# Theory PrioUniqueByAnnotatedList

```section ‹\isaheader{Implementing Unique Priority Queues by Annotated Lists}›
theory PrioUniqueByAnnotatedList
imports
"../spec/AnnotatedListSpec"
"../spec/PrioUniqueSpec"
begin

text ‹
In this theory we use annotated lists to implement unique priority queues
with totally ordered elements.

This theory is written as a generic adapter from the AnnotatedList interface
to the unique priority queue interface.

The annotated list stores a sequence of elements annotated with
priorities\footnote{Technically, the annotated list elements are of unit-type,
and the annotations hold both, the priority queue elements and the priorities.
This is required as we defined annotated lists to only sum up the elements
annotations.}

The monoids operations forms the maximum over the elements and
the minimum over the priorities.
The sequence of pairs is ordered by ascending elements' order.
The insertion point for a new element, or the priority of an existing element
can be found by splitting the
sequence at the point where the maximum of the elements read so far gets
bigger than the element to be inserted.

The minimum priority can be read out as the sum over the whole sequence.
Finding the element with minimum priority is done by splitting the sequence
at the point where the minimum priority of the elements read so far becomes
equal to the minimum priority of the whole sequence.
›

subsection "Definitions"

subsubsection "Monoid"
datatype ('e, 'a) LP = Infty | LP 'e 'a

fun p_unwrap :: "('e,'a) LP ⇒ ('e × 'a)" where
"p_unwrap (LP e a) = (e , a)"

fun p_min :: "('e::linorder, 'a::linorder) LP ⇒ ('e, 'a) LP ⇒ ('e, 'a) LP"  where
"p_min Infty Infty = Infty"|
"p_min Infty (LP e a) = LP e a"|
"p_min (LP e a) Infty = LP e a"|
"p_min (LP e1 a) (LP e2 b) = (LP (max e1 e2) (min a b))"

fun e_less_eq :: "'e ⇒ ('e::linorder, 'a::linorder) LP ⇒ bool"  where
"e_less_eq e Infty = False"|
"e_less_eq e (LP e' _) = (e ≤ e')"

text_raw‹\paragraph{Instantiation of classes}\ \\›
lemma p_min_re_neut[simp]: "p_min a Infty = a" by (induct a) auto
lemma p_min_le_neut[simp]: "p_min Infty a = a" by (induct a) auto
lemma p_min_asso: "p_min (p_min a b) c = p_min a (p_min b c)"
apply(induct a b  rule: p_min.induct )
apply (auto)
apply (induct c)
apply (auto)
apply (metis max.assoc)
apply (metis min.assoc)
done

begin
definition zero_def: "0 == Infty"
definition plus_def: "a+b == p_min a b"

instance by
intro_classes
(auto simp add: p_min_asso zero_def plus_def)
end

fun p_less_eq :: "('e, 'a::linorder) LP ⇒ ('e, 'a) LP ⇒ bool" where
"p_less_eq (LP e a) (LP f b) = (a ≤ b)"|
"p_less_eq  _ Infty = True"|
"p_less_eq Infty (LP e a) = False"

fun p_less :: "('e, 'a::linorder) LP ⇒ ('e, 'a) LP ⇒ bool" where
"p_less (LP e a) (LP f b) = (a < b)"|
"p_less (LP e a) Infty = True"|
"p_less Infty _ = False"

lemma p_less_le_not_le : "p_less x y ⟷ p_less_eq x y ∧ ¬ (p_less_eq y x)"
by (induct x y rule: p_less.induct) auto

lemma p_order_refl : "p_less_eq x x"
by (induct x) auto

lemma p_le_inf : "p_less_eq Infty x ⟹ x = Infty"
by (induct x) auto

lemma p_order_trans : "⟦p_less_eq x y; p_less_eq y z⟧ ⟹ p_less_eq x z"
apply (induct y z rule: p_less.induct)
apply auto
apply (induct x)
apply auto
apply (cases x)
apply auto
apply(induct x)
apply (metis p_le_inf p_less_eq.simps(2))
done

lemma p_linear2 : "p_less_eq x y ∨ p_less_eq y x"
apply (induct x y rule: p_less_eq.induct)
apply auto
done

instantiation LP :: (type, linorder) preorder
begin
definition plesseq_def: "less_eq = p_less_eq"
definition pless_def: "less = p_less"

instance
apply (intro_classes)
apply (simp only: p_less_le_not_le pless_def plesseq_def)
apply (simp only: p_order_refl plesseq_def pless_def)
apply (simp only: plesseq_def)
apply (metis p_order_trans)
done

end

subsubsection "Operations"

definition aluprio_α :: "('s ⇒ (unit × ('e::linorder,'a::linorder) LP) list)
⇒ 's ⇒ ('e::linorder ⇀  'a::linorder)"
where
"aluprio_α α ft == (map_of (map p_unwrap (map snd (α ft))))"

definition aluprio_invar :: "('s ⇒ (unit × ('c::linorder, 'd::linorder) LP) list)
⇒ ('s ⇒ bool) ⇒ 's ⇒ bool"
where
"aluprio_invar α invar ft ==
invar ft
∧ (∀ x∈set (α ft). snd x≠Infty)
∧ sorted (map fst (map p_unwrap (map snd (α ft))))
∧ distinct (map fst (map p_unwrap (map snd (α ft)))) "

definition aluprio_empty  where
"aluprio_empty empt = empt"

definition aluprio_isEmpty  where
"aluprio_isEmpty isEmpty = isEmpty"

definition aluprio_insert ::
"((('e::linorder,'a::linorder) LP ⇒ bool)
⇒ ('e,'a) LP ⇒ 's ⇒ ('s × (unit × ('e,'a) LP) × 's))
⇒ ('s ⇒ ('e,'a) LP)
⇒ ('s ⇒ bool)
⇒ ('s ⇒ 's ⇒ 's)
⇒ ('s ⇒ unit ⇒ ('e,'a) LP ⇒ 's)
⇒ 's ⇒ 'e ⇒ 'a ⇒ 's"
where
"
aluprio_insert splits annot isEmpty app consr s e a =
(if e_less_eq e (annot s) ∧ ¬ isEmpty s
then
(let (l, (_,lp) , r) = splits (e_less_eq e) Infty s in
(if e < fst (p_unwrap lp)
then
app (consr (consr l () (LP e a))  () lp) r
else
app (consr l () (LP e a)) r  ))
else
consr s () (LP e a))
"

definition aluprio_pop :: "((('e::linorder,'a::linorder) LP ⇒ bool) ⇒ ('e,'a) LP
⇒ 's ⇒ ('s × (unit × ('e,'a) LP) × 's))
⇒ ('s ⇒ ('e,'a) LP)
⇒ ('s ⇒ 's ⇒ 's)
⇒ 's
⇒ 'e ×'a ×'s"
where
"aluprio_pop splits annot app s =
(let (l, (_,lp) , r) = splits (λ x. x ≤ (annot s)) Infty s
in
(case lp of
(LP e a) ⇒
(e, a, app l r) ))"

definition aluprio_prio ::
"((('e::linorder,'a::linorder) LP ⇒ bool) ⇒ ('e,'a) LP ⇒ 's
⇒ ('s × (unit × ('e,'a) LP) × 's))
⇒ ('s ⇒ ('e,'a) LP)
⇒ ('s ⇒ bool)
⇒ 's ⇒ 'e ⇒ 'a option"
where
"
aluprio_prio splits annot isEmpty s e =
(if e_less_eq e (annot s) ∧ ¬ isEmpty s
then
(let (l, (_,lp) , r) = splits (e_less_eq e) Infty s in
(if e = fst (p_unwrap lp)
then
Some (snd (p_unwrap lp))
else
None))
else
None)
"

lemmas aluprio_defs =
aluprio_invar_def
aluprio_α_def
aluprio_empty_def
aluprio_isEmpty_def
aluprio_insert_def
aluprio_pop_def
aluprio_prio_def

subsection "Correctness"

subsubsection "Auxiliary Lemmas"

lemma p_linear: "(x::('e, 'a::linorder) LP) ≤ y ∨ y ≤ x"
by (unfold plesseq_def) (simp only: p_linear2)

lemma e_less_eq_mon1: "e_less_eq e x ⟹ e_less_eq e (x + y)"
apply (cases x)
apply (cases y)
done
lemma e_less_eq_mon2: "e_less_eq e y ⟹ e_less_eq e (x + y)"
apply (cases x)
apply (cases y)
done
lemmas e_less_eq_mon =
e_less_eq_mon1
e_less_eq_mon2

lemma p_less_eq_mon:
"(x::('e::linorder,'a::linorder) LP) ≤ z ⟹ (x + y) ≤ z"
apply(cases y)
apply (cases x)
apply (cases z)
apply (cases z)
done

lemma p_less_eq_lem1:
"⟦¬ (x::('e::linorder,'a::linorder) LP) ≤ z;
(x + y) ≤ z⟧
⟹ y ≤ z "
apply (cases x,auto simp add: plus_def)
apply (cases y, auto)
apply (cases z, auto simp add: plesseq_def)
apply (metis min_le_iff_disj)
done

lemma infadd: "x ≠ Infty ⟹x + y ≠ Infty"
apply (unfold plus_def)
apply (induct x y rule: p_min.induct)
apply auto
done

lemma e_less_eq_sum_list:
"⟦¬ e_less_eq e (sum_list xs)⟧ ⟹ ∀x ∈ set xs. ¬ e_less_eq e x"
proof (induct xs)
case Nil thus ?case by simp
next
case (Cons a xs)
hence "¬ e_less_eq e (sum_list xs)" by (auto simp add: e_less_eq_mon)
hence v1: "∀x∈set xs. ¬ e_less_eq e x" using Cons.hyps by simp
from Cons.prems have "¬ e_less_eq e a" by (auto simp add: e_less_eq_mon)
with v1 show "∀x∈set (a#xs). ¬ e_less_eq e x" by simp
qed

lemma e_less_eq_p_unwrap:
"⟦x ≠ Infty;¬ e_less_eq e x⟧ ⟹ fst (p_unwrap x) < e"
by (cases x) auto

lemma e_less_eq_refl :
"b ≠ Infty ⟹ e_less_eq (fst (p_unwrap b)) b"
by (cases b) auto

lemma e_less_eq_sum_list2:
assumes
"∀x∈set (αs). snd x ≠ Infty"
"((), b) ∈ set (αs)"
shows "e_less_eq (fst (p_unwrap b)) (sum_list (map snd (αs)))"
apply(insert assms)
apply (induct "αs")
apply (auto simp add: zero_def e_less_eq_mon e_less_eq_refl)
done

lemma e_less_eq_lem1:
"⟦¬ e_less_eq e a;e_less_eq e (a + b)⟧ ⟹ e_less_eq e b"
apply (cases a)
apply auto
apply (cases b)
apply auto
apply (metis le_max_iff_disj)
done

lemma p_unwrap_less_sum: "snd (p_unwrap ((LP e aa) + b)) ≤ aa"
apply (cases b)
done

lemma  sum_list_less_elems: "∀x∈set xs. snd x ≠ Infty ⟹
∀y∈set (map snd (map p_unwrap (map snd xs))).
snd (p_unwrap (sum_list (map snd xs))) ≤ y"
proof (induct xs)
case Nil thus ?case by simp
next
case (Cons a as) thus ?case
apply auto
apply (cases "(snd a)" rule: p_unwrap.cases)
apply auto
apply (cases "sum_list (map snd as)")
apply auto
apply (metis linorder_linear p_min_re_neut p_unwrap.simps
plus_def[abs_def] snd_eqD)
apply (unfold plus_def)
apply (cases "(snd a, sum_list (map snd as))" rule: p_min.cases)
apply auto
apply (cases "map snd as")
apply (metis min.coboundedI2 snd_conv)
done
qed

lemma distinct_sortet_list_app:
"⟦sorted xs; distinct xs; xs = as @ b # cs⟧
⟹ ∀ x∈ set cs. b < x"
by (metis distinct.simps(2) distinct_append
antisym_conv2 sorted_wrt.simps(2) sorted_append)

lemma distinct_sorted_list_lem1:
assumes
"sorted xs"
"sorted ys"
"distinct xs"
"distinct ys"
" ∀ x ∈ set xs. x < e"
" ∀ y ∈ set ys. e < y"
shows
"sorted (xs @ e # ys)"
"distinct (xs @ e # ys)"
proof -
from assms (5,6)
have "∀x∈set xs. ∀y∈set ys. x ≤ y" by force
thus "sorted (xs @ e # ys)"
using assms
have "set xs ∩ set ys = {}" using assms (5,6) by force
thus "distinct (xs @ e # ys)"
using assms
by (auto)
qed

lemma distinct_sorted_list_lem2:
assumes
"sorted xs"
"sorted ys"
"distinct xs"
"distinct ys"
"e < e'"
" ∀ x ∈ set xs. x < e"
" ∀ y ∈ set ys. e' < y"
shows
"sorted (xs @ e # e' # ys)"
"distinct (xs @ e # e' # ys)"
proof -
have "sorted (e' # ys)"
"distinct (e' # ys)"
"∀ y ∈ set (e' # ys). e < y"
using assms(2,4,5,7)
by (auto)
thus "sorted (xs @ e # e' # ys)"
"distinct (xs @ e # e' # ys)"
using assms(1,3,6) distinct_sorted_list_lem1[of xs "e' # ys" e]
by auto
qed

lemma map_of_distinct_upd:
"x ∉ set (map fst xs) ⟹ [x ↦ y] ++ map_of xs = (map_of xs) (x ↦ y)"
by (induct xs) (auto simp add: fun_upd_twist)

lemma map_of_distinct_upd2:
assumes "x ∉ set(map fst xs)"
"x ∉ set (map fst ys)"
shows "map_of (xs @ (x,y) # ys) = (map_of (xs @ ys))(x ↦ y)"
apply(insert assms)
apply(induct xs)
apply (auto intro: ext)
done

lemma map_of_distinct_upd3:
assumes "x ∉ set(map fst xs)"
"x ∉ set (map fst ys)"
shows "map_of (xs @ (x,y) # ys) = (map_of (xs @ (x,y') # ys))(x ↦ y)"
apply(insert assms)
apply(induct xs)
apply (auto intro: ext)
done

lemma map_of_distinct_upd4:
assumes "x ∉ set(map fst xs)"
"x ∉ set (map fst ys)"
shows "map_of (xs @ ys) = (map_of (xs @ (x,y) # ys))(x := None)"
apply(insert assms)
apply(induct xs)

apply clarsimp
apply (metis dom_map_of_conv_image_fst fun_upd_None_restrict
restrict_complement_singleton_eq restrict_map_self)

apply (auto simp add: map_of_eq_None_iff) []
done

lemma map_of_distinct_lookup:
assumes "x ∉ set(map fst xs)"
"x ∉ set (map fst ys)"
shows "map_of (xs @ (x,y) # ys) x = Some y"
proof -
have "map_of (xs @ (x,y) # ys) = (map_of (xs @ ys)) (x ↦ y)"
using assms map_of_distinct_upd2 by simp
thus ?thesis
by simp
qed

lemma ran_distinct:
assumes dist: "distinct (map fst al)"
shows "ran (map_of al) = snd ` set al"
using assms proof (induct al)
case Nil then show ?case by simp
next
case (Cons kv al)
then have "ran (map_of al) = snd ` set al" by simp
moreover from Cons.prems have "map_of al (fst kv) = None"
ultimately show ?case by (simp only: map_of.simps ran_map_upd) simp
qed

subsubsection "Finite"

lemma aluprio_finite_correct: "uprio_finite (aluprio_α α) (aluprio_invar α invar)"

subsubsection "Empty"
lemma aluprio_empty_correct:
assumes "al_empty α invar empt"
shows "uprio_empty (aluprio_α α) (aluprio_invar α invar) (aluprio_empty empt)"
proof -
interpret al_empty α invar empt by fact
show ?thesis
apply (unfold_locales)
apply (auto simp add: empty_correct aluprio_defs)
done
qed

subsubsection "Is Empty"

lemma aluprio_isEmpty_correct:
assumes "al_isEmpty α invar isEmpty"
shows "uprio_isEmpty (aluprio_α α) (aluprio_invar α invar) (aluprio_isEmpty isEmpty)"
proof -
interpret al_isEmpty α invar isEmpty by fact
show ?thesis
apply (unfold_locales)
apply (auto simp add: aluprio_defs isEmpty_correct)
done
qed

subsubsection "Insert"

lemma annot_inf:
assumes A: "invar s" "∀x∈set (α s). snd x ≠ Infty" "al_annot α invar annot"
shows "annot s = Infty ⟷ α s = [] "
proof -
from A have invs: "invar s" by (simp add: aluprio_defs)
interpret al_annot α invar annot by fact
show "annot s = Infty ⟷ α s = []"
proof (cases "α s = []")
case True
hence "map snd (α s) = []" by simp
hence "sum_list (map snd (α s)) = Infty"
with invs have  "annot s = Infty" by (auto simp add: annot_correct)
with True show ?thesis by simp
next
case False
hence " ∃x xs. (α s) = x # xs" by (cases "α s") auto
from this obtain x xs where [simp]: "(α s) = x # xs" by blast
from this assms(2) have "snd x ≠ Infty" by (auto simp add: aluprio_defs)
hence "sum_list (map snd (α s)) ≠ Infty" by (auto simp add: infadd)
thus ?thesis using annot_correct invs False by simp
qed
qed

lemma e_less_eq_annot:

assumes "al_annot α invar annot"
"invar s" "∀x∈set (α s). snd x ≠ Infty" "¬ e_less_eq e (annot s)"
shows "∀x ∈ set (map (fst ∘ (p_unwrap ∘ snd)) (α s)). x < e"
proof -
interpret al_annot α invar annot by fact
from assms(2) have "annot s = sum_list (map snd (α s))"
with assms(4) have
"∀x ∈ set (map snd (α s)). ¬ e_less_eq e x"
by (metis e_less_eq_sum_list)
with assms(3)
show ?thesis
qed

lemma aluprio_insert_correct:
assumes
"al_splits α invar splits"
"al_annot α invar annot"
"al_isEmpty α invar isEmpty"
"al_app α invar app"
"al_consr α invar consr"
shows
"uprio_insert (aluprio_α α) (aluprio_invar α invar)
(aluprio_insert splits annot isEmpty app consr)"
proof -
interpret al_splits α invar splits by fact
interpret al_annot α invar annot by fact
interpret al_isEmpty α invar isEmpty by fact
interpret al_app α invar app by fact
interpret al_consr α invar consr by fact
show ?thesis
proof (unfold_locales, unfold aluprio_defs, goal_cases)
case g1asms: (1 s e a)
thus ?case proof (cases "e_less_eq e (annot s) ∧ ¬ isEmpty s")
case False with g1asms show  ?thesis
apply (auto simp add: consr_correct )
proof goal_cases
case prems: 1
with assms(2) have
"∀x ∈ set (map (fst ∘ (p_unwrap ∘ snd)) (α s)). x < e"
with prems(3) show ?case
next
case prems: 2
hence "annot s = sum_list (map snd (α s))"
with prems
show ?case
next
case prems: 3
hence "α s = []" by (auto simp add: isEmpty_correct)
thus ?case by simp
next
case prems: 4
hence "α s = []" by (auto simp add: isEmpty_correct)
with prems show ?case by simp
qed
next
case True note T1 = this
obtain l uu lp r where
l_lp_r: "(splits (e_less_eq e) Infty s) = (l, ((), lp), r) "
by (cases "splits (e_less_eq e) Infty s", auto)
note v2 = splits_correct[of s "e_less_eq e" Infty l "()" lp r]
have
v3: "invar s"
"¬ e_less_eq e Infty"
"e_less_eq e (Infty + sum_list (map snd (α s)))"
using T1 g1asms annot_correct
have
v4: "α s = α l @ ((), lp) # α r"
"¬ e_less_eq e (Infty + sum_list (map snd (α l)))"
"e_less_eq e (Infty + sum_list (map snd (α l)) + lp)"
"invar l"
"invar r"
using v2[OF v3(1) _ v3(2) v3(3) l_lp_r] e_less_eq_mon(1) by auto
hence v5: "e_less_eq e lp"
by (metis e_less_eq_lem1)
hence v6: "e ≤ (fst (p_unwrap lp))"
by (cases lp) auto
have "(Infty + sum_list (map snd (α l))) = (annot l)"
by (metis add_0_left annot_correct v4(4) zero_def)
hence v7:"¬ e_less_eq e (annot l)"
using v4(2) by simp
have "∀x∈set (α l). snd x ≠ Infty"
using g1asms v4(1) by simp
hence v7: "∀x ∈ set (map (fst ∘ (p_unwrap ∘ snd)) (α l)). x < e"
using v4(4) v7 assms(2)
have v8:"map fst (map p_unwrap (map snd (α s))) =
map fst (map p_unwrap (map snd (α l))) @ fst(p_unwrap lp) #
map fst (map p_unwrap (map snd (α r)))"
using v4(1)
by simp
note distinct_sortet_list_app[of "map fst (map p_unwrap (map snd (α s)))"
"map fst (map p_unwrap (map snd (α l)))" "fst(p_unwrap lp)"
"map fst (map p_unwrap (map snd (α r)))"]
hence v9:
"∀ x∈set (map (fst ∘ (p_unwrap ∘ snd)) (α r)). fst(p_unwrap lp) < x"
using v4(1) g1asms v8
by auto
have v10:
"sorted (map fst (map p_unwrap (map snd (α l))))"
"distinct (map fst (map p_unwrap (map snd (α l))))"
"sorted (map fst (map p_unwrap (map snd (α r))))"
"distinct (map fst (map p_unwrap (map snd (α l))))"
using g1asms v8

from l_lp_r T1 g1asms show ?thesis
proof (fold aluprio_insert_def, cases "e < fst (p_unwrap lp)")
case True
hence v11:
"aluprio_insert splits annot isEmpty app consr s e a
= app (consr (consr l () (LP e a)) () lp) r"
using l_lp_r T1
have  v12: "invar (app (consr (consr l () (LP e a)) () lp) r)"
using v4(4,5)
by (auto simp add: app_correct consr_correct)
have v13:
"α (app (consr (consr l () (LP e a)) () lp) r)
= α l @ ((),(LP e a)) # ((), lp) # α r"
using v4(4,5) by (auto simp add: app_correct consr_correct)
hence v14:
"(∀x∈set (α (app (consr (consr l () (LP e a)) () lp) r)).
snd x ≠ Infty)"
using g1asms v4(1)
by auto
have v15: "e = fst(p_unwrap (LP e a))" by simp
hence v16:
"sorted (map fst (map p_unwrap
(map snd (α l @ ((),(LP e a)) # ((), lp) # α r))))"
"distinct (map fst (map p_unwrap
(map snd (α l @ ((),(LP e a)) # ((), lp) # α r))))"
using v10(1,3) v7 True v9 v4(1) g1asms distinct_sorted_list_lem2
thus "invar (aluprio_insert splits annot isEmpty app consr s e a) ∧
(∀x∈set (α (aluprio_insert splits annot isEmpty app consr s e a)).
snd x ≠ Infty) ∧
sorted (map fst (map p_unwrap (map snd (α
(aluprio_insert splits annot isEmpty app consr s e a))))) ∧
distinct (map fst (map p_unwrap (map snd (α
(aluprio_insert splits annot isEmpty app consr s e a)))))"
using v11 v12 v13 v14
by simp
next
case False
hence v11:
"aluprio_insert splits annot isEmpty app consr s e a
= app (consr l () (LP e a)) r"
using l_lp_r T1
have  v12: "invar (app (consr l () (LP e a)) r)" using v4(4,5)
by (auto simp add: app_correct consr_correct)
have v13: "α (app (consr l () (LP e a)) r) = α l @ ((),(LP e a)) # α r"
using v4(4,5) by (auto simp add: app_correct consr_correct)
hence v14: "(∀x∈set (α (app (consr l () (LP e a)) r)). snd x ≠ Infty)"
using g1asms v4(1)
by auto
have v15: "e = fst(p_unwrap (LP e a))" by simp
have v16: "e = fst(p_unwrap lp)"
using False v5 by (cases lp) auto
hence v17:
"sorted (map fst (map p_unwrap
(map snd (α l @ ((),(LP e a)) # α r))))"
"distinct (map fst (map p_unwrap
(map snd (α l @ ((),(LP e a)) # α r))))"
using v16 v15 v10(1,3) v7 True v9 v4(1)
g1asms distinct_sorted_list_lem1
thus "invar (aluprio_insert splits annot isEmpty app consr s e a) ∧
(∀x∈set (α (aluprio_insert splits annot isEmpty app consr s e a)).
snd x ≠ Infty) ∧
sorted (map fst (map p_unwrap (map snd (α
(aluprio_insert splits annot isEmpty app consr s e a))))) ∧
distinct (map fst (map p_unwrap (map snd (α
(aluprio_insert splits annot isEmpty app consr s e a)))))"
using v11 v12 v13 v14
by simp
qed
qed
next
case g1asms: (2 s e a)
thus ?case proof (cases "e_less_eq e (annot s) ∧ ¬ isEmpty s")
case False with g1asms show  ?thesis
proof goal_cases
case prems: 1
with assms(2) have
"∀x ∈ set (map (fst ∘ (p_unwrap ∘ snd)) (α s)). x < e"
hence "e ∉ set (map fst ((map (p_unwrap ∘ snd)) (α s)))"
by auto
thus ?case
next
case prems: 2
hence "α s = []" by (auto simp add: isEmpty_correct)
thus ?case
by simp
qed
next
case True note T1 = this
obtain l lp r where
l_lp_r: "(splits (e_less_eq e) Infty s) = (l, ((), lp), r) "
by (cases "splits (e_less_eq e) Infty s", auto)
note v2 = splits_correct[of s "e_less_eq e" Infty l "()" lp r]
have
v3: "invar s"
"¬ e_less_eq e Infty"
"e_less_eq e (Infty + sum_list (map snd (α s)))"
using T1 g1asms annot_correct
have
v4: "α s = α l @ ((), lp) # α r"
"¬ e_less_eq e (Infty + sum_list (map snd (α l)))"
"e_less_eq e (Infty + sum_list (map snd (α l)) + lp)"
"invar l"
"invar r"
using v2[OF v3(1) _ v3(2) v3(3) l_lp_r] e_less_eq_mon(1) by auto
hence v5: "e_less_eq e lp"
by (metis e_less_eq_lem1)
hence v6: "e ≤ (fst (p_unwrap lp))"
by (cases lp) auto
have "(Infty + sum_list (map snd (α l))) = (annot l)"
by (metis add_0_left annot_correct v4(4) zero_def)
hence v7:"¬ e_less_eq e (annot l)"
using v4(2) by simp
have "∀x∈set (α l). snd x ≠ Infty"
using g1asms v4(1) by simp
hence v7: "∀x ∈ set (map (fst ∘ (p_unwrap ∘ snd)) (α l)). x < e"
using v4(4) v7 assms(2)
have v8:"map fst (map p_unwrap (map snd (α s))) =
map fst (map p_unwrap (map snd (α l))) @ fst(p_unwrap lp) #
map fst (map p_unwrap (map snd (α r)))"
using v4(1)
by simp
note distinct_sortet_list_app[of "map fst (map p_unwrap (map snd (α s)))"
"map fst (map p_unwrap (map snd (α l)))" "fst(p_unwrap lp)"
"map fst (map p_unwrap (map snd (α r)))"]
hence v9: "
∀ x∈set (map (fst ∘ (p_unwrap ∘ snd)) (α r)). fst(p_unwrap lp) < x"
using v4(1) g1asms v8
by auto
hence v10: " ∀ x∈set (map (fst ∘ (p_unwrap ∘ snd)) (α r)). e < x"
using v6 by auto
have v11:
"e ∉ set (map fst (map p_unwrap (map snd (α l))))"
"e ∉ set (map fst (map p_unwrap (map snd (α r))))"
using v7 v10 v8 g1asms
by auto
from l_lp_r T1 g1asms show ?thesis
proof (fold aluprio_insert_def, cases "e < fst (p_unwrap lp)")
case True
hence v12:
"aluprio_insert splits annot isEmpty app consr s e a
= app (consr (consr l () (LP e a)) () lp) r"
using l_lp_r T1
have v13:
"α (app (consr (consr l () (LP e a)) () lp) r)
= α l @ ((),(LP e a)) # ((), lp) # α r"
using v4(4,5) by (auto simp add: app_correct consr_correct)
have v14: "e = fst(p_unwrap (LP e a))" by simp
have v15: "e ∉ set (map fst (map p_unwrap (map snd(((),lp)#α r))))"
using v11(2) True by auto
note map_of_distinct_upd2[OF v11(1) v15]
thus
"map_of (map p_unwrap (map snd (α
(aluprio_insert splits annot isEmpty app consr s e a))))
= (map_of (map p_unwrap (map snd (α s))))(e ↦ a)"
using v12 v13 v4(1)
by simp
next
case False
hence v12:
"aluprio_insert splits annot isEmpty app consr s e a
= app (consr l () (LP e a)) r"
using l_lp_r T1
have v13:
"α (app (consr l () (LP e a)) r) = α l @ ((),(LP e a)) # α r"
using v4(4,5) by (auto simp add: app_correct consr_correct)
have v14: "e = fst(p_unwrap lp)"
using False v5 by (cases lp) auto
note v15 = map_of_distinct_upd3[OF v11(1) v11(2)]
have v16:"(map p_unwrap (map snd (α s))) =
(map p_unwrap (map snd (α l))) @ (e,snd(p_unwrap lp)) #
(map p_unwrap (map snd (α r)))"
using v4(1) v14
by simp
note v15[of a "snd(p_unwrap lp)"]
thus
"map_of (map p_unwrap (map snd (α
(aluprio_insert splits annot isEmpty app consr s e a))))
= (map_of (map p_unwrap (map snd (α s))))(e ↦ a)"
using v12 v13 v16
by simp
qed
qed
qed
qed

subsubsection "Prio"
lemma aluprio_prio_correct:
assumes
"al_splits α invar splits"
"al_annot α invar annot"
"al_isEmpty α invar isEmpty"
shows
"uprio_prio (aluprio_α α) (aluprio_invar α invar) (aluprio_prio splits annot isEmpty)"
proof -
interpret al_splits α invar splits by fact
interpret al_annot α invar annot by fact
interpret al_isEmpty α invar isEmpty by fact
show ?thesis
proof (unfold_locales)
fix s e
assume inv1: "aluprio_invar α invar s"
hence sinv: "invar s"
"(∀ x∈set (α s). snd x≠Infty)"
"sorted (map fst (map p_unwrap (map snd (α s))))"
"distinct (map fst (map p_unwrap (map snd (α s))))"
show "aluprio_prio splits annot isEmpty s e = aluprio_α α s e"
proof(cases "e_less_eq e (annot s) ∧ ¬ isEmpty s")
case False note F1 = this
thus ?thesis
proof(cases "isEmpty s")
case True
hence "α s = []"
using sinv isEmpty_correct by simp
hence "aluprio_α α s = Map.empty" by (simp add:aluprio_defs)
hence "aluprio_α α s e = None" by simp
thus "aluprio_prio splits annot isEmpty s e = aluprio_α α s e"
using F1
next
case False
hence v3:"¬ e_less_eq e (annot s)"  using F1 by simp
note v4=e_less_eq_annot[OF assms(2)]
note v4[OF sinv(1) sinv(2) v3]
hence v5:"e∉set (map (fst ∘ (p_unwrap ∘ snd)) (α s))"
by auto
hence "map_of (map (p_unwrap ∘ snd) (α s)) e = None"
using map_of_eq_None_iff
by (metis map_map map_of_eq_None_iff set_map v5)
thus "aluprio_prio splits annot isEmpty s e = aluprio_α α s e"
using F1
qed
next
case True note T1 = this
obtain l uu lp r where
l_lp_r: "(splits (e_less_eq e) Infty s) = (l, ((), lp), r) "
by (cases "splits (e_less_eq e) Infty s", auto)
note v2 = splits_correct[of s "e_less_eq e" Infty l "()" lp r]
have
v3: "invar s"
"¬ e_less_eq e Infty"
"e_less_eq e (Infty + sum_list (map snd (α s)))"
using T1 sinv annot_correct
have
v4: "α s = α l @ ((), lp) # α r"
"¬ e_less_eq e (Infty + sum_list (map snd (α l)))"
"e_less_eq e (Infty + sum_list (map snd (α l)) + lp)"
"invar l"
"invar r"
using v2[OF v3(1) _ v3(2) v3(3) l_lp_r] e_less_eq_mon(1) by auto
hence v5: "e_less_eq e lp"
by (metis e_less_eq_lem1)
hence v6: "e ≤ (fst (p_unwrap lp))"
by (cases lp) auto
have "(Infty + sum_list (map snd (α l))) = (annot l)"
by (metis add_0_left annot_correct v4(4) zero_def)
hence v7:"¬ e_less_eq e (annot l)"
using v4(2) by simp
have "∀x∈set (α l). snd x ≠ Infty"
using sinv v4(1) by simp
hence v7: "∀x ∈ set (map (fst ∘ (p_unwrap ∘ snd)) (α l)). x < e"
using v4(4) v7 assms(2)
have v8:"map fst (map p_unwrap (map snd (α s))) =
map fst (map p_unwrap (map snd (α l))) @ fst(p_unwrap lp) #
map fst (map p_unwrap (map snd (α r)))"
using v4(1)
by simp
note distinct_sortet_list_app[of "map fst (map p_unwrap (map snd (α s)))"
"map fst (map p_unwrap (map snd (α l)))" "fst(p_unwrap lp)"
"map fst (map p_unwrap (map snd (α r)))"]
hence v9:
"∀ x∈set (map (fst ∘ (p_unwrap ∘ snd)) (α r)). fst(p_unwrap lp) < x"
using v4(1) sinv v8
by auto
hence v10: " ∀ x∈set (map (fst ∘ (p_unwrap ∘ snd)) (α r)). e < x"
using v6 by auto
have v11:
"e ∉ set (map fst (map p_unwrap (map snd (α l))))"
"e ∉ set (map fst (map p_unwrap (map snd (α r))))"
using v7 v10 v8 sinv
by auto
from l_lp_r T1 sinv show ?thesis
proof (cases "e = fst (p_unwrap lp)")
case False
have v12: "e ∉ set (map fst (map p_unwrap (map snd(α s))))"
using v11 False v4(1) by auto
hence "map_of (map (p_unwrap ∘ snd) (α s)) e = None"
using map_of_eq_None_iff
by (metis map_map map_of_eq_None_iff set_map v12)
thus ?thesis
using T1 False l_lp_r
next
case True
have v12: "map (p_unwrap ∘ snd) (α s) =
map p_unwrap (map snd (α l)) @ (e,snd (p_unwrap lp)) #
map p_unwrap (map snd (α r))"
using v4(1) True by simp
note map_of_distinct_lookup[OF v11]
hence
"map_of (map (p_unwrap ∘ snd) (α s)) e = Some (snd (p_unwrap lp))"
using v12 by simp
thus ?thesis
using T1 True l_lp_r
qed
qed
qed
qed

subsubsection "Pop"

lemma aluprio_pop_correct:
assumes "al_splits α invar splits"
"al_annot α invar annot"
"al_app α invar app"
shows
"uprio_pop (aluprio_α α) (aluprio_invar α invar) (aluprio_pop splits annot app)"
proof -
interpret al_splits α invar splits by fact
interpret al_annot α invar annot by fact
interpret al_app α invar app by fact
show ?thesis
proof (unfold_locales)
fix s e a s'
assume A: "aluprio_invar α invar s"
"aluprio_α α s ≠ Map.empty"
"aluprio_pop splits annot app s = (e, a, s')"
hence v1: "α s ≠ []"
obtain l lp r where
l_lp_r: "splits (λ x. x≤annot s) Infty s = (l,((),lp),r)"
by (cases "splits (λ x. x≤annot s) Infty s", auto)
have invs:
"invar s"
"(∀x∈set (α s). snd x ≠ Infty)"
"sorted (map fst (map p_unwrap (map snd (α s))))"
"distinct (map fst (map p_unwrap (map snd (α s))))"
using A by (auto simp add:aluprio_defs)
note a1 = annot_inf[of invar s α annot]
note a1[OF invs(1) invs(2) assms(2)]
hence v2: "annot s ≠ Infty"
using v1 by simp
hence v3:
"¬ Infty ≤ annot s"
by(cases "annot s") (auto simp add: plesseq_def)
have v4: "annot s = sum_list (map snd (α s))"
by (auto simp add: annot_correct invs(1))
hence
v5:
"(Infty + sum_list (map snd (α s))) ≤ annot s"
note p_mon = p_less_eq_mon[of _ "annot s"]
note v6 = splits_correct[OF invs(1)]
note v7 = v6[of "λ x. x ≤ annot s"]
note v7[OF _ v3 v5 l_lp_r] p_mon
hence v8:
" α s = α l @ ((), lp) # α r"
"¬ Infty + sum_list (map snd (α l)) ≤ annot s"
"Infty + sum_list (map snd (α l)) + lp ≤ annot s"
"invar l"
"invar r"
by auto
hence v9: "lp ≠ Infty"
using invs(2) by auto
hence v10:
"s' = app l r"
"(e,a) = p_unwrap lp"
using l_lp_r A(3)
apply (cases lp)
apply auto
apply (cases lp)
apply auto
done
have "lp ≤ annot s"
using v8(2,3) p_less_eq_lem1
by auto
hence v11: "a ≤ snd (p_unwrap (annot s))"
using v10(2) v2 v9
apply (cases "annot s")
apply auto
apply (cases lp)
done
note sum_list_less_elems[OF invs(2)]
hence v12: "∀y∈set (map snd (map p_unwrap (map snd (α s)))). a ≤ y"
using v4 v11 by auto
have "ran (aluprio_α α s) = set (map snd (map p_unwrap (map snd (α s))))"
using ran_distinct[OF invs(4)]
apply (unfold aluprio_defs)
apply (simp only: set_map)
done
hence ziel1: "∀y∈ran (aluprio_α α s). a ≤ y"
using v12 by simp
have v13:
"map p_unwrap (map snd (α s))
= map p_unwrap (map  snd (α l)) @ (e,a) # map p_unwrap (map snd (α r))"
using v8(1) v10 by auto
hence v14:
"map fst (map p_unwrap (map snd (α s)))
= map fst (map p_unwrap (map snd (α l))) @ e
# map fst (map p_unwrap (map snd (α r)))"
by auto
hence v15:
"e ∉ set (map fst (map p_unwrap (map snd (α l))))"
"e ∉ set (map fst (map p_unwrap (map snd (α r))))"
using invs(4) by auto
note map_of_distinct_lookup[OF v15]
note this[of a]
hence ziel2: "aluprio_α α s e = Some a"
using  v13
by (unfold aluprio_defs, auto)
have v16:
"α s' = α l @ α r"
"invar s'"
using v8(4,5) app_correct v10 by auto
note map_of_distinct_upd4[OF v15]
note this[of a]
hence
ziel3: "aluprio_α α s' = (aluprio_α α s)(e := None)"
unfolding aluprio_defs
using v16(1) v13 by auto
have ziel4: "aluprio_invar α invar s'"
using v16 v8(1) invs(2,3,4)
unfolding aluprio_defs

show "aluprio_invar α invar s' ∧
aluprio_α α s' = (aluprio_α α s)(e := None) ∧
aluprio_α α s e = Some a ∧ (∀y∈ran (aluprio_α α s). a ≤ y)"
using ziel1 ziel2 ziel3 ziel4 by simp
qed
qed

lemmas aluprio_correct =
aluprio_finite_correct
aluprio_empty_correct
aluprio_isEmpty_correct
aluprio_insert_correct
aluprio_pop_correct
aluprio_prio_correct

locale aluprio_defs = StdALDefs ops
for ops :: "(unit,('e::linorder,'a::linorder) LP,'s) alist_ops"
begin
definition [icf_rec_def]: "aluprio_ops ≡ ⦇
upr_α = aluprio_α α,
upr_invar = aluprio_invar α invar,
upr_empty = aluprio_empty empty,
upr_isEmpty = aluprio_isEmpty isEmpty,
upr_insert = aluprio_insert splits annot isEmpty app consr,
upr_pop = aluprio_pop splits annot app,
upr_prio = aluprio_prio splits annot isEmpty
⦈"

end

locale aluprio = aluprio_defs ops + StdAL ops
for ops :: "(unit,('e::linorder,'a::linorder) LP,'s) alist_ops"
begin
lemma aluprio_ops_impl: "StdUprio aluprio_ops"
apply (rule StdUprio.intro)