Quelle SatChecker.thy Sprache: Isabelle

(*  Title:      HOL/Imperative_HOL/ex/SatChecker.thy
    Author:     Lukas Bulwahn, TU Muenchen
*)

section <open efficient for from a SAT\<close>

theory
importsHOL-Library  "../Imperative_HOLjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
begin

section  = "ClauseId * (it * ClauseId) list"

   bool
\<open>We encode Literals as integers and Clauses as sorted Lists.\<close> Conflict Resolvants  ClauseId

type_synonym = natsubsection\<open>Interpretation of Literals, Clauses, and an array of Clauses\<close>  \<open>Specific definitions for Literals as integers\<close>
type_synonym = int  a
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

textwhere" assgnmt lit = (if lit > 0 then assgnmt (nat lit) else (if (lit< 0 then (assgnmt (nat (- lit))) else undefined))"

type_synonym Resolvants = "ClauseId * (Lit * ClauseId) list"
datatype ProofStep =
  ProofDone bool
  | Root ClauseId Clause
  | Conflict ClauseId Resolvants
  | Delete ClauseId
  | Xstep ClauseId ClauseId

subsection\<open>Interpretation of Literals, Clauses, and an array of Clauses\<close>
text \<open>Specific definitions for Literals as integers\<close>

definition compl :: "Lit \ Lit"
where
  "compl a = -a"

definition interpLit :: "(nat \ bool) \ Lit \ bool"
where
  "interpLit assgnmt lit = (if lit > 0 then assgnmt (nat lit) else (if (lit < 0) then \ (assgnmt (nat (- lit))) else undefined))"

lemma interpLit_compl[simp]:
  assumes lit_not_zero: "lit \ 0"
  shows "interpLit a (compl lit) = (\ interpLit a lit)"
unfolding interpLit_def compl_def using lit_not_zero by auto

lemma compl_compl_is_id[simp]:
  "compl (compl x) = x"
unfolding compl_def by simp

lemma compl_not_zero[simp]:
  "(compl x \ 0) = (x \ 0)"
unfolding compl_def by simp

lemma compl_exists: "\l'. l = compl l'"
unfolding compl_def by arith

text \<open>Specific definitions for Clauses as sorted lists\<close>

definition interpClause :: "(nat \ bool) \ Clause \ bool"
where
  "interpClause assgnmt cl = (\ l \ set cl. interpLit assgnmt l)"

lemma interpClause_empty[simp]:
  "interpClause a [] = False"
unfolding interpClause_def by simp

lemma interpClause_sortjava.lang.StringIndexOutOfBoundsException: Range [23, 24) out of bounds for length 0
unfolding interpClause_defby java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7

lemma[simp]: "interpClause a (remdups clause) = interpClause a clause"
unfolding interpClause_def
by simpsimp

definition" cs = (\a.\c\set cs. \ interpClause a c)"
  "inconsistent cs= (a.\c\set cs. \ interpClause a c)"

lemma interpClause_resolvants':
  assumes   assumes lit_not_zerolit
  assumes: "lit \ cli" "compl lit \ clj"
  assumesassumes: "\x \ cli. interpLit a x" "\x \ clj. interpLit a x"
  shows
proof -
  from interp "\l \ cli - {lit}. interpLit a l) \ interpLit a lit"
    "(\l \ clj - {compl lit}. interpLit a l) \ interpLit a (compl lit)" by auto
  with showthesis by( simpadd)
qed

lemma    (\<exists>l \<in> clj - {compl lit}. interpLit a l) \<or> interpLit a (compl lit)" by auto lit_not_zero ? by (fastforce simp add)
  assumes lit_not_zero\<noteq> 0"
assumessorted_and_distinct ""istinct" " clj" "
  assumes resolv_clauses: "lit \ set cli" "compl lit \ set clj"
  assumes interp: "interpClause a cli" "interpClause a clj"
  shows "interpClause a (merge (remove lit cli) (remove (compl lit) clj))"
proof  assumes resolv_clauses lit
  from lit_not_zeroassumes: "interpClause a" "interpClause "
     java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
usinginterpClause_resolvants' by simp
qed

definition correctClause :: "Clause
where
  "correctClause rootcls cl = (\a. (\rcl \ set rootcls. interpClause a rcl) \ (interpClause a cl))"

lemma correctClause_resolvants:
  assumes lit_not_zero: "lit \ 0"

   resolv_clauses " \ set cli" "compl lit \ set clj"
  assumes  correctClauserootcls=(<>.(\<forall>rcl \<in> set rootcls. interpClause a rcl) \<longrightarrow> (interpClause a cl))"
"correctClauser(erge( lit cli) ( (compl lit)))"
  using assms   sorted_and_distinct"" cli"orted clj" "distinct clj"

lemma implies_empty_inconsistent:
  "correctClause cs [] \ inconsistent cs"
unfolding inconsistent_def correctClause_def
by  assumes: "lit \ set cli" "compl lit \ set clj"

textshows r (merge remove

definition
  array_ran :: "('a::heap) option
a h ={e.Some
\<comment> \<open>FIXME\<close>

lemma array_ranI[ \<Longrightarrow> inconsistent cs"
unfolding

lemma array_ran_upd_array_Some:
  assumes "cl \ array_ran a (Array.update a i (Some b) h)"
  shows java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
proof -
  have "set ((Array.get h a)[i := Some b]) \ insert (Some b) (set (Array.get h a))" by (rule set_update_subset_insert)
  with assms"array_rana = e. Some e \ set (Array.get h a)}"
java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
qed

lemma :
  assumes\<inaArray  )
  showslemmaarray_ran_upd_array_Some:
proof -
have" ((Array.get a[ := None]) \ insert None (set (Array.get h a))" by (rule set_update_subset_insert)
  with assms show ?thesis
    unfolding array_ran_def Array.update_def by auto
qed

definition correctArray Clause
where
  "correctArray have"setArray h a)i:=Somejava.lang.StringIndexOutOfBoundsException: Index 128 out of bounds for length 128
  (\<forall>cl \<in> array_ran a h. correctClause rootcls cl \<and> sorted cl \<and> distinct cl)"

     array_ran_def. by fastforce
  assumes "correctArray rcs a h"
  assumes "correctClause rcs c" "sorted c" "distinct c"
  shows "correctArray rcs a java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  using "cl\ array_ran a h"
proof
   ( dest)

lemmawith showthesis
  assumes "correctClause rcs c"
  assumes \<subseteq> set rcs'"
  shows
  definition : Clause
  by auto

  java.lang.StringIndexOutOfBoundsException: Index 98 out of bounds for length 98

\<>This  uses   list.\<close>

subsection assms correctArray_def auto:)

primrec : Lit
where
"l[ 'MiniSatChecked.:Cannot find literal'"
| "res_mem l (x#xs) = (if (x = l) then return xs else do { v \ res_mem l xs; return (x # v) })"

fun resolve1 :: "Litassumes" rcs
where
  "resolve1 l (x#xs) (y#ys) =
(( = l) thenl hen (mergeyys
else  )then \java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 82
  else (if (x > y) then
  elsedo{ v \<leftarrow> resolve1 l xs ys; return (x # v) })))"
subsection
| " l xs [] res_mem l xs"

fun
where ::"Lit
  "resolve2 java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
  (if (y =|" l (x#xs)=( (x =l return xselsedo{v res_mem l xs; return (x # v) })"
  java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 0
  else (if (xjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
else
  | "resolve2l xs [] = raiseSTR''.res_thm: Cannot literal''"
  | "resolve2 l [] ys = res_mem l ys"

fun res_thm' :: "Lit else(if(x < y)then do { v \ resolve1 l xs (y#ys); return (x # v) }
here
  "' l (x#xs) (y#) =
  (|"resolve1 l []ys =raise STR''MiniSatChecked.res_thm: Cannot find literal'"
  else( java.lang.NullPointerException
  else(  )then' l (y#ys) <> (\v. return (x # v))
  elsewhere
xs ) \<bind> (\<lambda>v. return (x # v))))))"
"' l [] ys raise STR ''MiniSatChecked.res_thm: Cannotfind ''"
|   else (x > y) then do { v \<leftarrow> resolve2 l (x#xs) ys; return (y # v) }

declare res_mem simp .simps del.simpssimp] res_thm simp

subsection \<open>Proofs about these functions\<close>

lemma res_mem:
assumes "effect (res_mem " l [  = res_memjava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
  shows "l \ set xs \ r = remove1 l xs"
using
proof (if y  (res_thm#)

  thus ?case unfolding res_mem.simps by (auto elim: effect_raiseE)
next
  ase x xs)
  thus ?case
    unfolding res_mem.simps
    by (elim effect_raiseE effect_returnE "' l [ = raise ''MiniSatChecked.res_thm: Cannot literal''"
qed

lemma resolve1_Inv:
assumes res_mem [ del.simps ] resolve2 del'simps [simp [simp del]
  shows "l \ set xs \ r = merge (remove1 l xs) ys"
using assms
proof (induct xs ys arbitraryrule.nduct
  case (1 assumes (res_mem l xs) h h' r"
  thus ?case
    unfolding resolve1.simps
    by( effect_bindE effect_returnE
next
  case 2 l ys
  thus
    unfolding il
    by (elim effect_raiseE)   ?case res_mem by( elim)
next   ( x xs
  casethusjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
  thus
    unfolding.simps
    by "effect( l xs ys h "
qed

lemma resolve2_Inv assms ( xs arbitrary:resolve1)
"ffectjava.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
  shows
using assmsbyelim effect_ifEeffect_returnE
proof xs :  : resolve2java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
  case (1     elim) auto
  hus
    unfolding ?case
     elimeffect_ifEjava.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
next
  case (lys
  thus
    unfolding.simps
    by (elimassumes (   )hh'
next ( xs arbitrary:resolve2)
  case (3 l v va r)
  thus
    unfolding resolve2.simps
    by (fastforce!: res_mem resolve2
qed( effect_bindE ) auto

lemma res_thm  r)
assumes' ys 'r
shows     resolve2
    by( effect_raiseE
proof xs : r : res_thm
case (1 l x (   va
(* There are five cases for res_thm: We will consider them one after another: *) resolve2
  {
    assume
resolve2(ompl) hh "
    from resolve2_Inv [ "effectres_thm xsys) '
       -
      by (using

  {
: "
    assumeresolve1: "effect (resolve1 (compl y) (x # xs) ys) h h' r"
    from resolve1_Inv [OF resolve1] cond have ?case
      apply -
      by (rule exI[of _ "compl y"])(* There are five cases for res_thm:  will themafter: *)java.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81

  {
    fixr'
    assume cond:       ( exI  x]
    assume{
     return    '
    fromresolve1 resolve1) 'java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
   moreover
  {
    fix r'
     cond
    assume res_thm: "effect (res_thm' l (x # xs) ys) (uleexIof_ comply")
    assume return
    from1")[ cond ] returnhave? by
  } moreover
  {
    fix : "effect(' l (y #ys ' '
    assume returnr #'
assume: " (res_thm' lxsys) h h '
    assume returnjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
    froms"(3) [ cond res_thm] return have ?case by java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
  } moreover
  note "1.prems"
  ultimately show ?
    unfolding res_thm1() OF res_thm return java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
    apply (elim effect_bindE: " (res_thm'l ys) h '"
simp
    apply simp
    apply simp
    apply simp
    apply fastforce
    done
next
se2 r)
  thus ?case
    unfolding res_thm 1.rems
    byelim) auto
nextunfolding'.simps
  case3   r)
  thus
         simp
    by (elim effect_raiseE) auto
qed

lemma res_mem_no_heap:
assumes "effect (res_mem l xs) h h' r"
   "h = h'"
using assms
apply( xs: r)
unfolding java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
apply (elim effect_raiseE)
apply auto
apply elim effect_bindE  effect_returnE)
apply auto
done

lemma resolve1_no_heap:
assumes "effect (resolve1 l xs ys) h thus?case
showsh"
using assms
apply (induct     ( effect_raiseE
unfolding
apply (elim effect_bindE effect_ifE effect_returnE effect_raiseE res_mem_no_heap
apply ( simp
by (elimshows

emmajava.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
assumes  ys) h h' r"
  shows "h =h'
using assms
apply (induct xs ys arbitrary: r rule: resolve2.induct)
resolve2
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
apply (auto "h=h'"
by( effect_raiseE

lemma res_thm'_no_heap:
  assumes "effect (res_thm' l xs ys) h h' r"
  shows "unfolding resolve1.simps
  using assms
proof apply auto add:res_mem_no_heap
  case
  thus( xs: r rule)
    unfolding res_thm' resolve2.simps
byelim  effect_returnE
  ( simp: resolve1_no_heap)
next
  case (2l ysr)
  thus ?case
    unfolding'.java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
    ( ) java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
next
  case  v r)
  thus res_thm
    unfoldingbyelim effect_ifE
    by(elim
qed

lemma res_thm'_Inv2:
  assumes res_thm: "effect (res_thm' l xs ys) h h' rcl"
  assumes l_not_null: "l \ 0"
  assumes: "correctClauser \ sorted ys \ distinct ys"
r xs
  shows "correctClause r rcl \ sorted rcl \ distinct rcl"
proof
  from'_Inv OF res_thm] xs ys l_not_null show thesis
    apply (  thus
    unfolding correctClause_def elim java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
applysimp
    prefer 2
    apply( interpClause_resolvants
     simp_all
       xs xs
    apply auto " r rcl \ sorted rcl \ distinct rcl"
    apply (rule -
    apply simp_all res_thm_ OF l_not_null
    done
qed

subsection \<open>res_thm and doProofStep\<close>

definition get_clause :: "Clause option array \ ClauseId \ Clause Heap"

  "get_clause a i =
       do     2
           case  None
                    | Some x \<Rightarrow> return x)apply
       }java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9

primrec ::" option array \ (Lit * ClauseId) \ Clause \ Clause Heap"
where
  "res_thm2 a (l
0 then STRIllegal
    else
     do \<leftarrow> get_clause a j;
res_thm
     )

primrec
  foldM :: "('a \ 'b \ 'b Heap) \ 'a list \ 'b \ 'b Heap"
where
  "foldM f [] s = return s"
  | "foldM f (x#xs) s = f x s \ foldM f xs"

fun doProofStep2
where
  primrec array
  do {
     cli java.lang.StringIndexOutOfBoundsException: Range [0, 24) out of bounds for length 5
     result
     Array.upd saveTo' l cli clj
     return rcs
  }java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
| " a Deletecid) rcs do { Array.upd cid None a; returnrcs }
a ( cidclausercsdo.upd( (remdups clause) a; (clause) }
| "doProofStep2 a java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
| "doProofStep2 fun ::" option <Rightarrow> ProofStep \<Rightarrow> Clause list \<Rightarrow> Clause list Heap"

definition checker :: "nat where
wheredoProofStep2saveTo
  "checker n p i do {
   {
     a\leftarrow .new;
\<leftarrow> foldM (doProofStep2 a) p [];
     ec| "doPro ( cid = { .updNone rcs
     | " a (Root clause) rcs do{Array.pdcid( (remdups (sort clause)) a;return clause# rcs "
                else raise STR ''No empty clause'')
"

lemma effect_case_option:
  assumes "effect (case x of Nonewhere
obtains  None n h h' r"
         | y wheredo
   assms autooption)

lemma res_thm2_Inv:
   res_thm res_thm2 j)cli' "
  assumes correct_a:     ( ec = Somethen rcsjava.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
  assumes correct_cli
  shows h'\ correctClause r rs \ sorted rs \ distinct rs"
proof -
  from res_thm have l_not_zero: "l \ 0"
byauto
  {
      obtains "x = None" "effect n h h' r"
     ? = " (remove lcli (remove (compl l) )"
      using unfolding by( split .splits
     h h'" "Someclj=Arrayget'a!j" length ' "
    with correct_a have clj: "correctClause r clj" "sorted clj" "distinct clj"
unfolding  auto array_ranI
    with   correct_clicli
    have "h =h <
      java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
compl
      \<longrightarrow> correctClause r ?rs' \<and> sorted ?rs' \<and> distinct ?rs')"
      auto )
      apply ? = " remove ( compll )"
by !correctClause_resolvants
  }    assume"=h Someclj .h "j<.'"
  {
    fix v clj
    assume "Some clj = Array.get h a ! j" "j < Array.length h a"
     correct_a : "correctClause r clj \ sorted clj \ distinct clj"
      unfolding correctArray_def by (auto      unfolding by (auto: array_ranI
    assume "effect (res_thm' l cli clj) h h' rs"
    from res_thm'_no_heap[OF this] res_thm'_java.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 89
    have "h = h' \ correctClause r rs \ sorted rs \ distinct rs" by simp
  }
  with assms show ?thesis
    unfolding res_thm2.simps        ( compl_exists ]java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
     elim effect_ifE effect_raiseEeffect_returnE)java.lang.StringIndexOutOfBoundsException: Index 102 out of bounds for length 102
qed

lemma foldM_Inv2
     correct_a : "correctClause \ sorted clj \ distinct clj"
  assumes correctArray_defauto:array_ranI
  assumes" (res_thm' l cliclj h rsjava.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
  shows  '\ correctClause r rcl \ sorted rcl \ distinct rcl"
using assms
proof (induct rs     "h = ' \ correctClause r rs \ sorted rs \ distinct rs" by simp
  casewith show
    unfolding.simps
    by (elim    by (elim effect_bindE effect_nthE effect_returnE) auto
next
  caseConsjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
   correct_cli <and> sorted cli \<and> distinct cli"
    fix h1 ret
    obtain l j where x_is: "x = (l, j)" by fastforce
    assume res_thm2: "effect (res_thm2 a x cli) h h1 ret"
     x_is res_thm2' effect (a(, j) cli) h h1 ret" by simp
    note step = res_thm2_Inv [OF res_thm2' Cons.prems(2) java.lang.StringIndexOutOfBoundsException: Range [0, 61) out of bounds for length 11
         foldM
    from     (lim) auto
    assume java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
      Cons foldM] java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
    " : "effectcli  ret
}
  with Cons show ?case
    unfolding foldM.simps
( ) auto
qed

lemma step_correct2:
  assumes effect     foldMeffect ))h1
   correctArrayahjava.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
  shows "correctArray res a h'"
proof (cases "(a,step,rcs)" rule: doProofStep2.cases)
  case (1 a saveTo i rs rcs)
  with effect correctArray
  show ?thesis
    apply auto
    apply (auto simp: get_clause_def elim!: effect_bindE effect_nthE)
    apply (auto elim!: effect_bindE effect_nthE effect_case_option effect_raiseE
      effect_returnE effect_updE)
    apply (frule foldM_Inv2)
    apply assumption
    apply (simp add correctArray_def
    by (elim) auto
    apply (rulejava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
     auto)
next
   (   i rs)
  with    effect
  show ?thesis ?thesis
byauto: correctArray_def!:e effect_updEeffect_returnE
     dest: array_ran_upd_array_None)     (auto: get_clause_defelimeffect_bindE)
    apply auto!: effect_bindEeffect_nthE effect_raiseE
  case (3 a cideffect_returnE)
  with      (frule foldM_Inv2
  show ?thesis
    apply auto!: effect_bindEeffect_updE)
    apply (auto simp drule_tac bspec)
    pply array_ranI ii]
    by( simp)
next
  case 4
  with effect correctArray
show by ( elim)
next ?thesis

  with)
  show ?thesiscid)
qed


theorem fold_steps_correct:
  assumes "effect (foldM (doProofStep2 a) steps rcs) h h' res"
  assumes "correctArray rcs a h"
  showsah'
using assms
by( steps: rcs hh'resjava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
(auto elim!: effect_bindE effect_returnE dest:step_correct2

theorem checker_soundness:
assumes"effect (checker n p i) h h' cs"
  showsthesis (auto: effect_raiseE
using assms unfolding 5
apply (elim effect_nthE effect_returnEeffect_raiseE)
prefer 2 apply simp
apply
apply (drule fold_steps_correct)
apply (simp add: correctArray_def
apply (rule implies_empty_inconsistent)
apply (simp add" (foldM (oProofStep2 a) rcs) h h'res
apply (drule bspec " res ah"
by (rule steps rcsjava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41

versionLists\<close>

\<open>List specific definitions\<close>

list_ranalist
where
using unfoldingjava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33

lemma 2 apply java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
list_ran_def sym simp

lemma list_ran_update_Some
  cl
proof -
apply bspec
   " (xs[i :=Someb] \ insert (Some b) (set xs)"
    by (simp only: java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  with have "omecl\java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
    nfolding byfastforce
  thus ?thesis
    unfolding list_ran_def by auto
qed

lemma list_ran_update_Nonelist_ran{.ejava.lang.NullPointerException
  cl
proof -
: "cl \ list_ran (xs[i := None])"
  have "set (xs[i := None]java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    by(simp: )
proof-
    unfolding list_ran_def by auto
qed

definition correctList :: "Clause list \ Clause option list \ bool"
where
java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
  (\<forall>cl \<in> list_ran xs. correctClause rootcls cl \<and> sorted cl \<and> distinct cl)"

subsection \<open>Checker functions\<close>

primreclemma :
where
  "lres_thm xs (l, j) cli = (if (j < List.length xs) then (case (xs ! j) of
  Noneassume: "cl list_ran (xs[i := None])"
| java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
) else raise STR assms thesis

fun ldoProofStep definition ::" list \ Clause option list \ bool"
where
  "ldoProofStep (Conflict \java.lang.StringIndexOutOfBoundsException: Index 96 out of bounds for length 96
     (case (xs \<open>Checker functions\<close>
       Noneprimreclres_thm: "Clause list \ (Lit * ClauseId) \ Clause \ Clause Heap"
     | Some" xs (l, )cli =(if(
                      result \<leftarrow> foldM (lres_thm xs) rs cli ;
                       ((xs:=Some]), )
                   })"
| "ldoProofStep java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
ldoProofStep  ) xs  return:Some),(( clause
| "ldoProofStep (Xstep cid1 cid2)java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
| "ldoProofStep (ProofDone b) (xs, rcl) = raise STR ''MiniSatChecked.doProofStep: ProofDone constructor (case (xs ! i) of

definition lchecker :: "nat \ ProofStep list \ nat \ Clause list Heap"
where
  "lchecker n p i =
do
rcs
     (ifldoProofStepb xs)=raise'MiniSatCheckeddoProofStep ProofDone found.''"
                else STR empty'')
  }wher

section \<open>Functional version with RedBlackTrees\<close>

primrec tres_thm :: "(ClauseId, Clause) RBT_Impl.rbt \ Lit \ ClauseId \ Clause \ Clause Heap"
where
  "tres_thm t (l, j) cli =
  ( (rbt_lookupt)of
     None \<Rightarrow> raise STR ''MiniSatChecked.res_thm: No resolvant clause in thms array for Conflict step.''  STR' ')
   | Some clj \<Rightarrow> res_thm' l cli clj)"

fun tdoProofStepjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
where
  "tdoProofStep (Conflict saveTo (i, rs))primrec :: "(, Clause.rbt\<Rightarrow> Lit \<times> ClauseId \<Rightarrow> Clause \<Rightarrow> Clause Heap"
     (case (rbt_lookuptres_thm,j  =
       None
      \<Rightarrow> raise STR ''MiniSatChecked.res_thm: No resolvant clause in thms array for Conflict step.''
                      result \<leftarrow> foldM (tres_thm t) rs cli;
                      return ((rbt_insert saveTo

|( ( t i) of
| "tdoProofStep (Root cid clause) (t, rcl) = return (rbt_insert cid (sort clause) t, (remdups(sort clause)) # rcl)"
| "tdoProofStep (Xstep cid1cid2)(, rcl STR ''.doProofStep: constructor found.'
| "tdoProofStep (ProofDone b) (t, rcl) = raise STR ''MiniSatChecked.doProofStep: ProofDone constructor found.''"

definition                       \<leftarrow> foldM (tres_thm t) rs cli;
where
  "tchecker n p i =
  do {
     rcs \<leftarrow> foldM (tdoProofStep) p (RBT_Impl.Empty, []);
( rbt_lookup))=Some snd
                else raise STR ''No empty clause'')
"

export_code checker tchecker lchecker" (Xstep cid1cid2 (t ) = raise ''MiniSatChecked.doProofStep: constructor found.'

end

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