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products/sources/formale Sprachen/Isabelle/HOL/TPTP/CASC/ (Beweissystem Isabelle Version 2025-1^©) Datei vom 16.11.2025 mit Größe 3 kB

Quellcode-Bibliothek SatChecker.thy Sprache: Isabelle

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

section\open>An checker proofs solver

theory SatChecker ".RBT_Impl" Sorted_List"
imports "HOL-Library.RBT_Impl" Sorted_List "../Imperative_HOL"
begin

section\<open>General settings and functions for our representation of clauses\<close>

subsection\<open>Types for Literals, Clauses and ProofSteps\<close>
text \<open>We encode Literals as integers and Clauses as sorted Lists.\<close>

type_synonym ClauseId = nat
type_synonym Lit = int
type_synonym Clause = "Lit list"

text \<open>This resembles exactly to Isat's Proof Steps\<close>

type_synonym Resolvants*(it
datatype
ProofDone
text
  |  ClauseId
  |Delete
  | Xstep ClauseId ClauseId ClauseId

subsection
text

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

definition interpLit :: "(nat \ bool) \ Lit \ bool"

  interpLitassgnmt 0java.lang.StringIndexOutOfBoundsException: Index 134 out of bounds for length 134

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_sort[simp]: "interpClause a (sort clause) = interpClause a clause"
unfolding
bysimp

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

definition
  inconsistent

lemma interpClause_resolvants  java.lang.NullPointerException
: "lit \ 0"
  assumes resolv_clauses
   interp
  shows  "\x \ (cli - {lit} \ (clj - {compl lit})). interpLit a x"
proof   resolv_clauses have(l
  from resolv_clauses lit_not_zero ?thesis fastforce : bex_Un
"
  with showthesis simp: bex_Un
qed

lemma interpClause_resolvants:
  assumes: "lit \ 0"
  assumes : "sortedcli " clisorted "distinctcljjava.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86
assumes:" \ set cli" "compl lit \ set clj"
   interp cli aclj
  shows "interpClause a (merge (remove unfoldinginterpClause_def
proof -
  from lit_not_zero resolv_clauses interp sorted_and_distinct show    using interpClause_resolvants simp

    java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
qed

assumes:lit
where
" cl =\foralla \rcl \ set rootcls. interpClause a rcl) \ (interpClause a cl))"

lemma correctClause_resolvants:
  assumes lit_not_zero: "lit \ 0"  shows  ( removeremove)clj
  assumes: "sorted cli "distinct" "java.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86
resolv_clauses
  assumes
   "correctClause (remove lit cli)(remove (compl lit) clj))"
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma   "array_ran e  e \<in> set (Array.get h a)}"
  "correctClause cs ]\ inconsistent cs"
unfolding inconsistent_def correctClause_def
by auto

text \<open>Specific definition for derived clauses in the Array\<close>

definition
  array_ran ::proofjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
   h{e
        unfolding array_ran_def Array.update_def by fastforce

lemma array_ranI
unfoldingarray_ran_upd_array_None "cl <> array_ran (.update a iNoneh)

:
  proof
   seth)ijava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
proof ::" list \ Clause option array \ heap \ bool"
  ((.get[   b]) <subseteq> insert (Some b) (set (Array.get h a))" by (rule set_update_subset_insert)
  with
unfolding Arrayupdate_defjava.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
qed

lemma array_ran_upd_array_None:
  assumes "cl \ array_ran a (Array.update a i None h)"
  shows
proof -
byauto:array_ran_upd_array_Some
   assms ?java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
    unfolding "set rcs\ set rcs'"
qed

correctArray:" list \ Clause option array \ heap \ bool"


(<forall>cl \<in> array_ran a h. correctClause rootcls cl \<and> sorted cl \<and> distinct cl)"

lemma correctArray_update:
  assumes "correctArray rcs a h"
  textopen version justasingle traversal
  shows "correctArray
  using
  unfolding
  by( destarray_ran_upd_array_Some

lemma res_mem: " \ Clause \ Clause Heap"
  assumes  "res_mem [ =raiseSTR''res_thm literal'java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76
   set \<subseteq> set rcs'"
  shows "java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 5
  using  (f x = )t return xs (#ys))
  by   (if (x<y  do {v <leftarrow> resolve1 l xs (y#ys); return (x # v) }

section\<open>Improved version of SatChecker\<close>

text\<open>This version just uses a single list traversal.\<close>

subsection\<open>Function definitions\<close>|resolve1=res_mem

primrec res_mem "Lit \ Clause \ Clause Heap"
where
  "res_mem l [] = raise STR ''MiniSatChecked.res_thm: Cannot find literal''"
res_mem  if=)then    v\<

fun resolve1 :: "Lit \ Clause \ Clause \ Clause Heap"
where
  "resolve1 l (x#xs) (y#ys) =
  (if (x = l   do {v\<leftarrow> resolve2 l xs ys; return (x # v) })))" xs  MiniSatCheckedfind
  < do
  else (if (x >w
  elseres_thmys
|l =  MiniSatChecked'
| "resolve1 l (if ( =l y = compl l) then resolve1 (compl y) (x#xs) ys

fun (if x<y then (res_thmxs#))\bind
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
  "resolve2 l (x#xs) (y#ys) =
  (if (y = l)   else (res_thm' lys
  else|res_thm=raise literal
(if
  else.simps[ del]resolve1 [simp] resolve2 [simp delres_thm'.simps[ del]
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
|resolve2]ys l ys"

fun res_thm' :: "Lit \ Clause \ Clause \ Clause Heap"
where
  "res_thm' l (x#xs) (y#ys) =
  (if (x = l \<or> x = compl l) then resolve2 (compl x) xs (y#ys)
  else using assms
  else (x <) then' l xs (y#s)) \ (\v. return (x # v))
  else (if
c (Cons'java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
|res_thm]ysSTRMiniSatChecked find
| "res_thm' l xs [] = raise STR ''java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 3

declare.simpssimp] resolve1 [simpdel.simps [simp] res_thm. java.lang.StringIndexOutOfBoundsException: Index 110 out of bounds for length 110

subsection: r : resolve1.nduct)

lemma res_mem:
"effectjava.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
  shows elim effect_ifE) auto
using assms( r)
proof   ?case
  caseNjava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
  thus unfolding.simpsby auto: effect_raiseE
next
  caseCons')
  thus ?case
    unfolding res_mem.simps
    by (elim effect_raiseE effect_returnE effect_ifE effect_bindE) auto
qed

lemma     unfolding resolve1
assumes resolve1)hh'r
  java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
using
proofinduct ys: r rule resolve1.induct
  case (1 l   assumes"ffect (resolve2 l xs ys) h h' r"
  thus ?case
    unfolding "l \ set ys \ r = merge xs (remove1 l ys)"
     ( effect_bindE ) auto
next (induct ysarbitraryrrule.induct)
  case (2 l ys r)
  thus ?case
    unfolding resolve1.simps
by( effect_raiseE
next
  case (hus ?case
  thus
    unfolding resolve1.simpsby( effect_bindE  effect_returnE) auto
    by (fastforce
qed( l ys r)

resolve2
   "effect resolve2lxsys h r"
  shows "l \ set ys \ r = merge xs (remove1 l ys)"
using assms
proofinduct ys: r rule .induct
  case  ?case
  thus dest)
    unfolding resolve2.simps
    by elim effect_ifEeffect_returnE
next
  case (2 lys
  thus "effect (res_thm l xs ys)hh'"
    unfolding.simps
by (lim) auto
next (induct ys arbitraryrule'.induct)
  case(lv  r)
  thus ?case
    unfolding .simps
    by (fastforce dest!: res_mem)
qed

lemma    assume: "effect (resolve2 ( x) xs (y # ys) h 'r
assumes ('l xs ys)hh r"
shows      apply-
using assms
proof (induct   }moreover
case (1 l     assumecond\<not> (x = l \<or> x = compl l)" "y = l \<or> y = compl l" assume java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
We consider one after another
  {
    assume cond: "x = l \ x = compl l"
    assume resolve2: "effect (resolve2 (compl x) xs (y # ys)) h h' r"
    from resolve2_Inv [OF resolve2] cond have
     rjava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
byrule[of _ "") fastforce
  } moreover
  {
    assumeassumereturn: "r=x#r"
    assume resolve1: "effect (resolve1 (compl y) (x # xs) ys) hh r"
    from resolve1_Inv [OF resolve1]  }moreover
      apply -assume: "\ (x = l \ x = compl l)" "\ (y = l \ y = compl l)" "\ x < y" "y < x"
by( [ _" ])fastforce
  } moreover
  {
    fix     ".hyps(2 OFcondres_thmcond return case by auto
    assume cond
    assume res_thm res_thmxs ))hh r"
    assumereturn: "r =x #r'"
    from "1.hyps"(1)     res_thmeffect  ys'r"
  } moreover
  {
    fix r'
    assume cond OFcondauto
    assume res_thm: "effect (res_thm' l (x
    assume ultimatelyshowcase
    from ".hyps"2[OF cond] cond have ?case byauto
  } moreover
  {
    fix r'
    assume cond: "\ (x = l \ x = compl l)" "\ (y = l \ y = compl l)" "\ x < y" "\ y < x"
    assume res_thmeffect lxsh h'r'"
        applyjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
    fromnext
  } se (2l ysjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
  note"."
  ultimately ( effect_raiseE
     res_thm
     ( lv va
    apply simp
    applyjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
    applyshows
     induct arbitrary
    apply fastforce
    done
next
  case (2 l ys r)
  thus ?case
    unfolding
    byapply( effect_ifEeffect_raiseE
next
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   case
    unfolding    "h = h'java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
byelim) auto
qed

lemma:
assumes "effect (res_mem l xs) hauto add: res_mem_no_heap)
   "h = h'"
using
applyl resolve2_no_heap:
unfolding res_mem.simps
apply (elim effect_raiseE)
apply auto
apply (elim effect_ifE effect_bindE effect_raiseE effect_returnE "effect (resolve2 l xsh '
apply auto
doneunfolding.simps

lemma resolve1_no_heap:
assumes "effect (resolve1 l xs ys) h h' r"
  showsh  '"
using elim) auto
apply (inductjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 0
.
applyusingassms
( simp )
by (elim effect_raiseE) auto

lemma resolve2_no_heap:
assumes "effect (resolve2 l xs ys) h h' r"
  shows "h = h'"
usingassms
apply (nduct ys arbitrary: resolve2.induct
unfolding
apply      ( effect_bindEeffect_ifE)
apply (auto simp(uto add resolve2_no_heap
bycase2

     res_thmsimps
  assumes "effect (res_thm' l xs ys) h h' r"
  shows "h = h'"
  using assms
proof (inductby elimeffect_raiseEauto
  case (1 l x xs y ys r)
  thus ?thesis (3l v va
    unfolding'.simps
     (elim effect_bindE effect_returnE)
  (auto simp add  effect_raiseE) auto
next
  case java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  thus ys ys
    unfolding   assumes xs: "correctClause \<and> sorted xs \<and> distinct xs"
    by -
next res_thm[ res_thmys?
  case (3 l v va r)
?case
    unfolding res_thm'.simps
    by ( effect_raiseE)auto
qed

lemma res_thm'_Inv2:
  assumes res_thm: "effect (res_thm' l xs ys) h h' (auto add: remove1_eq_remove)
  assumes rule)
  assumes ysapply simp_all
assumes: "correctClause r \ sorted xs \ distinct xs"
  showscorrectClause \<and> sorted rcl \<and> distinct rcl"
proof
  from'_nv[ res_thm] xsys show ?thesis
    apply (
    java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    apply (auto simp where
prefer
    apply            ( cof \<Rightarrow> raise STR ''Clause not found''
     simp_all
    apply (insert compl_exists       "
    apply
    apply (rule res_thm2 Clausearray
    apply simp_all
one
qed

  ( if l = raiseSTR '' literal''

definition get_clause { clj
where
  "get_clause ' l cli clj
       }"
           (case cjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
                    |
       }"

res_thm2 :: "Clauseoption \ (Lit * ClauseId) \ Clause \ Clause Heap"
where
  "res_thm2 a (l, j) cli =
  ( if l = 0 then raise STR ''Illegal literal''
    else
     do { clj \<leftarrow> get_clause a j; \<leftarrow> foldM (res_thm2 a) rs cli;
         res_thm"
     })"

primrecdoProofStep2( =  }
  | "doProofStep2Root ) rcs = do { Array cid Some (sortclause))a return # rcs"
where
  "foldM f [] s = return s"
  | "foldM f (x#xs) s = f x s \ foldM f xs"

fundoProofStep2 Clause arrayjava.lang.StringIndexOutOfBoundsException: Index 122 out of bounds for length 122
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
  "doProofStep2 a (Conflict saveTo (i, rs)) rcs =
  do
  do
     result <>Array n None
     Array.     rcs
     return rcs
  }"
ofStep2a Delete) rcs= do Array cid  a; return }"
doProofStep2cid=  .  Some)  ( # )}java.lang.StringIndexOutOfBoundsException: Index 119 out of bounds for length 119
| "doProofStep2 a (Xstep cid1 }java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
| "doProofStep2 a (ProofDone b) rcs = raise STR ''MiniSatChecked.doProofStep: ProofDone constructor found.''"

definition checker :: "nat \ ProofStep list \ nat \ Clause list Heap"

  "checker n p i "x=" "effect
   {
     a \<leftarrow> Array.new n None;
     rcs \<leftarrow> foldM (doProofStep2 a) p [];using unfolding effect_defby (auto split: option.splits
  assumes: "effect( a (l, ) h h'rs
if  []  return
                else raise STR ''No empty clause'')
  }"

lemma effect_case_option "h =h
  assumes "effect (case x of ( elim: effect_raiseE)

         | y whereletrsmergel )(removeclj
  using assms effect_def (auto:option)

lemma res_thm2_Inv:
  assumesassume" =Some = . h a ! " j < Array.h'ajava.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
  assumes correct_a       correctArray_defby( intro:)
assumes: "correctClause r cli \ sorted cli \ distinct cli"
  shows ='\correctClauserrs\ sorted rs \ distinct rs"
-
  from res_thm have l_not_zero: "l \ 0"
    by (auto       (compl l \<in> set cli \<and> l \<in> set clj
  {
    apply ( intro!: correctClause_resolvants
    letrs"merge (remove lcli) (emove(compl )clj"
    let ?rs' (auto intro!: )
assume " h'"" clj =Arrayget 'a!j "  Arraylength h' ajava.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
    withwith havecljcorrectClause
correctArray_def intro)
    with clj l_not_zero correct_cli
    have "(l \ set cli \ compl l \ set clj
      <longrightarrow> correctClause r ?rs \<and> sorted ?rs \<and> distinct ?rs) \<and>
      (compl l \<in> set cli \<and> l \<in> set clj
      \<longrightarrow> correctClause r ?rs' \<and> sorted ?rs' \<and> distinct ?rs')"
      apply (auto intro!: java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 3
apply(insert [ofl)
      by (auto intro!: correctClause_resolvants)
  }
  {
    fix v cljby( effect_bindE effect_nthE  effect_case_option auto
    lemma:
with havecljrclj
      unfolding by ( intro )
    assume effect )h'"
    from res_thm'_no_heap[OF this] res_thm'_Inv2[OF this l_not_zero "h =h'
haveh
  }
   assms ?thesis
    unfolding res_thm2 foldMsimps
  effect_ifE effect_raiseE effect_case_option
qed

lemma foldM_Inv2 ( x xs)
  assumes "effect (foldM (res_thm2 a) rs cli) h h' rcl"
  assumes correct_a: "correctArray r a h"
assumes: "correctClause r cli\
      with haveres_thm2'"(res_thm2 (java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
using assms
proof (induct rs arbitrary: h h' cli)
  case Nil thus ?case
    unfolding.simps
    by ( effect_returnE
next
  case (Cons x xs)
  {
    fix h1 ret
    obtain l j where x_is: "x = (l, j)" by fastforcefromstep.hyps [OF correct_a rethave
assumeres_thm2 (res_thm2 a x cli) hh1"
    with  java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
    note step =     by elimeffect_bindE

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
assume: " (foldM (res_thm2 a xs ret h1 h' rcl"
    fromassumes: "correctArray rcs h"
    "h = h' \ correctClause r rcl \ sorted rcl \ distinct rcl" by auto
  }
  with Cons show ?case
    unfolding:)
by effect_bindE
qed

lemma step_correct2:
  assumes effect: "effect (doProofStep2 a step rcs) h h' res"
  assumes correctArray: "correctArray rcs a h"
  shows "correctArray res a h'"
proof (casesby( intro: correctArray_update
case(asaveTo rcs
with correctArray
  show
         ( simp elim ffect_bindE effect_returnE
apply simp !:  effect_nthE
    apply( elim  effect_case_option
       effect_updE
applyfrule)
    apply assumption
    apply (simp( elim  effect_returnE
    apply ( x="y" in)
a (rule[where=]
    by (auto auto: correctClause_def
next
  case (2 a cid rcs)
  with effect    ?thesisauto: effect_raiseE
  show
    by (auto simp  case5
     dest: array_ran_upd_array_None
next
  case (3 a  c rcs
  with effect correctArray
  showjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    apply (auto elim!: effect_bindE "correctArray res h'
    apply (auto simp induct arbitrary h )
    apply (auto intro: correctClause_mono)
    by (auto simp: correctClause_def)
next
  case 4
   effect
? by elim)
next
  case5
  with effect effect_bindE effect_ifE  effect_newE
  show ?thesis by (auto elim: effect_raiseEauto
qed


theorem fold_steps_correct:
  assumes effect (oProofStep2steps h'"
  assumes "correctArray rcs a h"
  showscorrectArray '
using assms
by (induct arbitrary: rcs h h' res)
(auto elim!: effect_bindE effect_returnE dest:step_correct2)

theorem checker_soundness with Lists as Array
  assumessubsection\<open>List specific definitions\<close>
  shows on :: "' option list \ 'a set"
assms checker_def
apply (elim effect_bindE effect_nthE effect_ifE effect_returnE effect_raiseE effect_newE)
prefer simp
apply auto
apply (druleunfolding by (drule, )
apply (simplist_ran_update_Some:
apply  " \ list_ran (xs[i := (Some b)]) \ cl \ list_ran xs \ cl = b"
apply (simp add:correctArray_def)
(drule)
byhaveset  b)\<subseteq> insert (Some b) (set xs)"

section \<open>Functional version with Lists as Array\<close>

subsection assms" insert (Some b) (set xs)"

definition list_ranu list_ran_def java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
where
  "list_ran xs = e Some set xs }"

lemma" \ list_ran (xs[i := None]) \ cl \ list_ran xs"
unfolding list_ran_def by  assumeassmscl

lemma list_ran_update_Some:
       ( onlyset_update_subset_insert
proof java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
  assume assms: "cl \ list_ran (xs[i := (Some b)])"
  have "set (xs[i := Some b]) \ insert (Some b) (set xs)"
    by (simp only: set_update_subset_insert)
  with assms have "Some s =
    unfolding list_ran_def by fastforce
  thus ?thesis
    unfolding list_ran_def by auto
qed

lemmalist_ran_update_None
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
proof -
   assmsjava.lang.NullPointerException
  have "set (xs[i Someclj res_thm' l cli clj
    by (simp only: set_update_subset_insert)
  with assms show?
    unfolding list_ran_def
qed

correctList Clause\<Rightarrow> Clause option list \<Rightarrow> bool"
where
  "correctList rootcls xs =
(<forall>cl \<in> list_ran xs. correctClause rootcls cl \<and> sorted cl \<and> distinct cl)"

subsection

  :option
where
  lres_thmj    j  . )then  j)of
  None \<Rightarrow> raise STR ''MiniSatChecked.res_thm: No resolvant clause in thms array for Conflict step.''
  | Somereturn[saveTo resultrcl
) else raise

fun |" (Rootcid clause) (, rcl)= (xs[cid= (sort clause] remdupssort)) # rcl)"
where
  "ldoProofStep (Conflict saveTo (i, rs)) (xs, rcl) =
casexs
       None \<Rightarrow> raise STR ''MiniSatChecked.doProofStep: No starting clause in thms array for Conflict step.''
     | Some cli \<Rightarrow> do {
                      result \<leftarrow> foldM (lres_thm xs) rs cli ;
                      return ((xs[saveTo:=Some result]), rcl)
                   })"
| "ldoProofStep (Delete cid) (xs, rcl) = return (xs[cid:=None], rcl)"
| "ldoProofStep (Root cid clause) (xs, rcl) = return do {
| "ldoProofStep \ foldM (ldoProofStep) p ([], []);
| " (ProofDone )(, rcl) STR '.:constructor

definition lchecker raise ''No clause
where
  "lchecker n p i =
  do {
     rcs \<leftarrow> foldM (ldoProofStep) p ([], []);
     (if (fst rcs !  (caserbt_lookup t j
                else raise 'Noemptyclause'java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
  }"

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

tres_thmClauseId) RBT_Impl \<Rightarrow> Lit \<times> ClauseId \<Rightarrow> Clause \<Rightarrow> Clause Heap"
where
  " t (l,)cli=
  (case (rbt_lookup \<Rightarrow> raise STR ''MiniSatChecked.doProofStep: No starting clause in thms array for Conflict step.''
     None
   | Some clj \<Rightarrow> res_thm' l cli clj)"

fun tdoProofStep :: " ProofStep \ ((ClauseId, Clause) RBT_Impl.rbt * Clause list) \ ((ClauseId, Clause) RBT_Impl.rbt * Clause list) Heap"
where
  "tdoProofStep (Conflict saveTo (i, })"
     case(bt_lookup java.lang.StringIndexOutOfBoundsException: Range [30, 31) out of bounds for length 30
       None \<Rightarrow> raise STR ''MiniSatChecked.doProofStep: No starting clause in thms array for Conflict step.''  ()=raiseMiniSatCheckedXstep'"
     | Some cli \<Rightarrow> do {
                      result
                      return ((rbt_insert saveTo result t), rcljava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
                   })"
| "tdoProofStep (Delete cid) (t, rcl) = return ((rbt_delete cid if ( (fst rcs)i [] then return ( rcs)
| "tdoProofStep (Root cid clause) (t, rcl) = return (rbt_insert cid (sort clause) t, (remdups(sort clause)) # rcl) }java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
| tdoProofStep ),rclSTRMiniSatCheckedXstep'"
| "tdoProofStep (ProofDone b) (t

where
  "tchecker n p i =
  do {
     rcs \<leftarrow> foldM (tdoProofStep) p (RBT_Impl.Empty, []);
     (if (rbt_lookup (fst rcs) i) = Some [] then return (snd rcs)
                else raise STR ''No empty clause'')
  }"

export_code checker tchecker lchecker checking SML

end

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