finI
intro: invar_from_cscc_invarI outer_invar_from_cscc_invarI
dest!: sym[of "pop A" for A]
simp: pE_fin'[OF invar_from_cscc_invarI] finite_V0, auto)
end

end

section ‹Refinement to Gabow's Data Structure›

context GS begin
definition "seg_set_impl l u ≡ do {
(_,res) ← WHILET
(λ(l,_). l<u)
(λ(l,res). do {
ASSERT (l<length S);
let x = S!l;
ASSERT (x∉res);
RETURN (Suc l,insert x res)
})
(l,{});

RETURN res
}"

lemma seg_set_impl_aux:
fixes l u
shows "⟦l<u; u≤length S; distinct S⟧ ⟹ seg_set_impl l u
≤ SPEC (λr. r = {S!j | j. l≤j ∧ j<u})"
unfolding seg_set_impl_def
apply (refine_rcg
WHILET_rule[where
I="λ(l',res). res = {S!j | j. l≤j ∧ j<l'} ∧ l≤l' ∧ l'≤u"
and R="measure (λ(l',_). u-l')"
]
refine_vcg)

apply (auto simp: less_Suc_eq nth_eq_iff_index_eq)
done

lemma (in GS_invar) seg_set_impl_correct:
assumes "i<length B"
shows "seg_set_impl (seg_start i) (seg_end i) ≤ SPEC (λr. r=p_α!i)"
apply (refine_rcg order_trans[OF seg_set_impl_aux] refine_vcg)

using assms
apply (simp_all add: seg_start_less_end seg_end_bound S_distinct) [3]

apply (auto simp: p_α_def assms seg_def) []
done

definition "last_seg_impl
≡ do {
ASSERT (length B - 1 < length B);
seg_set_impl (seg_start (length B - 1)) (seg_end (length B - 1))
}"

lemma (in GS_invar) last_seg_impl_correct:
assumes "p_α ≠ []"
shows "last_seg_impl ≤ SPEC (λr. r=last p_α)"
unfolding last_seg_impl_def
apply (refine_rcg order_trans[OF seg_set_impl_correct] refine_vcg)
using assms apply (auto simp add: p_α_def last_conv_nth)
done

end

context fr_graph
begin

definition "last_seg_impl s ≡ GS.last_seg_impl s"
lemmas last_seg_impl_def_opt =
last_seg_impl_def[abs_def, THEN opt_GSdef,
unfolded GS.last_seg_impl_def GS.seg_set_impl_def
GS.seg_start_def GS.seg_end_def GS_sel_simps]
(* TODO: Some potential for optimization here: the assertion
guarantees that length B - 1 + 1 = length B !*)

lemma last_seg_impl_refine:
assumes A: "(s,(p,D,pE))∈GS_rel"
assumes NE: "p≠[]"
shows "last_seg_impl s ≤ ⇓Id (RETURN (last p))"
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 last_seg_impl_def[abs_def]
apply (rule order_trans[OF GS_invar.last_seg_impl_correct])
using INV NE
apply (simp_all)
done
qed

definition compute_SCC_impl :: "'v set list nres" where
"compute_SCC_impl ≡ do {
stat_start_nres;
let so = ([],Map.empty);
(l,D) ← FOREACHi (λit (l,s). cscc_outer_invar it (l,oGS_α s))
V0 (λv0 (l,I0). do {
if ¬is_done_oimpl v0 I0 then do {
let ls = (l,initial_impl v0 I0);

(l,(S,B,I,P))←WHILEIT (λ(l,s). cscc_invar v0 (oGS_α I0) (l,GS.α s))
(λ(l,s). ¬path_is_empty_impl s) (λ(l,s).
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 {
s←collapse_impl v s;
RETURN (l,s)
} else if ¬is_done_impl v s then do {
― ‹Edge to new node. Append to path›
RETURN (l,push_impl v s)
} else do {
― ‹Edge to done node. Skip›
RETURN (l,s)
}
}
| None ⇒ do {
― ‹No more outgoing edges from current node on path›
scc ← last_seg_impl s;
s←pop_impl s;
let l = scc#l;
RETURN (l,s)
}
}) (ls);
RETURN (l,I)
} else RETURN (l,I0)
}) so;
stat_stop_nres;
RETURN l
}"

lemma compute_SCC_impl_refine: "compute_SCC_impl ≤ ⇓Id compute_SCC"
proof -
note [refine2] = bind_Let_refine2[OF last_seg_impl_refine]

have [refine2]: "⋀s' p D pE l' l v' v. ⟦
(s',(p,D,pE))∈GS_rel;
(l',l)∈Id;
(v',v)∈Id;
v∈⋃(set p)
⟧ ⟹ do { s'←collapse_impl v' s'; RETURN (l',s') }
≤ ⇓(Id ×⇩r GS_rel) (RETURN (l,collapse v (p,D,pE)))"
apply (refine_rcg order_trans[OF collapse_refine] refine_vcg)
apply assumption+
apply (auto simp add: pw_le_iff refine_pw_simps)
done

note [[goals_limit = 1]]
show ?thesis
unfolding compute_SCC_impl_def compute_SCC_def
apply (refine_rcg
bind_refine'
select_edge_refine push_refine
pop_refine
(*collapse_refine*)
initial_refine
oinitial_refine
(*last_seg_impl_refine*)
prod_relI IdI
inj_on_id
)

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
qed

end

end


# Theory Find_Path

section ‹Safety-Property Model-Checker\label{sec:find_path}›
theory Find_Path
imports
CAVA_Automata.Digraph
CAVA_Base.CAVA_Code_Target
begin
section ‹Finding Path to Error›
text ‹
This function searches a graph and a set of start nodes for a reachable
node that satisfies some property, and returns a path to such a node iff it
exists.
›
definition "find_path E U0 P ≡ do {
ASSERT (finite U0);
ASSERT (finite (E⇧*U0));
SPEC (λp. case p of
Some (p,v) ⇒ ∃u0∈U0. path E u0 p v ∧ P v ∧ (∀v∈set p. ¬ P v)
| None ⇒ ∀u0∈U0. ∀v∈E⇧*{u0}. ¬P v)
}"

lemma find_path_ex_rule:
assumes "finite U0"
assumes "finite (E⇧*U0)"
assumes "∃v∈E⇧*U0. P v"
shows "find_path E U0 P ≤ SPEC (λr.
∃p v. r = Some (p,v) ∧ P v ∧ (∀v∈set p. ¬P v) ∧ (∃u0∈U0. path E u0 p v))"
unfolding find_path_def
using assms
by (fastforce split: option.splits)

subsection ‹Nontrivial Paths›

definition "find_path1 E u0 P ≡ do {
ASSERT (finite (E⇧*{u0}));
SPEC (λp. case p of
Some (p,v) ⇒ path E u0 p v ∧ P v ∧ p≠[]
| None ⇒ ∀v∈E⇧+{u0}. ¬P v)}"

lemma (in -) find_path1_ex_rule:
assumes "finite (E⇧*{u0})"
assumes "∃v∈E⇧+{u0}. P v"
shows "find_path1 E u0 P ≤ SPEC (λr.
∃p v. r = Some (p,v) ∧ p≠[] ∧ P v ∧ path E u0 p v)"
unfolding find_path1_def
using assms
by (fastforce split: option.splits)

end


# Theory Gabow_GBG

section ‹Lasso Finding Algorithm for Generalized B\"uchi Graphs \label{sec:gbg}›
theory Gabow_GBG
imports
Gabow_Skeleton
CAVA_Automata.Lasso
Find_Path
begin

(* TODO: convenience locale, consider merging this with invariants *)
locale igb_fr_graph =
igb_graph G + fr_graph G
for G :: "('Q,'more) igb_graph_rec_scheme"

lemma igb_fr_graphI:
assumes "igb_graph G"
assumes "finite ((g_E G)⇧*  g_V0 G)"
shows "igb_fr_graph G"
proof -
interpret igb_graph G by fact
show ?thesis using assms(2) by unfold_locales
qed

text ‹
We implement an algorithm that computes witnesses for the
non-emptiness of Generalized B\"uchi Graphs (GBG).
›

section ‹Specification›
context igb_graph
begin
definition ce_correct
― ‹Specifies a correct counter-example›
where
"ce_correct Vr Vl ≡ (∃pr pl.
Vr ⊆ E⇧*V0 ∧ Vl ⊆ E⇧*V0 ― ‹Only reachable nodes are covered›
∧ set pr⊆Vr ∧ set pl⊆Vl     ― ‹The paths are inside the specified sets›
∧ Vl×Vl ⊆ (E ∩ Vl×Vl)⇧*      ― ‹‹Vl› is mutually connected›
∧ Vl×Vl ∩ E ≠ {}            ― ‹‹Vl› is non-trivial›
∧ is_lasso_prpl (pr,pl))    ― ‹Paths form a lasso›
"

definition find_ce_spec :: "('Q set × 'Q set) option nres" where
"find_ce_spec ≡ SPEC (λr. case r of
None ⇒ (∀prpl. ¬is_lasso_prpl prpl)
| Some (Vr,Vl) ⇒ ce_correct Vr Vl
)"

definition find_lasso_spec :: "('Q list × 'Q list) option nres" where
"find_lasso_spec ≡ SPEC (λr. case r of
None ⇒ (∀prpl. ¬is_lasso_prpl prpl)
| Some prpl ⇒ is_lasso_prpl prpl
)"

end

section ‹Invariant Extension›

text ‹Extension of the outer invariant:›
context igb_fr_graph
begin
definition no_acc_over
― ‹Specifies that there is no accepting cycle touching a set of nodes›
where
"no_acc_over D ≡ ¬(∃v∈D. ∃pl. pl≠[] ∧ path E v pl v ∧
(∀i<num_acc. ∃q∈set pl. i∈acc q))"

definition "fgl_outer_invar_ext ≡ λit (brk,D).
case brk of None ⇒ no_acc_over D | Some (Vr,Vl) ⇒ ce_correct Vr Vl"

definition "fgl_outer_invar ≡ λit (brk,D). case brk of
None ⇒ outer_invar it D ∧ no_acc_over D
| Some (Vr,Vl) ⇒ ce_correct Vr Vl"

end

text ‹Extension of the inner invariant:›
locale fgl_invar_loc =
invar_loc G v0 D0 p D pE
+ igb_graph G
for G :: "('Q, 'more) igb_graph_rec_scheme"
and v0 D0 and brk :: "('Q set × 'Q set) option" and p D pE +
assumes no_acc: "brk=None ⟹ ¬(∃v pl. pl≠[] ∧ path lvE v pl v ∧
(∀i<num_acc. ∃q∈set pl. i∈acc q))" ― ‹No accepting cycle over
visited edges›
assumes acc: "brk=Some (Vr,Vl) ⟹ ce_correct Vr Vl"
begin
lemma locale_this: "fgl_invar_loc G v0 D0 brk p D pE"
by unfold_locales
lemma invar_loc_this: "invar_loc G v0 D0 p D pE" by unfold_locales
lemma eas_gba_graph_this: "igb_graph G" by unfold_locales
end

definition (in igb_graph) "fgl_invar v0 D0 ≡
λ(brk, p, D, pE). fgl_invar_loc G v0 D0 brk p D pE"

section ‹Definition of the Lasso-Finding Algorithm›

context igb_fr_graph
begin
definition find_ce :: "('Q set × 'Q set) option nres" where
"find_ce ≡ do {
let D = {};
(brk,_)←FOREACHci fgl_outer_invar V0
(λ(brk,_). brk=None)
(λv0 (brk,D0). do {
if v0∉D0 then do {
let s = (None,initial v0 D0);

(brk,p,D,pE) ← WHILEIT (fgl_invar v0 D0)
(λ(brk,p,D,pE). brk=None ∧ 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 {
if v ∈ ⋃(set p) then do {
― ‹Collapse›
let (p,D,pE) = collapse v (p,D,pE);

ASSERT (p≠[]);

if ∀i<num_acc. ∃q∈last p. i∈acc q then
RETURN (Some (⋃(set (butlast p)),last p),p,D,pE)
else
RETURN (None,p,D,pE)
} else if v∉D then do {
― ‹Edge to new node. Append to path›
RETURN (None,push v (p,D,pE))
} else RETURN (None,p,D,pE)
}
| None ⇒ do {
― ‹No more outgoing edges from current node on path›
ASSERT (pE ∩ last p × UNIV = {});
RETURN (None,pop (p,D,pE))
}
}) s;
ASSERT (brk=None ⟶ (p=[] ∧ pE={}));
RETURN (brk,D)
} else
RETURN (brk,D0)
}) (None,D);
RETURN brk
}"
end

section ‹Invariant Preservation›

context igb_fr_graph
begin

definition "fgl_invar_part ≡ λ(brk, p, D, pE).
fgl_invar_loc_axioms G brk p D pE"

lemma fgl_outer_invarI[intro?]:
"⟦
brk=None ⟹ outer_invar it D;
⟦brk=None ⟹ outer_invar it D⟧ ⟹ fgl_outer_invar_ext it (brk,D)⟧
⟹ fgl_outer_invar it (brk,D)"
unfolding outer_invar_def fgl_outer_invar_ext_def fgl_outer_invar_def
apply (auto split: prod.splits option.splits)
done

lemma fgl_invarI[intro?]:
"⟦ invar v0 D0 PDPE;
invar v0 D0 PDPE ⟹ fgl_invar_part (B,PDPE)⟧
⟹ fgl_invar v0 D0 (B,PDPE)"
unfolding invar_def fgl_invar_part_def fgl_invar_def
apply (simp split: prod.split_asm)
apply intro_locales
apply assumption
done

lemma fgl_invar_initial:
assumes OINV: "fgl_outer_invar it (None,D0)"
assumes A: "v0∈it" "v0∉D0"
shows "fgl_invar_part (None, initial v0 D0)"
proof -
from OINV interpret outer_invar_loc G it D0

from OINV have no_acc: "no_acc_over D0"

{
fix v pl

assume "pl ≠ []" and P: "path (vE [{v0}] D0 (E ∩ {v0} × UNIV)) v pl v"
hence 1: "v∈D0"
by (cases pl) (auto simp: path_cons_conv vE_def touched_def)
have 2: "path E v pl v" using path_mono[OF vE_ss_E P] .
note 1 2
} note AUX1=this

show ?thesis
unfolding fgl_invar_part_def
apply (simp split: prod.splits add: initial_def)
apply unfold_locales
using ‹v0∉D0›
using AUX1 no_acc unfolding no_acc_over_def apply blast
by simp
qed

lemma fgl_invar_pop:
assumes INV: "fgl_invar v0 D0 (None,p,D,pE)"
assumes INV': "invar v0 D0 (pop (p,D,pE))"
assumes NE[simp]: "p≠[]"
assumes NO': "pE ∩ last p × UNIV = {}"
shows "fgl_invar_part (None, pop (p,D,pE))"
proof -
from INV interpret fgl_invar_loc G v0 D0 None p D pE

show ?thesis
apply (unfold fgl_invar_part_def pop_def)
apply (simp split: prod.splits)
apply unfold_locales
unfolding vE_pop[OF NE]

using no_acc apply auto []
apply simp
done
qed

lemma fgl_invar_collapse_ce_aux:
assumes INV: "invar v0 D0 (p, D, pE)"
assumes NE[simp]: "p≠[]"
assumes NONTRIV: "vE p D pE ∩ (last p × last p) ≠ {}"
assumes ACC: "∀i<num_acc. ∃q∈last p. i∈acc q"
shows "fgl_invar_part (Some (⋃(set (butlast p)), last p), p, D, pE)"
proof -
from INV interpret invar_loc G v0 D0 p D pE by (simp add: invar_def)
txt ‹The last collapsed node on the path contains states from all
accepting sets.
As it is strongly connected and reachable, we get a counter-example.
Here, we explicitely construct the lasso.›

let ?Er = "E ∩ (⋃(set (butlast p)) × UNIV)"

txt ‹We choose a node in the last Cnode, that is reachable only using
former Cnodes.›

obtain w where "(v0,w)∈?Er⇧*" "w∈last p"
proof cases
assume "length p = 1"
hence "v0∈last p"
using root_v0
by (cases p) auto
thus thesis by (auto intro: that)
next
assume "length p≠1"
hence "length p > 1" by (cases p) auto
hence "Suc (length p - 2) < length p" by auto
from p_connected'[OF this] obtain u v where
UIP: "u∈p!(length p - 2)" and VIP: "v∈p!(length p - 1)" and "(u,v)∈lvE"
using ‹length p > 1› by auto
from root_v0 have V0IP: "v0∈p!0" by (cases p) auto

from VIP have "v∈last p" by (cases p rule: rev_cases) auto

from pathI[OF V0IP UIP] ‹length p > 1› have
"(v0,u)∈(lvE ∩ ⋃(set (butlast p)) × ⋃(set (butlast p)))⇧*"
(is "_ ∈ …⇧*")
also have "… ⊆ ?Er" using lvE_ss_E by auto
finally (rtrancl_mono_mp[rotated]) have "(v0,u)∈?Er⇧*" .
also note ‹(u,v)∈lvE› UIP hence "(u,v)∈?Er" using lvE_ss_E ‹length p > 1›
apply (auto simp: Bex_def in_set_conv_nth)
by (metis One_nat_def Suc_lessE ‹Suc (length p - 2) < length p›
diff_Suc_1 length_butlast nth_butlast)
finally show ?thesis by (rule that) fact
qed
then obtain "pr" where
P_REACH: "path E v0 pr w" and
R_SS: "set pr ⊆ ⋃(set (butlast p))"
apply -
apply (erule rtrancl_is_path)
apply (frule path_nodes_edges)
apply (auto
dest!: order_trans[OF _ image_Int_subset]
dest: path_mono[of _ E, rotated])
done

have [simp]: "last p = p!(length p - 1)" by (cases p rule: rev_cases) auto

txt ‹From that node, we construct a lasso by inductively appending a path
for each accepting set›
{
fix na
assume na_def: "na = num_acc"

have "∃pl. pl≠[]
∧ path (lvE ∩ last p×last p) w pl w
∧ (∀i<num_acc. ∃q∈set pl. i∈acc q)"
using ACC
unfolding na_def[symmetric]
proof (induction na)
case 0

from NONTRIV obtain u v
where "(u,v)∈lvE ∩ last p × last p" "u∈last p" "v∈last p"
by auto
from cnode_connectedI ‹w∈last p› ‹u∈last p›
have "(w,u)∈(lvE ∩ last p × last p)⇧*"
by auto
also note ‹(u,v)∈lvE ∩ last p × last p›
also (rtrancl_into_trancl1) from cnode_connectedI ‹v∈last p› ‹w∈last p›
have "(v,w)∈(lvE ∩ last p × last p)⇧*"
by auto
finally obtain pl where "pl≠[]" "path (lvE ∩ last p × last p) w pl w"
by (rule trancl_is_path)
thus ?case by auto
next
case (Suc n)
from Suc.prems have "∀i<n. ∃q∈last p. i∈acc q" by auto
with Suc.IH obtain pl where IH:
"pl≠[]"
"path (lvE ∩ last p × last p) w pl w"
"∀i<n. ∃q∈set pl. i∈acc q"
by blast

from Suc.prems obtain v where "v∈last p" and "n∈acc v" by auto
from cnode_connectedI ‹w∈last p› ‹v∈last p›
have "(w,v)∈(lvE ∩ last p × last p)⇧*" by auto
then obtain pl1 where P1: "path (lvE ∩ last p × last p) w pl1 v"
by (rule rtrancl_is_path)
also from cnode_connectedI ‹w∈last p› ‹v∈last p›
have "(v,w)∈(lvE ∩ last p × last p)⇧*" by auto
then obtain pl2 where P2: "path (lvE ∩ last p × last p) v pl2 w"
by (rule rtrancl_is_path)
also (path_conc) note IH(2)
finally (path_conc) have
P: "path (lvE ∩ last p × last p) w (pl1@pl2@pl) w"
by simp
moreover from IH(1) have "pl1@pl2@pl ≠ []" by simp
moreover have "∀i'<n. ∃q∈set (pl1@pl2@pl). i'∈acc q" using IH(3) by auto
moreover have "v∈set (pl1@pl2@pl)" using P1 P2 P IH(1)
apply (cases pl2, simp_all add: path_cons_conv path_conc_conv)
apply (cases pl, simp_all add: path_cons_conv)
apply (cases pl1, simp_all add: path_cons_conv)
done
with ‹n∈acc v› have "∃q∈set (pl1@pl2@pl). n∈acc q" by auto
ultimately show ?case
apply (intro exI conjI)
apply assumption+
apply (auto elim: less_SucE)
done
qed
}
then obtain pl where pl: "pl≠[]" "path (lvE ∩ last p×last p) w pl w"
"∀i<num_acc. ∃q∈set pl. i∈acc q" by blast
hence "path E w pl w" and L_SS: "set pl ⊆ last p"
apply -
apply (frule path_mono[of _ E, rotated])
using lvE_ss_E
apply auto [2]

apply (drule path_nodes_edges)
apply (drule order_trans[OF _ image_Int_subset])
apply auto []
done

have LASSO: "is_lasso_prpl (pr,pl)"
unfolding is_lasso_prpl_def is_lasso_prpl_pre_def
using ‹path E w pl w› P_REACH pl by auto

from p_sc have "last p × last p ⊆ (lvE ∩ last p × last p)⇧*" by auto
with lvE_ss_E have VL_CLOSED: "last p × last p ⊆ (E ∩ last p × last p)⇧*"
apply (erule_tac order_trans)
apply (rule rtrancl_mono)
by blast

have NONTRIV': "last p × last p ∩ E ≠ {}"
by (metis Int_commute NONTRIV disjoint_mono lvE_ss_E subset_refl)

from order_trans[OF path_touched touched_reachable]
have LP_REACH: "last p ⊆ E⇧*V0"
and BLP_REACH: "⋃(set (butlast p)) ⊆ E⇧*V0"
apply -
apply (cases p rule: rev_cases)
apply simp
apply auto []

apply (cases p rule: rev_cases)
apply simp
apply auto []
done

show ?thesis
apply unfold_locales
apply simp

using LASSO R_SS L_SS VL_CLOSED NONTRIV' LP_REACH BLP_REACH
unfolding ce_correct_def
apply simp
apply blast
done

qed

lemma fgl_invar_collapse_ce:
fixes u v
assumes INV: "fgl_invar v0 D0 (None,p,D,pE)"
defines "pE' ≡ pE - {(u,v)}"
assumes CFMT: "(p',D',pE'') = collapse v (p,D,pE')"
assumes INV': "invar v0 D0 (p',D',pE'')"
assumes NE[simp]: "p≠[]"
assumes E: "(u,v)∈pE" and "u∈last p"
assumes BACK: "v∈⋃(set p)"
assumes ACC: "∀i<num_acc. ∃q∈last p'. i∈acc q"
defines i_def: "i ≡ idx_of p v"
shows "fgl_invar_part (
Some (⋃(set (butlast p')), last p'),
collapse v (p,D,pE'))"
proof -

from CFMT have p'_def: "p' = collapse_aux p i" and [simp]: "D'=D" "pE''=pE'"

from INV interpret fgl_invar_loc G v0 D0 None p D pE

from idx_of_props[OF BACK] have "i<length p" and "v∈p!i"

have "u∈last p'"
using ‹u∈last p› ‹i<length p›
unfolding p'_def collapse_aux_def
by (metis Misc.last_in_set drop_eq_Nil last_drop not_le)
moreover have "v∈last p'"
using ‹v∈p!i› ‹i<length p›
unfolding p'_def collapse_aux_def
by (metis UnionI append_Nil Cons_nth_drop_Suc in_set_conv_decomp last_snoc)
ultimately have "vE p' D pE' ∩ last p' × last p' ≠ {}"
unfolding p'_def pE'_def by (auto simp: E)

have "p'≠[]" by (simp add: p'_def collapse_aux_def)

have [simp]: "collapse v (p,D,pE') = (p',D,pE')"
unfolding collapse_def p'_def i_def
by simp

show ?thesis
apply simp
apply (rule fgl_invar_collapse_ce_aux)
using INV' apply simp
apply fact+
done
qed

lemma fgl_invar_collapse_nce:
fixes u v
assumes INV: "fgl_invar v0 D0 (None,p,D,pE)"
defines "pE' ≡ pE - {(u,v)}"
assumes CFMT: "(p',D',pE'') = collapse v (p,D,pE')"
assumes INV': "invar v0 D0 (p',D',pE'')"
assumes NE[simp]: "p≠[]"
assumes E: "(u,v)∈pE" and "u∈last p"
assumes BACK: "v∈⋃(set p)"
assumes NACC: "j<num_acc" "∀q∈last p'. j∉acc q"
defines "i ≡ idx_of p v"
shows "fgl_invar_part (None, collapse v (p,D,pE'))"
proof -
from CFMT have p'_def: "p' = collapse_aux p i" and [simp]: "D'=D" "pE''=pE'"

have [simp]: "collapse v (p,D,pE') = (p',D,pE')"
by (simp add: collapse_def p'_def i_def)

from INV interpret fgl_invar_loc G v0 D0 None p D pE

from INV' interpret inv': invar_loc G v0 D0 p' D pE' by (simp add: invar_def)

define vE' where "vE' = vE p' D pE'"

have vE'_alt: "vE' = insert (u,v) lvE"
by (simp add: vE'_def p'_def pE'_def E)

from idx_of_props[OF BACK] have "i<length p" and "v∈p!i"

have "u∈last p'"
using ‹u∈last p› ‹i<length p›
unfolding p'_def collapse_aux_def
by (metis Misc.last_in_set drop_eq_Nil last_drop leD)
moreover have "v∈last p'"
using ‹v∈p!i› ‹i<length p›
unfolding p'_def collapse_aux_def
by (metis UnionI append_Nil Cons_nth_drop_Suc in_set_conv_decomp last_snoc)
ultimately have "vE' ∩ last p' × last p' ≠ {}"
unfolding vE'_alt by (auto)

have "p'≠[]" by (simp add: p'_def collapse_aux_def)

{
txt ‹
We show that no visited strongly connected component contains states
from all acceptance sets.›
fix w pl
txt ‹For this, we chose a non-trivial loop inside the visited edges›
assume P: "path vE' w pl w" and NT: "pl≠[]"
txt ‹And show that there is one acceptance set disjoint with the nodes
of the loop›
have "∃i<num_acc. ∀q∈set pl. i∉acc q"
proof cases
assume "set pl ∩ last p' = {}"
― ‹Case: The loop is outside the last Cnode›
with path_restrict[OF P] ‹u∈last p'› ‹v∈last p'› have "path lvE w pl w"
apply -
apply (drule path_mono[of _ lvE, rotated])
unfolding vE'_alt
by auto
with no_acc NT show ?thesis by auto
next
assume "set pl ∩ last p' ≠ {}"
― ‹Case: The loop touches the last Cnode›
txt ‹Then, the loop must be completely inside the last CNode›
from inv'.loop_in_lastnode[folded vE'_def, OF P ‹p'≠[]› this]
have "w∈last p'" "set pl ⊆ last p'" .
with NACC show ?thesis by blast
qed
} note AUX_no_acc = this

show ?thesis
apply unfold_locales
using AUX_no_acc[unfolded vE'_def] apply auto []

apply simp
done
qed

lemma collapse_ne: "([],D',pE') ≠ collapse v (p,D,pE)"
by (simp add: collapse_def collapse_aux_def Let_def)

lemma fgl_invar_push:
assumes INV: "fgl_invar v0 D0 (None,p,D,pE)"
assumes BRK[simp]: "brk=None"
assumes NE[simp]: "p≠[]"
assumes E: "(u,v)∈pE" and UIL: "u∈last p"
assumes VNE: "v∉⋃(set p)" "v∉D"
assumes INV': "invar v0 D0 (push v (p,D,pE - {(u,v)}))"
shows "fgl_invar_part (None, push v (p,D,pE - {(u,v)}))"
proof -
from INV interpret fgl_invar_loc G v0 D0 None p D pE

define pE' where "pE' = (pE - {(u,v)} ∪ E∩{v}×UNIV)"

have [simp]: "push v (p,D,pE - {(u,v)}) = (p@[{v}],D,pE')"

from INV' interpret inv': invar_loc G v0 D0 "(p@[{v}])" D "pE'"

note defs_fold = vE_push[OF E UIL VNE, folded pE'_def]

{
txt ‹We show that there still is no loop that contains all accepting
nodes. For this, we choose some loop.›
fix w pl
assume P: "path (insert (u,v) lvE) w pl w" and [simp]: "pl≠[]"
have "∃i<num_acc. ∀q∈set pl. i∉acc q"
proof cases
assume "v∈set pl" ― ‹Case: The newly pushed last cnode is on the loop›
txt ‹Then the loop is entirely on the last cnode›
with inv'.loop_in_lastnode[unfolded defs_fold, OF P]
have [simp]: "w=v" and SPL: "set pl = {v}" by auto
txt ‹However, we then either have that the last cnode is contained in
the last but one cnode, or that there is a visited edge inside the
last cnode.›
from P SPL have "u=v ∨ (v,v)∈lvE"
apply (cases pl) apply (auto simp: path_cons_conv)
apply (case_tac list)
apply (auto simp: path_cons_conv)
done
hence False proof
assume "u=v" ― ‹This is impossible, as @{text "u"} was on the
original path, but @{text "v"} was not›
with UIL VNE show False by auto
next
assume "(v,v)∈lvE" ― ‹This is impossible, as all visited edges are
from touched nodes, but @{text "v"} was untouched›
with vE_touched VNE show False unfolding touched_def by auto
qed
thus ?thesis ..
next
assume A: "v∉set pl"
― ‹Case: The newly pushed last cnode is not on the loop›
txt ‹Then, the path lays inside the old visited edges›
have "path lvE w pl w"
proof -
have "w∈set pl" using P by (cases pl) (auto simp: path_cons_conv)
with A show ?thesis using path_restrict[OF P]
apply -
apply (drule path_mono[of _ lvE, rotated])
apply (cases pl, auto) []

apply assumption
done
qed
txt ‹And thus, the proposition follows from the invariant on the old
state›
with no_acc show ?thesis
apply simp
using ‹pl≠[]›
by blast
qed
} note AUX_no_acc = this

show ?thesis
unfolding fgl_invar_part_def
apply simp
apply unfold_locales
unfolding defs_fold

using AUX_no_acc apply auto []

apply simp
done
qed

lemma fgl_invar_skip:
assumes INV: "fgl_invar v0 D0 (None,p,D,pE)"
assumes BRK[simp]: "brk=None"
assumes NE[simp]: "p≠[]"
assumes E: "(u,v)∈pE" and UIL: "u∈last p"
assumes VID: "v∈D"
assumes INV': "invar v0 D0 (p, D, (pE - {(u,v)}))"
shows "fgl_invar_part (None, p, D, (pE - {(u,v)}))"
proof -
from INV interpret fgl_invar_loc G v0 D0 None p D pE
from INV' interpret inv': invar_loc G v0 D0 p D "(pE - {(u,v)})"

{
txt ‹We show that there still is no loop that contains all accepting
nodes. For this, we choose some loop.›
fix w pl
assume P: "path (insert (u,v) lvE) w pl w" and [simp]: "pl≠[]"
from P have "∃i<num_acc. ∀q∈set pl. i∉acc q"
proof (cases rule: path_edge_rev_cases)
case no_use ― ‹Case: The loop does not use the new edge›
txt ‹The proposition follows from the invariant for the old state›
with no_acc show ?thesis
apply simp
using ‹pl≠[]›
by blast
next
case (split p1 p2) ― ‹Case: The loop uses the new edge›
txt ‹As done is closed under transitions, the nodes of the edge have
from split(2) D_closed_vE_rtrancl
have WID: "w∈D"
using VID by (auto dest!: path_is_rtrancl)
from split(1) WID D_closed_vE_rtrancl have "u∈D"
apply (cases rule: path_edge_cases)
apply (auto dest!: path_is_rtrancl)
done
txt ‹Which is a contradition to the assumptions›
with UIL p_not_D have False by (cases p rule: rev_cases) auto
thus ?thesis ..
qed
} note AUX_no_acc = this

show ?thesis
apply unfold_locales
unfolding vE_remove[OF NE E]

using AUX_no_acc apply auto []

apply simp
done

qed

lemma fgl_outer_invar_initial:
"outer_invar V0 {} ⟹ fgl_outer_invar_ext V0 (None, {})"
unfolding fgl_outer_invar_ext_def
done

lemma fgl_outer_invar_brk:
assumes INV: "fgl_invar v0 D0 (Some (Vr,Vl),p,D,pE)"
shows "fgl_outer_invar_ext anyIt (Some (Vr,Vl), anyD)"
proof -
from INV interpret fgl_invar_loc G v0 D0 "Some (Vr,Vl)" p D pE

from acc show ?thesis by (simp add: fgl_outer_invar_ext_def)
qed

lemma fgl_outer_invar_newnode_nobrk:
assumes A: "v0∉D0" "v0∈it"
assumes OINV: "fgl_outer_invar it (None,D0)"
assumes INV: "fgl_invar v0 D0 (None,[],D',pE)"
shows "fgl_outer_invar_ext (it-{v0}) (None,D')"
proof -
from OINV interpret outer_invar_loc G it D0
unfolding fgl_outer_invar_def outer_invar_def by simp

from INV interpret inv: fgl_invar_loc G v0 D0 None "[]" D' pE
unfolding fgl_invar_def by simp

from inv.pE_fin have [simp]: "pE = {}" by simp

{ fix v pl
assume A: "v∈D'" "path E v pl v"
have "path (E ∩ D' × UNIV) v pl v"
apply (rule path_mono[OF _ path_restrict_closed[OF inv.D_closed A]])
by auto
} note AUX1=this

show ?thesis
unfolding fgl_outer_invar_ext_def
apply simp
using inv.no_acc AUX1
apply (auto simp add: vE_def touched_def no_acc_over_def) []
done
qed

lemma fgl_outer_invar_newnode:
assumes A: "v0∉D0" "v0∈it"
assumes OINV: "fgl_outer_invar it (None,D0)"
assumes INV: "fgl_invar v0 D0 (brk,p,D',pE)"
assumes CASES: "(∃Vr Vl. brk = Some (Vr, Vl)) ∨ p = []"
shows "fgl_outer_invar_ext (it-{v0}) (brk,D')"
using CASES
apply (elim disjE1)
using fgl_outer_invar_brk[of v0 D0 _ _ p D' pE] INV
apply -
apply (auto, assumption) []
using fgl_outer_invar_newnode_nobrk[OF A] OINV INV apply auto []
done

lemma fgl_outer_invar_Dnode:
assumes "fgl_outer_invar it (None, D)" "v∈D"
shows "fgl_outer_invar_ext (it - {v}) (None, D)"
using assms
by (auto simp: fgl_outer_invar_def fgl_outer_invar_ext_def)

lemma fgl_fin_no_lasso:
assumes A: "fgl_outer_invar {} (None, D)"
assumes B: "is_lasso_prpl prpl"
shows "False"
proof -
obtain "pr" pl where [simp]: "prpl = (pr,pl)" by (cases prpl)
from A have NA: "no_acc_over D"

from A have "outer_invar {} D" by (simp add: fgl_outer_invar_def)
with fin_outer_D_is_reachable have [simp]: "D=E⇧*V0" by simp

from NA B show False
apply (simp add: no_acc_over_def is_lasso_prpl_def is_lasso_prpl_pre_def)
apply clarsimp
apply (blast dest: path_is_rtrancl)
done
qed

lemma fgl_fin_lasso:
assumes A: "fgl_outer_invar it (Some (Vr,Vl), D)"
shows "ce_correct Vr Vl"
using A by (simp add: fgl_outer_invar_def fgl_outer_invar_ext_def)

lemmas fgl_invar_preserve =
fgl_invar_initial fgl_invar_push fgl_invar_pop
fgl_invar_collapse_ce fgl_invar_collapse_nce fgl_invar_skip
fgl_outer_invar_newnode fgl_outer_invar_Dnode
invar_initial outer_invar_initial fgl_invar_initial fgl_outer_invar_initial
fgl_fin_no_lasso fgl_fin_lasso

end

section ‹Main Correctness Proof›

context igb_fr_graph
begin
lemma outer_invar_from_fgl_invarI:
"fgl_outer_invar it (None,D) ⟹ outer_invar it D"
unfolding fgl_outer_invar_def outer_invar_def
by (simp split: prod.splits)

lemma invar_from_fgl_invarI: "fgl_invar v0 D0 (B,PDPE) ⟹ invar v0 D0 PDPE"
unfolding fgl_invar_def invar_def
apply (simp split: prod.splits)
unfolding fgl_invar_loc_def by simp

theorem find_ce_correct: "find_ce ≤ find_ce_spec"
proof -
note [simp del] = Union_iff

show ?thesis
unfolding find_ce_def find_ce_spec_def select_edge_def select_def
apply (refine_rcg
WHILEIT_rule[where R="inv_image (abs_wf_rel v0) snd" for v0]
refine_vcg
)

using [[goals_limit = 5]]

apply (vc_solve
rec: fgl_invarI fgl_outer_invarI
intro: invar_from_fgl_invarI outer_invar_from_fgl_invarI
dest!: sym[of "collapse a b" for a b]
simp: collapse_ne
simp: pE_fin'[OF invar_from_fgl_invarI] finite_V0
solve: invar_preserve
solve: asm_rl[of "_ ∩ _ = {}"]
solve: fgl_invar_preserve)
done
qed
end

section "Emptiness Check"
text ‹Using the lasso-finding algorithm, we can define an emptiness check›

context igb_fr_graph
begin
definition "abs_is_empty ≡ do {
ce ← find_ce;
RETURN (ce = None)
}"

theorem abs_is_empty_correct:
"abs_is_empty ≤ SPEC (λres. res ⟷ (∀r. ¬is_acc_run r))"
unfolding abs_is_empty_def
apply (refine_rcg refine_vcg
order_trans[OF find_ce_correct, unfolded find_ce_spec_def])
unfolding ce_correct_def
using lasso_accepted accepted_lasso
apply (clarsimp split: option.splits)
apply (metis is_lasso_prpl_of_lasso surj_pair)
by (metis is_lasso_prpl_conv)

definition "abs_is_empty_ce ≡ do {
ce ← find_ce;
case ce of
None ⇒ RETURN None
| Some (Vr,Vl) ⇒ do {
ASSERT (∃pr pl. set pr ⊆ Vr ∧ set pl ⊆ Vl ∧ Vl × Vl ⊆ (E ∩ Vl×Vl)⇧*
∧ is_lasso_prpl (pr,pl));
(pr,pl) ← SPEC (λ(pr,pl).
set pr ⊆ Vr
∧ set pl ⊆ Vl
∧ Vl × Vl ⊆ (E ∩ Vl×Vl)⇧*
∧ is_lasso_prpl (pr,pl));
RETURN (Some (pr,pl))
}
}"

theorem abs_is_empty_ce_correct: "abs_is_empty_ce ≤ SPEC (λres. case res of
None ⇒ (∀r. ¬is_acc_run r)
| Some (pr,pl) ⇒ is_acc_run (pr⌢pl⇧ω)
)"
unfolding abs_is_empty_ce_def
apply (refine_rcg refine_vcg
order_trans[OF find_ce_correct, unfolded find_ce_spec_def])

apply (clarsimp_all simp: ce_correct_def)

using accepted_lasso finite_reachableE_V0 apply (metis is_lasso_prpl_of_lasso surj_pair)
apply blast
done

end

section ‹Refinement›
text ‹
In this section, we refine the lasso finding algorithm to use efficient
data structures. First, we explicitely keep track of the set of acceptance
classes for every c-node on the path. Second, we use Gabow's data structure
to represent the path.
›

subsection ‹Addition of Explicit Accepting Sets›
text ‹In a first step, we explicitely keep track of the current set of
acceptance classes for every c-node on the path.›

type_synonym 'a abs_gstate = "nat set list × 'a abs_state"
type_synonym 'a ce = "('a set × 'a set) option"
type_synonym 'a abs_gostate = "'a ce × 'a set"

context igb_fr_graph
begin

definition gstate_invar :: "'Q abs_gstate ⇒ bool" where
"gstate_invar ≡ λ(a,p,D,pE). a = map (λV. ⋃(accV)) p"

definition "gstate_rel ≡ br snd gstate_invar"

lemma gstate_rel_sv[relator_props,simp,intro!]: "single_valued gstate_rel"

definition (in -) gcollapse_aux
:: "nat set list ⇒ 'a set list ⇒ nat ⇒ nat set list × 'a set list"
where "gcollapse_aux a p i ≡
(take i a @ [⋃(set (drop i a))],take i p @ [⋃(set (drop i p))])"

definition (in -) gcollapse :: "'a ⇒ 'a abs_gstate ⇒ 'a abs_gstate"
where "gcollapse v APDPE ≡
let
(a,p,D,pE)=APDPE;
i=idx_of p v;
(a,p) = gcollapse_aux a p i
in (a,p,D,pE)"

definition "gpush v s ≡
let
(a,s) = s
in
(a@[acc v],push v s)"

definition "gpop s ≡
let (a,s) = s in (butlast a,pop s)"

definition ginitial :: "'Q ⇒ 'Q abs_gostate ⇒ 'Q abs_gstate"
where "ginitial v0 s0 ≡ ([acc v0], initial v0 (snd s0))"

definition goinitial :: "'Q abs_gostate" where "goinitial ≡ (None,{})"
definition go_is_no_brk :: "'Q abs_gostate ⇒ bool"
where "go_is_no_brk s ≡ fst s = None"
definition goD :: "'Q abs_gostate ⇒ 'Q set" where "goD s ≡ snd s"
definition goBrk :: "'Q abs_gostate ⇒ 'Q ce" where "goBrk s ≡ fst s"
definition gto_outer :: "'Q ce ⇒ 'Q abs_gstate ⇒ 'Q abs_gostate"
where "gto_outer brk s ≡ let (A,p,D,pE)=s in (brk,D)"

definition "gselect_edge s ≡ do {
let (a,s)=s;
(r,s)←select_edge s;
RETURN (r,a,s)
}"

definition gfind_ce :: "('Q set × 'Q set) option nres" where
"gfind_ce ≡ do {
let os = goinitial;
os←FOREACHci fgl_outer_invar V0 (go_is_no_brk) (λv0 s0. do {
if v0∉goD s0 then do {
let s = (None,ginitial v0 s0);

(brk,a,p,D,pE) ← WHILEIT (λ(brk,a,s). fgl_invar v0 (goD s0) (brk,s))
(λ(brk,a,p,D,pE). brk=None ∧ p ≠ []) (λ(_,a,p,D,pE).
do {
― ‹Select edge from end of path›
(vo,(a,p,D,pE)) ← gselect_edge (a,p,D,pE);

ASSERT (p≠[]);
case vo of
Some v ⇒ do {
if v ∈ ⋃(set p) then do {
― ‹Collapse›
let (a,p,D,pE) = gcollapse v (a,p,D,pE);

ASSERT (p≠[]);
ASSERT (a≠[]);

if last a = {0..<num_acc} then
RETURN (Some (⋃(set (butlast p)),last p),a,p,D,pE)
else
RETURN (None,a,p,D,pE)
} else if v∉D then do {
― ‹Edge to new node. Append to path›
RETURN (None,gpush v (a,p,D,pE))
} else RETURN (None,a,p,D,pE)
}
| None ⇒ do {
― ‹No more outgoing edges from current node on path›
ASSERT (pE ∩ last p × UNIV = {});
RETURN (None,gpop (a,p,D,pE))
}
}) s;
ASSERT (brk=None ⟶ (p=[] ∧ pE={}));
RETURN (gto_outer brk (a,p,D,pE))
} else RETURN s0
}) os;
RETURN (goBrk os)
}"

lemma gcollapse_refine:
"⟦(v',v)∈Id; (s',s)∈gstate_rel⟧
⟹ (gcollapse v' s',collapse v s)∈gstate_rel"
unfolding gcollapse_def collapse_def collapse_aux_def gcollapse_aux_def
apply (simp add: gstate_rel_def br_def Let_def)
unfolding gstate_invar_def[abs_def]
apply (auto split: prod.splits simp: take_map drop_map)
done

lemma gpush_refine:
"⟦(v',v)∈Id; (s',s)∈gstate_rel⟧ ⟹ (gpush v' s',push v s)∈gstate_rel"
unfolding gpush_def push_def
unfolding gstate_invar_def[abs_def]
apply (auto split: prod.splits)
done

lemma gpop_refine:
"⟦(s',s)∈gstate_rel⟧ ⟹ (gpop s',pop s)∈gstate_rel"
unfolding gpop_def pop_def
unfolding gstate_invar_def[abs_def]
apply (auto split: prod.splits simp: map_butlast)
done

lemma ginitial_refine:
"(ginitial x (None, b), initial x b) ∈ gstate_rel"
unfolding ginitial_def gstate_rel_def br_def gstate_invar_def initial_def
by auto

lemma oinitial_b_refine: "((None,{}),(None,{}))∈Id×⇩rId" by simp

lemma gselect_edge_refine: "⟦(s',s)∈gstate_rel⟧ ⟹ gselect_edge s'
≤⇓(⟨Id⟩option_rel ×⇩r gstate_rel) (select_edge s)"
unfolding gselect_edge_def select_edge_def
apply (simp add: pw_le_iff refine_pw_simps prod_rel_sv
split: prod.splits option.splits)

apply (auto simp: gstate_rel_def br_def gstate_invar_def)
done

lemma last_acc_impl:
assumes "p≠[]"
assumes "((a',p',D',pE'),(p,D,pE))∈gstate_rel"
shows "(last a' = {0..<num_acc}) = (∀i<num_acc. ∃q∈last p. i∈acc q)"
using assms acc_bound unfolding gstate_rel_def br_def gstate_invar_def
by (auto simp: last_map)

lemma fglr_aux1:
assumes V: "(v',v)∈Id" and S: "(s',s)∈gstate_rel"
and P: "⋀a' p' D' pE' p D pE. ((a',p',D',pE'),(p,D,pE))∈gstate_rel
⟹ f' a' p' D' pE' ≤⇓R (f p D pE)"
shows "(let (a',p',D',pE') = gcollapse v' s' in f' a' p' D' pE')
≤ ⇓R (let (p,D,pE) = collapse v s in f p D pE)"
apply (auto split: prod.splits)
apply (rule P)
using gcollapse_refine[OF V S]
apply simp
done

lemma gstate_invar_empty:
"gstate_invar (a,[],D,pE) ⟹ a=[]"
"gstate_invar ([],p,D,pE) ⟹ p=[]"

lemma find_ce_refine: "gfind_ce ≤⇓Id find_ce"
unfolding gfind_ce_def find_ce_def
unfolding goinitial_def go_is_no_brk_def[abs_def] goD_def goBrk_def
gto_outer_def
using [[goals_limit = 1]]
apply (refine_rcg
gselect_edge_refine prod_relI[OF IdI gpop_refine]
prod_relI[OF IdI gpush_refine]
fglr_aux1 last_acc_impl oinitial_b_refine
inj_on_id
)
apply refine_dref_type
apply (vc_solve (nopre)
solve: asm_rl
simp: gstate_rel_def br_def gstate_invar_empty)
done
end

subsection ‹Refinement to Gabow's Data Structure›

subsubsection ‹Preliminaries›
definition Un_set_drop_impl :: "nat ⇒ 'a set list ⇒ 'a set nres"
― ‹Executable version of @{text "⋃set (drop i A)"}, using indexing to
access @{text "A"}›
where "Un_set_drop_impl i A ≡
do {
(_,res) ← WHILET (λ(i,res). i < length A) (λ(i,res). do {
ASSERT (i<length A);
let res = A!i ∪ res;
let i = i + 1;
RETURN (i,res)
}) (i,{});
RETURN res
}"

lemma Un_set_drop_impl_correct:
"Un_set_drop_impl i A ≤ SPEC (λr. r=⋃(set (drop i A)))"
unfolding Un_set_drop_impl_def
apply (refine_rcg
WHILET_rule[where I="λ(i',res). res=⋃(set ((drop i (take i' A)))) ∧ i≤i'"
and R="measure (λ(i',_). length A - i')"]
refine_vcg)
apply (auto simp: take_Suc_conv_app_nth)
done

schematic_goal Un_set_drop_code_aux:
assumes [autoref_rules]: "(es_impl,{})∈⟨R⟩Rs"
assumes [autoref_rules]: "(un_impl,(∪))∈⟨R⟩Rs→⟨R⟩Rs→⟨R⟩Rs"
shows "(?c,Un_set_drop_impl)∈nat_rel → ⟨⟨R⟩Rs⟩as_rel → ⟨⟨R⟩Rs⟩nres_rel"
unfolding Un_set_drop_impl_def[abs_def]
apply (autoref (trace, keep_goal))
done
concrete_definition Un_set_drop_code uses Un_set_drop_code_aux

schematic_goal Un_set_drop_tr_aux:
"RETURN ?c ≤ Un_set_drop_code es_impl un_impl i A"
unfolding Un_set_drop_code_def
by refine_transfer
concrete_definition Un_set_drop_tr for es_impl un_impl i A
uses Un_set_drop_tr_aux

lemma Un_set_drop_autoref[autoref_rules]:
assumes "GEN_OP es_impl {} (⟨R⟩Rs)"
assumes "GEN_OP un_impl (∪) (⟨R⟩Rs→⟨R⟩Rs→⟨R⟩Rs)"
shows "(λi A. RETURN (Un_set_drop_tr es_impl un_impl i A),Un_set_drop_impl)
∈nat_rel → ⟨⟨R⟩Rs⟩as_rel → ⟨⟨R⟩Rs⟩nres_rel"
apply (intro fun_relI nres_relI)
apply (rule order_trans[OF Un_set_drop_tr.refine])
using Un_set_drop_code.refine[of es_impl Rs R un_impl,
param_fo, THEN nres_relD]
using assms
by simp

subsubsection ‹Actual Refinement›

type_synonym 'Q gGS = "nat set list × 'Q GS"

type_synonym 'Q goGS = "'Q ce × 'Q oGS"

context igb_graph
begin

definition gGS_invar :: "'Q gGS ⇒ bool"
where "gGS_invar s ≡
let (a,S,B,I,P) = s in
GS_invar (S,B,I,P)
∧ length a = length B
∧ ⋃(set a) ⊆ {0..<num_acc}
"

definition gGS_α :: "'Q gGS ⇒ 'Q abs_gstate"
where "gGS_α s ≡ let (a,s)=s in (a,GS.α s)"

definition "gGS_rel ≡ br gGS_α gGS_invar"

lemma gGS_rel_sv[relator_props,intro!,simp]: "single_valued gGS_rel"
unfolding gGS_rel_def by auto

definition goGS_invar :: "'Q goGS ⇒ bool" where
"goGS_invar s ≡ let (brk,ogs)=s in brk=None ⟶ oGS_invar ogs"

definition "goGS_α s ≡ let (brk,ogs)=s in (brk,oGS_α ogs)"

definition "goGS_rel ≡ br goGS_α goGS_invar"

lemma goGS_rel_sv[relator_props,intro!,simp]: "single_valued goGS_rel"
unfolding goGS_rel_def by auto

end

context igb_fr_graph
begin
lemma gGS_relE:
assumes "(s',(a,p,D,pE))∈gGS_rel"
obtains S' B' I' P' where "s'=(a,S',B',I',P')"
and "((S',B',I',P'),(p,D,pE))∈GS_rel"
and "length a = length B'"
and "⋃(set a) ⊆ {0..<num_acc}"
using assms
apply (cases s')
apply (simp add: gGS_rel_def br_def gGS_α_def GS.α_def)
apply (rule that)
apply (simp only:)
apply (auto simp: GS_rel_def br_def gGS_invar_def GS.α_def)
done

definition goinitial_impl :: "'Q goGS"
where "goinitial_impl ≡ (None,Map.empty)"
lemma goinitial_impl_refine: "(goinitial_impl,goinitial)∈goGS_rel"
by (auto
simp: goinitial_impl_def goinitial_def goGS_rel_def br_def
simp: goGS_α_def goGS_invar_def oGS_α_def oGS_invar_def)

definition gto_outer_impl :: "'Q ce ⇒ 'Q gGS ⇒ 'Q goGS"
where "gto_outer_impl brk s ≡ let (A,S,B,I,P)=s in (brk,I)"

lemma gto_outer_refine:
assumes A: "brk = None ⟶ (p=[] ∧ pE={})"
assumes B: "(s, (A,p, D, pE)) ∈ gGS_rel"
assumes C: "(brk',brk)∈Id"
shows "(gto_outer_impl brk' s,gto_outer brk (A,p,D,pE))∈goGS_rel"
proof (cases s)
fix A S B I P
assume [simp]: "s=(A,S,B,I,P)"
show ?thesis
using C
apply (cases brk)
using assms I_to_outer[of S B I P D]
apply (auto
simp: goGS_rel_def br_def goGS_α_def gto_outer_def
gto_outer_impl_def goGS_invar_def
simp: gGS_rel_def oGS_rel_def GS_rel_def gGS_α_def gGS_invar_def
GS.α_def) []

using B apply (auto
simp: gto_outer_def gto_outer_impl_def
simp: br_def goGS_rel_def goGS_invar_def goGS_α_def oGS_α_def
simp: gGS_rel_def gGS_α_def GS.α_def GS.D_α_def
)

done
qed

definition "gpush_impl v s ≡ let (a,s)=s in (a@[acc v], push_impl v s)"

lemma gpush_impl_refine:
assumes B: "(s',(a,p,D,pE))∈gGS_rel"
assumes A: "(v',v)∈Id"
assumes PRE: "v' ∉ ⋃(set p)" "v' ∉ D"
shows "(gpush_impl v' s', gpush v (a,p,D,pE))∈gGS_rel"
proof -
from B obtain S' B' I' P' where [simp]: "s'=(a,S',B',I',P')"
and OSR: "((S',B',I',P'),(p,D,pE))∈GS_rel" and L: "length a = length B'"
and R: "⋃(set a) ⊆ {0..<num_acc}"
by (rule gGS_relE)
{
fix S B I P S' B' I' P'
assume "push_impl v (S, B, I, P) = (S', B', I', P')"
hence "length B' = Suc (length B)"
by (auto simp add: push_impl_def GS.push_impl_def Let_def)
} note AUX1=this

from push_refine[OF OSR A PRE] A L acc_bound R show ?thesis
unfolding gpush_impl_def gpush_def
gGS_rel_def gGS_invar_def gGS_α_def GS_rel_def br_def
apply (auto dest: AUX1)
done
qed

definition gpop_impl :: "'Q gGS ⇒ 'Q gGS nres"
where "gpop_impl s ≡ do {
let (a,s)=s;
s←pop_impl s;
ASSERT (a≠[]);
let a = butlast a;
RETURN (a,s)
}"

lemma gpop_impl_refine:
assumes A: "(s',(a,p,D,pE))∈gGS_rel"
assumes PRE: "p ≠ []" "pE ∩ last p × UNIV = {}"
shows "gpop_impl s' ≤ ⇓gGS_rel (RETURN (gpop (a,p,D,pE)))"
proof -
from A obtain S' B' I' P' where [simp]: "s'=(a,S',B',I',P')"
and OSR: "((S',B',I',P'),(p,D,pE))∈GS_rel" and L: "length a = length B'"
and R: "⋃(set a) ⊆ {0..<num_acc}"
by (rule gGS_relE)

from PRE OSR have [simp]: "a≠[]" using L
by (auto simp add: GS_rel_def br_def GS.α_def GS.p_α_def)

{
fix S B I P S' B' I' P'
assume "nofail (pop_impl ((S, B, I, P)::'a GS))"
"inres (pop_impl ((S, B, I, P)::'a GS)) (S', B', I', P')"
hence "length B' = length B - Suc 0"
apply (simp add: pop_impl_def GS.pop_impl_def Let_def
refine_pw_simps)
apply auto
done
} note AUX1=this

from A L show ?thesis
unfolding gpop_impl_def gpop_def gGS_rel_def gGS_α_def br_def
using pop_refine[OF OSR PRE]
apply (simp add: pw_le_iff refine_pw_simps split: prod.splits)
unfolding gGS_rel_def gGS_invar_def gGS_α_def GS_rel_def GS.α_def br_def
apply (auto dest!: AUX1 in_set_butlastD iff: Sup_le_iff)
done
qed

definition gselect_edge_impl :: "'Q gGS ⇒ ('Q option × 'Q gGS) nres"
where "gselect_edge_impl s ≡
do {
let (a,s)=s;
(vo,s)←select_edge_impl s;
RETURN (vo,a,s)
}"

thm select_edge_refine
lemma gselect_edge_impl_refine:
assumes A: "(s', a, p, D, pE) ∈ gGS_rel"
assumes PRE: "p ≠ []"
shows "gselect_edge_impl s' ≤ ⇓(Id ×⇩r gGS_rel) (gselect_edge (a, p, D, pE))"
proof -
from A obtain S' B' I' P' where [simp]: "s'=(a,S',B',I',P')"
and OSR: "((S',B',I',P'),(p,D,pE))∈GS_rel" and L: "length a = length B'"
and R: "⋃(set a) ⊆ {0..<num_acc}"
by (rule gGS_relE)

{
fix S B I P S' B' I' P' vo
assume "nofail (select_edge_impl ((S, B, I, P)::'a GS))"
"inres (select_edge_impl ((S, B, I, P)::'a GS)) (vo, (S', B', I', P'))"
hence "length B' = length B"
apply (simp add: select_edge_impl_def GS.sel_rem_last_def refine_pw_simps
split: if_split_asm prod.splits)
apply auto
done
} note AUX1=this

show ?thesis
using select_edge_refine[OF OSR PRE]
unfolding gselect_edge_impl_def gselect_edge_def
apply (simp add: refine_pw_simps pw_le_iff prod_rel_sv)

unfolding gGS_rel_def br_def gGS_α_def gGS_invar_def GS_rel_def GS.α_def
apply (simp split: prod.splits)
apply clarsimp
using R
apply (auto simp: L dest: AUX1)
done
qed

term GS.idx_of_impl

thm GS_invar.idx_of_correct

definition gcollapse_impl_aux :: "'Q ⇒ 'Q gGS ⇒ 'Q gGS nres" where
"gcollapse_impl_aux v s ≡
do {
let (A,s)=s;
⌦‹ASSERT (v∈⋃set (GS.p_α s));›
i ← GS.idx_of_impl s v;
s ← collapse_impl v s;
ASSERT (i < length A);
us ← Un_set_drop_impl i A;
let A = take i A @ [us];
RETURN (A,s)
}"

term collapse
lemma gcollapse_alt:
"gcollapse v APDPE = (
let
(a,p,D,pE)=APDPE;
i=idx_of p v;
s=collapse v (p,D,pE);
us=⋃(set (drop i a));
a = take i a @ [us]
in (a,s))"
unfolding gcollapse_def gcollapse_aux_def collapse_def collapse_aux_def
by auto

thm collapse_refine
lemma gcollapse_impl_aux_refine:
assumes A: "(s', a, p, D, pE) ∈ gGS_rel"
assumes B: "(v',v)∈Id"
assumes PRE: "v∈⋃(set p)"
shows "gcollapse_impl_aux v' s'
≤ ⇓ gGS_rel (RETURN (gcollapse v (a, p, D, pE)))"
proof -
note [simp] = Let_def

from A obtain S' B' I' P' where [simp]: "s'=(a,S',B',I',P')"
and OSR: "((S',B',I',P'),(p,D,pE))∈GS_rel" and L: "length a = length B'"
and R: "⋃(set a) ⊆ {0..<num_acc}"
by (rule gGS_relE)

from B have [simp]: "v'=v" by simp

from OSR have [simp]: "GS.p_α (S',B',I',P') = p"
by (simp add: GS_rel_def br_def GS.α_def)

from OSR PRE have PRE': "v ∈ ⋃(set (GS.p_α (S',B',I',P')))"
by (simp add: GS_rel_def br_def GS.α_def)

from OSR have GS_invar: "GS_invar (S',B',I',P')"

term GS.B
{
fix s
assume "collapse v (p, D, pE) = (GS.p_α s, GS.D_α s, GS.pE_α s)"
hence "length (GS.B s) = Suc (idx_of p v)"
unfolding collapse_def collapse_aux_def Let_def
apply (cases s)
apply (auto simp: GS.p_α_def)
apply (drule arg_cong[where f=length])
using GS_invar.p_α_disjoint_sym[OF GS_invar]
and PRE ‹GS.p_α (S', B', I', P') = p› idx_of_props(1)[of p v]
by simp
} note AUX1 = this

show ?thesis
unfolding gcollapse_alt gcollapse_impl_aux_def
apply simp
apply (rule RETURN_as_SPEC_refine)
apply (refine_rcg
order_trans[OF GS_invar.idx_of_correct[OF GS_invar PRE']]
order_trans[OF collapse_refine[OF OSR B PRE, simplified]]
refine_vcg
)
using PRE' apply simp

unfolding gGS_rel_def