# Theory AnnotatedListGAPrioUniqueImpl

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theory AnnotatedListGAPrioUniqueImpl
imports AnnotatedListSpec PrioUniqueSpec
`header {*\isaheader{Implementing Unique Priority Queues by Annotated Lists}*}theory AnnotatedListGAPrioUniqueImplimports   AnnotatedListSpec   PrioUniqueSpec begintext {*  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 'afun 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) autolemma p_min_le_neut[simp]: "p_min Infty a = a" by (induct a) autolemma 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 min_max.sup_assoc)apply (metis min_max.inf_assoc)  donelemma lp_mono: "class.monoid_add p_min Infty" by  unfold_locales  (auto simp add: p_min_asso)instantiation LP :: (linorder,linorder) monoid_addbegindefinition 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)endfun 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) autolemma p_order_refl : "p_less_eq x x"  by (induct x) autolemma p_le_inf : "p_less_eq Infty x ==> x = Infty"  by (induct x) autolemma 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 (auto simp add: p_le_inf)  apply (metis p_le_inf p_less_eq.simps(2))  apply (metis p_le_inf p_less_eq.simps(2))  donelemma p_linear2 : "p_less_eq x y ∨ p_less_eq y x"  apply (induct x y rule: p_less_eq.induct)  apply auto  doneinstantiation LP :: (type, linorder) preorderbegindefinition 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)  doneendsubsubsection "Operations"definition aluprio_α :: "('s => (unit × ('e::linorder,'a::linorder) LP) list)   => 's => ('e::linorder \<rightharpoonup>  '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_defaluprio_α_defaluprio_empty_defaluprio_isEmpty_defaluprio_insert_defaluprio_pop_defaluprio_prio_defsubsection "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 (auto simp add: plus_def)   apply (cases y)   apply (auto simp add: min_max.le_supI1)  donelemma e_less_eq_mon2: "e_less_eq e y ==> e_less_eq e (x + y)"  apply (cases x)   apply (auto simp add: plus_def)   apply (cases y)   apply (auto simp add: min_max.le_supI2)  donelemmas e_less_eq_mon =   e_less_eq_mon1  e_less_eq_mon2lemma p_less_eq_mon:  "(x::('e::linorder,'a::linorder) LP) ≤ z ==> (x + y) ≤ z"  apply(cases y)  apply(auto simp add: plus_def)  apply (cases x)  apply (cases z)  apply (auto simp add: plesseq_def)  apply (cases z)  apply (auto simp add: min_max.le_infI1)  donelemma 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  donelemma e_less_eq_listsum:   "[|¬ e_less_eq e (listsum xs)|] ==> ∀x ∈ set xs. ¬ e_less_eq e x"proof (induct xs)  case Nil thus ?case by simpnext  case (Cons a xs)  hence "¬ e_less_eq e (listsum 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 simpqedlemma e_less_eq_p_unwrap:   "[|x ≠ Infty;¬ e_less_eq e x|] ==> fst (p_unwrap x) < e"  by (cases x) autolemma e_less_eq_refl :  "b ≠ Infty ==> e_less_eq (fst (p_unwrap b)) b"  by (cases b) autolemma e_less_eq_listsum2:  assumes   "∀x∈set (αs). snd x ≠ Infty"  "((), b) ∈ set (αs)"  shows "e_less_eq (fst (p_unwrap b)) (listsum (map snd (αs)))"  apply(insert assms)  apply (induct "αs")  apply (auto simp add: zero_def e_less_eq_mon e_less_eq_refl)   donelemma e_less_eq_lem1:  "[|¬ e_less_eq e a;e_less_eq e (a + b)|] ==> e_less_eq e b"  apply (auto simp add: plus_def)  apply (cases a)  apply auto  apply (cases b)  apply auto  apply (metis le_max_iff_disj)  donelemma p_unwrap_less_sum: "snd (p_unwrap ((LP e aa) + b)) ≤ aa"  apply (cases b)  apply (auto simp add: plus_def)donelemma  listsum_less_elems: "∀x∈set xs. snd x ≠ Infty ==>  ∀y∈set (map snd (map p_unwrap (map snd xs))).              snd (p_unwrap (listsum (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 "listsum (map snd as)")      apply auto      apply (metis linorder_linear p_min_re_neut p_unwrap.simps plus_def [abs_def] snd_eqD)      apply (auto simp add: p_unwrap_less_sum)      apply (unfold plus_def)      apply (cases "(snd a, listsum (map snd as))" rule: p_min.cases)      apply auto      apply (cases "map snd as")      apply (auto simp add: infadd)      apply (metis min_max.le_infI2 snd_conv)      doneqedlemma distinct_sortet_list_app:  "[|sorted xs; distinct xs; xs = as @ b # cs|]  ==> ∀ x∈ set cs. b < x"by(auto simp add: sorted_append sorted_Cons dest: bspec intro: xt1(11))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    by (auto simp add: sorted_append sorted_Cons)  have "set xs ∩ set ys = {}" using assms (5,6) by force  thus "distinct (xs @ e # ys)"    using assms    by (auto simp add: distinct_append)qedlemma 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 simp add: sorted_Cons)  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 autoqedlemma map_of_distinct_upd:  "x ∉ set (map fst xs) ==> [x \<mapsto> y] ++ map_of xs = (map_of xs) (x \<mapsto> 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 \<mapsto> y)"  apply(insert assms)  apply(induct xs)  apply (auto intro: ext)  donelemma 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 \<mapsto> y)"  apply(insert assms)  apply(induct xs)  apply (auto intro: ext)  donelemma 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 (auto simp add: map_of_eq_None_iff intro: ext)  donelemma 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 \<mapsto> y)"    using assms map_of_distinct_upd2 by simp  thus ?thesis    by simpqedlemma 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 simpnext  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"    by (simp add: map_of_eq_None_iff)  ultimately show ?case by (simp only: map_of.simps ran_map_upd) simpqedsubsubsection "Finite"lemma aluprio_finite_correct: "uprio_finite (aluprio_α α) (aluprio_invar α invar)"   by(unfold_locales) (simp add: aluprio_defs finite_dom_map_of)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)    doneqedsubsubsection "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)    apply (metis Nil_is_map_conv hd_in_set map_map map_of_eq_None_iff set_map)    doneqedsubsubsection "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 "listsum (map snd (α s)) = Infty"        by (auto simp add: zero_def)    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 "listsum (map snd (α s)) ≠ Infty" by (auto simp add: infadd)    thus ?thesis using annot_correct invs False by simp  qedqedlemma 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 o (p_unwrap o snd)) (α s)). x < e"proof -  interpret al_annot α invar annot by fact  from assms(2) have "annot s = listsum (map snd (α s))"    by (auto simp add: annot_correct)  with assms(4) have     "∀x ∈ set (map snd (α s)). ¬ e_less_eq e x"    by (metis e_less_eq_listsum)  with assms(3)   show ?thesis    by (auto simp add: e_less_eq_p_unwrap)qedlemma 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)    case goal1 note g1asms = this    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 -        case goal1        with assms(2) have            "∀x ∈ set (map (fst o (p_unwrap o snd)) (α s)). x < e"          by (simp add: e_less_eq_annot)        with goal1(3) show ?case          by(auto simp add: sorted_append)      next        case goal2        hence "annot s = listsum (map snd (α s))"           by (simp add: annot_correct)        with goal2        show ?case           by (auto simp add: e_less_eq_listsum2)      next        case goal3        hence "α s = []" by (auto simp add: isEmpty_correct)        thus ?case by simp      next        case goal4        hence "α s = []" by (auto simp add: isEmpty_correct)        with goal4(6) 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 + listsum (map snd (α s)))"        using T1 g1asms annot_correct        by (auto simp add: plus_def)      have         v4: "α s = α l @ ((), lp) # α r"          "¬ e_less_eq e (Infty + listsum (map snd (α l)))"        "e_less_eq e (Infty + listsum (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 + listsum (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 o (p_unwrap o snd)) (α l)). x < e"        using v4(4) v7 assms(2)        by(simp add: e_less_eq_annot)      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 o (p_unwrap o 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        by (auto simp add: sorted_append sorted_Cons)            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          by (auto simp add: aluprio_defs)        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          by (auto simp add: sorted_append sorted_Cons)                      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          by (auto simp add: aluprio_defs)        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          by (auto simp add: sorted_append sorted_Cons)                      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 goal2 note g1asms = this    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 -        case goal1        with assms(2) have            "∀x ∈ set (map (fst o (p_unwrap o snd)) (α s)). x < e"          by (simp add: e_less_eq_annot)        hence "e ∉ set (map fst ((map (p_unwrap o snd)) (α s)))"          by auto        thus ?case          by (auto simp add: map_of_distinct_upd)      next        case goal2        hence "α s = []" by (auto simp add: isEmpty_correct)        thus ?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 + listsum (map snd (α s)))"        using T1 g1asms annot_correct        by (auto simp add: plus_def)      have         v4: "α s = α l @ ((), lp) # α r"          "¬ e_less_eq e (Infty + listsum (map snd (α l)))"        "e_less_eq e (Infty + listsum (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 + listsum (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 o (p_unwrap o snd)) (α l)). x < e"        using v4(4) v7 assms(2)        by(simp add: e_less_eq_annot)      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 o (p_unwrap o snd)) (α r)). fst(p_unwrap lp) < x"        using v4(1) g1asms v8        by auto      hence v10: " ∀ x∈set (map (fst o (p_unwrap o 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          by (auto simp add: aluprio_defs)        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 \<mapsto> 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          by (auto simp add: aluprio_defs)        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 \<mapsto> a)"          using v12 v13 v16          by simp      qed    qed  qedqedsubsubsection "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))))"      by (auto simp add: aluprio_defs)    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 = 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           by (auto simp add: aluprio_defs)      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 o (p_unwrap o snd)) (α s))"          by auto        hence "map_of (map (p_unwrap o 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           by (auto simp add: aluprio_defs)      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 + listsum (map snd (α s)))"        using T1 sinv annot_correct        by (auto simp add: plus_def)      have         v4: "α s = α l @ ((), lp) # α r"          "¬ e_less_eq e (Infty + listsum (map snd (α l)))"        "e_less_eq e (Infty + listsum (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 + listsum (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 o (p_unwrap o snd)) (α l)). x < e"        using v4(4) v7 assms(2)        by(simp add: e_less_eq_annot)      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 o (p_unwrap o snd)) (α r)). fst(p_unwrap lp) < x"        using v4(1) sinv v8        by auto      hence v10: " ∀ x∈set (map (fst o (p_unwrap o 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 o 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          by (auto simp add: aluprio_defs)      next        case True        have v12: "map (p_unwrap o 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 o snd) (α s)) e = Some (snd (p_unwrap lp))"          using v12 by simp        thus ?thesis          using T1 True l_lp_r          by (auto simp add: aluprio_defs)      qed    qed  qedqedsubsubsection "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 ≠ empty"       "aluprio_pop splits annot app s = (e, a, s')"    hence v1: "α s ≠ []"      by (auto simp add: aluprio_defs)    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 = listsum (map snd (α s))"      by (auto simp add: annot_correct invs(1))    hence       v5:      "(Infty + listsum (map snd (α s))) ≤ annot s"      by (auto simp add: plus_def)    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 + listsum (map snd (α l)) ≤ annot s"      "Infty + listsum (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 (auto simp add: aluprio_defs)      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)      apply (auto simp add: plesseq_def)      done     note listsum_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      by (auto simp add: sorted_Cons sorted_append)        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  qedqed    lemmas aluprio_correct =  aluprio_finite_correct  aluprio_empty_correct  aluprio_isEmpty_correct  aluprio_insert_correct  aluprio_pop_correct  aluprio_prio_correct    end`