src/HOL/Tools/Qelim/cooper.ML
author haftmann
Wed May 05 18:25:34 2010 +0200 (2010-05-05)
changeset 36692 54b64d4ad524
parent 36527 68a837d1a754
child 36717 2a72455be88b
permissions -rw-r--r--
farewell to old-style mem infixes -- type inference in situations with mem_int and mem_string should provide enough information to resolve the type of (op =)
     1 (*  Title:      HOL/Tools/Qelim/cooper.ML
     2     Author:     Amine Chaieb, TU Muenchen
     3 *)
     4 
     5 signature COOPER =
     6  sig
     7   val cooper_conv : Proof.context -> conv
     8   exception COOPER of string * exn
     9 end;
    10 
    11 structure Cooper: COOPER =
    12 struct
    13 
    14 open Conv;
    15 open Normalizer;
    16 
    17 exception COOPER of string * exn;
    18 fun simp_thms_conv ctxt =
    19   Simplifier.rewrite (Simplifier.context ctxt HOL_basic_ss addsimps @{thms simp_thms});
    20 val FWD = Drule.implies_elim_list;
    21 
    22 val true_tm = @{cterm "True"};
    23 val false_tm = @{cterm "False"};
    24 val zdvd1_eq = @{thm "zdvd1_eq"};
    25 val presburger_ss = @{simpset} addsimps [zdvd1_eq];
    26 val lin_ss = presburger_ss addsimps (@{thm dvd_eq_mod_eq_0} :: zdvd1_eq :: @{thms zadd_ac});
    27 
    28 val iT = HOLogic.intT
    29 val bT = HOLogic.boolT;
    30 val dest_numeral = HOLogic.dest_number #> snd;
    31 
    32 val [miconj, midisj, mieq, mineq, milt, mile, migt, mige, midvd, mindvd, miP] =
    33     map(instantiate' [SOME @{ctyp "int"}] []) @{thms "minf"};
    34 
    35 val [infDconj, infDdisj, infDdvd,infDndvd,infDP] =
    36     map(instantiate' [SOME @{ctyp "int"}] []) @{thms "inf_period"};
    37 
    38 val [piconj, pidisj, pieq,pineq,pilt,pile,pigt,pige,pidvd,pindvd,piP] =
    39     map (instantiate' [SOME @{ctyp "int"}] []) @{thms "pinf"};
    40 
    41 val [miP, piP] = map (instantiate' [SOME @{ctyp "bool"}] []) [miP, piP];
    42 
    43 val infDP = instantiate' (map SOME [@{ctyp "int"}, @{ctyp "bool"}]) [] infDP;
    44 
    45 val [[asetconj, asetdisj, aseteq, asetneq, asetlt, asetle,
    46       asetgt, asetge, asetdvd, asetndvd,asetP],
    47      [bsetconj, bsetdisj, bseteq, bsetneq, bsetlt, bsetle,
    48       bsetgt, bsetge, bsetdvd, bsetndvd,bsetP]]  = [@{thms "aset"}, @{thms "bset"}];
    49 
    50 val [miex, cpmi, piex, cppi] = [@{thm "minusinfinity"}, @{thm "cpmi"},
    51                                 @{thm "plusinfinity"}, @{thm "cppi"}];
    52 
    53 val unity_coeff_ex = instantiate' [SOME @{ctyp "int"}] [] @{thm "unity_coeff_ex"};
    54 
    55 val [zdvd_mono,simp_from_to,all_not_ex] =
    56      [@{thm "zdvd_mono"}, @{thm "simp_from_to"}, @{thm "all_not_ex"}];
    57 
    58 val [dvd_uminus, dvd_uminus'] = @{thms "uminus_dvd_conv"};
    59 
    60 val eval_ss = presburger_ss addsimps [simp_from_to] delsimps [insert_iff,bex_triv];
    61 val eval_conv = Simplifier.rewrite eval_ss;
    62 
    63 (* recognising cterm without moving to terms *)
    64 
    65 datatype fm = And of cterm*cterm| Or of cterm*cterm| Eq of cterm | NEq of cterm
    66             | Lt of cterm | Le of cterm | Gt of cterm | Ge of cterm
    67             | Dvd of cterm*cterm | NDvd of cterm*cterm | Nox
    68 
    69 fun whatis x ct =
    70 ( case (term_of ct) of
    71   Const("op &",_)$_$_ => And (Thm.dest_binop ct)
    72 | Const ("op |",_)$_$_ => Or (Thm.dest_binop ct)
    73 | Const ("op =",ty)$y$_ => if term_of x aconv y then Eq (Thm.dest_arg ct) else Nox
    74 | Const (@{const_name Not},_) $ (Const ("op =",_)$y$_) =>
    75   if term_of x aconv y then NEq (funpow 2 Thm.dest_arg ct) else Nox
    76 | Const (@{const_name Orderings.less}, _) $ y$ z =>
    77    if term_of x aconv y then Lt (Thm.dest_arg ct)
    78    else if term_of x aconv z then Gt (Thm.dest_arg1 ct) else Nox
    79 | Const (@{const_name Orderings.less_eq}, _) $ y $ z =>
    80    if term_of x aconv y then Le (Thm.dest_arg ct)
    81    else if term_of x aconv z then Ge (Thm.dest_arg1 ct) else Nox
    82 | Const (@{const_name Rings.dvd},_)$_$(Const(@{const_name Groups.plus},_)$y$_) =>
    83    if term_of x aconv y then Dvd (Thm.dest_binop ct ||> Thm.dest_arg) else Nox
    84 | Const (@{const_name Not},_) $ (Const (@{const_name Rings.dvd},_)$_$(Const(@{const_name Groups.plus},_)$y$_)) =>
    85    if term_of x aconv y then
    86    NDvd (Thm.dest_binop (Thm.dest_arg ct) ||> Thm.dest_arg) else Nox
    87 | _ => Nox)
    88   handle CTERM _ => Nox;
    89 
    90 fun get_pmi_term t =
    91   let val (x,eq) =
    92      (Thm.dest_abs NONE o Thm.dest_arg o snd o Thm.dest_abs NONE o Thm.dest_arg)
    93         (Thm.dest_arg t)
    94 in (Thm.cabs x o Thm.dest_arg o Thm.dest_arg) eq end;
    95 
    96 val get_pmi = get_pmi_term o cprop_of;
    97 
    98 val p_v' = @{cpat "?P' :: int => bool"};
    99 val q_v' = @{cpat "?Q' :: int => bool"};
   100 val p_v = @{cpat "?P:: int => bool"};
   101 val q_v = @{cpat "?Q:: int => bool"};
   102 
   103 fun myfwd (th1, th2, th3) p q
   104       [(th_1,th_2,th_3), (th_1',th_2',th_3')] =
   105   let
   106    val (mp', mq') = (get_pmi th_1, get_pmi th_1')
   107    val mi_th = FWD (instantiate ([],[(p_v,p),(q_v,q), (p_v',mp'),(q_v',mq')]) th1)
   108                    [th_1, th_1']
   109    val infD_th = FWD (instantiate ([],[(p_v,mp'), (q_v, mq')]) th3) [th_3,th_3']
   110    val set_th = FWD (instantiate ([],[(p_v,p), (q_v,q)]) th2) [th_2, th_2']
   111   in (mi_th, set_th, infD_th)
   112   end;
   113 
   114 val inst' = fn cts => instantiate' [] (map SOME cts);
   115 val infDTrue = instantiate' [] [SOME true_tm] infDP;
   116 val infDFalse = instantiate' [] [SOME false_tm] infDP;
   117 
   118 val cadd =  @{cterm "op + :: int => _"}
   119 val cmulC =  @{cterm "op * :: int => _"}
   120 val cminus =  @{cterm "op - :: int => _"}
   121 val cone =  @{cterm "1 :: int"}
   122 val cneg = @{cterm "uminus :: int => _"}
   123 val [addC, mulC, subC, negC] = map term_of [cadd, cmulC, cminus, cneg]
   124 val [zero, one] = [@{term "0 :: int"}, @{term "1 :: int"}];
   125 
   126 val is_numeral = can dest_numeral;
   127 
   128 fun numeral1 f n = HOLogic.mk_number iT (f (dest_numeral n));
   129 fun numeral2 f m n = HOLogic.mk_number iT (f (dest_numeral m) (dest_numeral n));
   130 
   131 val [minus1,plus1] =
   132     map (fn c => fn t => Thm.capply (Thm.capply c t) cone) [cminus,cadd];
   133 
   134 fun decomp_pinf x dvd inS [aseteq, asetneq, asetlt, asetle,
   135                            asetgt, asetge,asetdvd,asetndvd,asetP,
   136                            infDdvd, infDndvd, asetconj,
   137                            asetdisj, infDconj, infDdisj] cp =
   138  case (whatis x cp) of
   139   And (p,q) => ([p,q], myfwd (piconj, asetconj, infDconj) (Thm.cabs x p) (Thm.cabs x q))
   140 | Or (p,q) => ([p,q], myfwd (pidisj, asetdisj, infDdisj) (Thm.cabs x p) (Thm.cabs x q))
   141 | Eq t => ([], K (inst' [t] pieq, FWD (inst' [t] aseteq) [inS (plus1 t)], infDFalse))
   142 | NEq t => ([], K (inst' [t] pineq, FWD (inst' [t] asetneq) [inS t], infDTrue))
   143 | Lt t => ([], K (inst' [t] pilt, FWD (inst' [t] asetlt) [inS t], infDFalse))
   144 | Le t => ([], K (inst' [t] pile, FWD (inst' [t] asetle) [inS (plus1 t)], infDFalse))
   145 | Gt t => ([], K (inst' [t] pigt, (inst' [t] asetgt), infDTrue))
   146 | Ge t => ([], K (inst' [t] pige, (inst' [t] asetge), infDTrue))
   147 | Dvd (d,s) =>
   148    ([],let val dd = dvd d
   149        in K (inst' [d,s] pidvd, FWD (inst' [d,s] asetdvd) [dd],FWD (inst' [d,s] infDdvd) [dd]) end)
   150 | NDvd(d,s) => ([],let val dd = dvd d
   151         in K (inst' [d,s] pindvd, FWD (inst' [d,s] asetndvd) [dd], FWD (inst' [d,s] infDndvd) [dd]) end)
   152 | _ => ([], K (inst' [cp] piP, inst' [cp] asetP, inst' [cp] infDP));
   153 
   154 fun decomp_minf x dvd inS [bseteq,bsetneq,bsetlt, bsetle, bsetgt,
   155                            bsetge,bsetdvd,bsetndvd,bsetP,
   156                            infDdvd, infDndvd, bsetconj,
   157                            bsetdisj, infDconj, infDdisj] cp =
   158  case (whatis x cp) of
   159   And (p,q) => ([p,q], myfwd (miconj, bsetconj, infDconj) (Thm.cabs x p) (Thm.cabs x q))
   160 | Or (p,q) => ([p,q], myfwd (midisj, bsetdisj, infDdisj) (Thm.cabs x p) (Thm.cabs x q))
   161 | Eq t => ([], K (inst' [t] mieq, FWD (inst' [t] bseteq) [inS (minus1 t)], infDFalse))
   162 | NEq t => ([], K (inst' [t] mineq, FWD (inst' [t] bsetneq) [inS t], infDTrue))
   163 | Lt t => ([], K (inst' [t] milt, (inst' [t] bsetlt), infDTrue))
   164 | Le t => ([], K (inst' [t] mile, (inst' [t] bsetle), infDTrue))
   165 | Gt t => ([], K (inst' [t] migt, FWD (inst' [t] bsetgt) [inS t], infDFalse))
   166 | Ge t => ([], K (inst' [t] mige,FWD (inst' [t] bsetge) [inS (minus1 t)], infDFalse))
   167 | Dvd (d,s) => ([],let val dd = dvd d
   168         in K (inst' [d,s] midvd, FWD (inst' [d,s] bsetdvd) [dd] , FWD (inst' [d,s] infDdvd) [dd]) end)
   169 | NDvd (d,s) => ([],let val dd = dvd d
   170         in K (inst' [d,s] mindvd, FWD (inst' [d,s] bsetndvd) [dd], FWD (inst' [d,s] infDndvd) [dd]) end)
   171 | _ => ([], K (inst' [cp] miP, inst' [cp] bsetP, inst' [cp] infDP))
   172 
   173     (* Canonical linear form for terms, formulae etc.. *)
   174 fun provelin ctxt t = Goal.prove ctxt [] [] t
   175   (fn _ => EVERY [simp_tac lin_ss 1, TRY (Lin_Arith.tac ctxt 1)]);
   176 fun linear_cmul 0 tm = zero
   177   | linear_cmul n tm = case tm of
   178       Const (@{const_name Groups.plus}, _) $ a $ b => addC $ linear_cmul n a $ linear_cmul n b
   179     | Const (@{const_name Groups.times}, _) $ c $ x => mulC $ numeral1 (fn m => n * m) c $ x
   180     | Const (@{const_name Groups.minus}, _) $ a $ b => subC $ linear_cmul n a $ linear_cmul n b
   181     | (m as Const (@{const_name Groups.uminus}, _)) $ a => m $ linear_cmul n a
   182     | _ => numeral1 (fn m => n * m) tm;
   183 fun earlier [] x y = false
   184   | earlier (h::t) x y =
   185     if h aconv y then false else if h aconv x then true else earlier t x y;
   186 
   187 fun linear_add vars tm1 tm2 = case (tm1, tm2) of
   188     (Const (@{const_name Groups.plus}, _) $ (Const (@{const_name Groups.times}, _) $ c1 $ x1) $ r1,
   189     Const (@{const_name Groups.plus}, _) $ (Const (@{const_name Groups.times}, _) $ c2 $ x2) $ r2) =>
   190    if x1 = x2 then
   191      let val c = numeral2 Integer.add c1 c2
   192       in if c = zero then linear_add vars r1 r2
   193          else addC$(mulC$c$x1)$(linear_add vars r1 r2)
   194      end
   195      else if earlier vars x1 x2 then addC $ (mulC $ c1 $ x1) $ linear_add vars r1 tm2
   196    else addC $ (mulC $ c2 $ x2) $ linear_add vars tm1 r2
   197  | (Const (@{const_name Groups.plus}, _) $ (Const (@{const_name Groups.times}, _) $ c1 $ x1) $ r1, _) =>
   198       addC $ (mulC $ c1 $ x1) $ linear_add vars r1 tm2
   199  | (_, Const (@{const_name Groups.plus}, _) $ (Const (@{const_name Groups.times}, _) $ c2 $ x2) $ r2) =>
   200       addC $ (mulC $ c2 $ x2) $ linear_add vars tm1 r2
   201  | (_, _) => numeral2 Integer.add tm1 tm2;
   202 
   203 fun linear_neg tm = linear_cmul ~1 tm;
   204 fun linear_sub vars tm1 tm2 = linear_add vars tm1 (linear_neg tm2);
   205 
   206 
   207 fun lint vars tm =  if is_numeral tm then tm  else case tm of
   208   Const (@{const_name Groups.uminus}, _) $ t => linear_neg (lint vars t)
   209 | Const (@{const_name Groups.plus}, _) $ s $ t => linear_add vars (lint vars s) (lint vars t)
   210 | Const (@{const_name Groups.minus}, _) $ s $ t => linear_sub vars (lint vars s) (lint vars t)
   211 | Const (@{const_name Groups.times}, _) $ s $ t =>
   212   let val s' = lint vars s
   213       val t' = lint vars t
   214   in if is_numeral s' then (linear_cmul (dest_numeral s') t')
   215      else if is_numeral t' then (linear_cmul (dest_numeral t') s')
   216      else raise COOPER ("Cooper Failed", TERM ("lint: not linear",[tm]))
   217   end
   218  | _ => addC $ (mulC $ one $ tm) $ zero;
   219 
   220 fun lin (vs as x::_) (Const (@{const_name Not}, _) $ (Const (@{const_name Orderings.less}, T) $ s $ t)) =
   221     lin vs (Const (@{const_name Orderings.less_eq}, T) $ t $ s)
   222   | lin (vs as x::_) (Const (@{const_name Not},_) $ (Const(@{const_name Orderings.less_eq}, T) $ s $ t)) =
   223     lin vs (Const (@{const_name Orderings.less}, T) $ t $ s)
   224   | lin vs (Const (@{const_name Not},T)$t) = Const (@{const_name Not},T)$ (lin vs t)
   225   | lin (vs as x::_) (Const(@{const_name Rings.dvd},_)$d$t) =
   226     HOLogic.mk_binrel @{const_name Rings.dvd} (numeral1 abs d, lint vs t)
   227   | lin (vs as x::_) ((b as Const("op =",_))$s$t) =
   228      (case lint vs (subC$t$s) of
   229       (t as a$(m$c$y)$r) =>
   230         if x <> y then b$zero$t
   231         else if dest_numeral c < 0 then b$(m$(numeral1 ~ c)$y)$r
   232         else b$(m$c$y)$(linear_neg r)
   233       | t => b$zero$t)
   234   | lin (vs as x::_) (b$s$t) =
   235      (case lint vs (subC$t$s) of
   236       (t as a$(m$c$y)$r) =>
   237         if x <> y then b$zero$t
   238         else if dest_numeral c < 0 then b$(m$(numeral1 ~ c)$y)$r
   239         else b$(linear_neg r)$(m$c$y)
   240       | t => b$zero$t)
   241   | lin vs fm = fm;
   242 
   243 fun lint_conv ctxt vs ct =
   244 let val t = term_of ct
   245 in (provelin ctxt ((HOLogic.eq_const iT)$t$(lint vs t) |> HOLogic.mk_Trueprop))
   246              RS eq_reflection
   247 end;
   248 
   249 fun is_intrel_type T = T = @{typ "int => int => bool"};
   250 
   251 fun is_intrel (b$_$_) = is_intrel_type (fastype_of b)
   252   | is_intrel (@{term "Not"}$(b$_$_)) = is_intrel_type (fastype_of b)
   253   | is_intrel _ = false;
   254 
   255 fun linearize_conv ctxt vs ct = case term_of ct of
   256   Const(@{const_name Rings.dvd},_)$d$t =>
   257   let
   258     val th = binop_conv (lint_conv ctxt vs) ct
   259     val (d',t') = Thm.dest_binop (Thm.rhs_of th)
   260     val (dt',tt') = (term_of d', term_of t')
   261   in if is_numeral dt' andalso is_numeral tt'
   262      then Conv.fconv_rule (arg_conv (Simplifier.rewrite presburger_ss)) th
   263      else
   264      let
   265       val dth =
   266       ((if dest_numeral (term_of d') < 0 then
   267           Conv.fconv_rule (arg_conv (arg1_conv (lint_conv ctxt vs)))
   268                            (Thm.transitive th (inst' [d',t'] dvd_uminus))
   269         else th) handle TERM _ => th)
   270       val d'' = Thm.rhs_of dth |> Thm.dest_arg1
   271      in
   272       case tt' of
   273         Const(@{const_name Groups.plus},_)$(Const(@{const_name Groups.times},_)$c$_)$_ =>
   274         let val x = dest_numeral c
   275         in if x < 0 then Conv.fconv_rule (arg_conv (arg_conv (lint_conv ctxt vs)))
   276                                        (Thm.transitive dth (inst' [d'',t'] dvd_uminus'))
   277         else dth end
   278       | _ => dth
   279      end
   280   end
   281 | Const (@{const_name Not},_)$(Const(@{const_name Rings.dvd},_)$_$_) => arg_conv (linearize_conv ctxt vs) ct
   282 | t => if is_intrel t
   283       then (provelin ctxt ((HOLogic.eq_const bT)$t$(lin vs t) |> HOLogic.mk_Trueprop))
   284        RS eq_reflection
   285       else reflexive ct;
   286 
   287 val dvdc = @{cterm "op dvd :: int => _"};
   288 
   289 fun unify ctxt q =
   290  let
   291   val (e,(cx,p)) = q |> Thm.dest_comb ||> Thm.dest_abs NONE
   292   val x = term_of cx
   293   val ins = insert (op = : int * int -> bool)
   294   fun h (acc,dacc) t =
   295    case (term_of t) of
   296     Const(s,_)$(Const(@{const_name Groups.times},_)$c$y)$ _ =>
   297     if x aconv y andalso member (op =)
   298       ["op =", @{const_name Orderings.less}, @{const_name Orderings.less_eq}] s
   299     then (ins (dest_numeral c) acc,dacc) else (acc,dacc)
   300   | Const(s,_)$_$(Const(@{const_name Groups.times},_)$c$y) =>
   301     if x aconv y andalso member (op =)
   302        [@{const_name Orderings.less}, @{const_name Orderings.less_eq}] s
   303     then (ins (dest_numeral c) acc, dacc) else (acc,dacc)
   304   | Const(@{const_name Rings.dvd},_)$_$(Const(@{const_name Groups.plus},_)$(Const(@{const_name Groups.times},_)$c$y)$_) =>
   305     if x aconv y then (acc,ins (dest_numeral c) dacc) else (acc,dacc)
   306   | Const("op &",_)$_$_ => h (h (acc,dacc) (Thm.dest_arg1 t)) (Thm.dest_arg t)
   307   | Const("op |",_)$_$_ => h (h (acc,dacc) (Thm.dest_arg1 t)) (Thm.dest_arg t)
   308   | Const (@{const_name Not},_)$_ => h (acc,dacc) (Thm.dest_arg t)
   309   | _ => (acc, dacc)
   310   val (cs,ds) = h ([],[]) p
   311   val l = Integer.lcms (union (op =) cs ds)
   312   fun cv k ct =
   313     let val (tm as b$s$t) = term_of ct
   314     in ((HOLogic.eq_const bT)$tm$(b$(linear_cmul k s)$(linear_cmul k t))
   315          |> HOLogic.mk_Trueprop |> provelin ctxt) RS eq_reflection end
   316   fun nzprop x =
   317    let
   318     val th =
   319      Simplifier.rewrite lin_ss
   320       (Thm.capply @{cterm Trueprop} (Thm.capply @{cterm "Not"}
   321            (Thm.capply (Thm.capply @{cterm "op = :: int => _"} (Numeral.mk_cnumber @{ctyp "int"} x))
   322            @{cterm "0::int"})))
   323    in equal_elim (Thm.symmetric th) TrueI end;
   324   val notz =
   325     let val tab = fold Inttab.update
   326           (ds ~~ (map (fn x => nzprop (l div x)) ds)) Inttab.empty
   327     in
   328       fn ct => the (Inttab.lookup tab (ct |> term_of |> dest_numeral))
   329         handle Option =>
   330           (writeln ("noz: Theorems-Table contains no entry for " ^
   331               Syntax.string_of_term ctxt (Thm.term_of ct)); raise Option)
   332     end
   333   fun unit_conv t =
   334    case (term_of t) of
   335    Const("op &",_)$_$_ => binop_conv unit_conv t
   336   | Const("op |",_)$_$_ => binop_conv unit_conv t
   337   | Const (@{const_name Not},_)$_ => arg_conv unit_conv t
   338   | Const(s,_)$(Const(@{const_name Groups.times},_)$c$y)$ _ =>
   339     if x=y andalso member (op =)
   340       ["op =", @{const_name Orderings.less}, @{const_name Orderings.less_eq}] s
   341     then cv (l div dest_numeral c) t else Thm.reflexive t
   342   | Const(s,_)$_$(Const(@{const_name Groups.times},_)$c$y) =>
   343     if x=y andalso member (op =)
   344       [@{const_name Orderings.less}, @{const_name Orderings.less_eq}] s
   345     then cv (l div dest_numeral c) t else Thm.reflexive t
   346   | Const(@{const_name Rings.dvd},_)$d$(r as (Const(@{const_name Groups.plus},_)$(Const(@{const_name Groups.times},_)$c$y)$_)) =>
   347     if x=y then
   348       let
   349        val k = l div dest_numeral c
   350        val kt = HOLogic.mk_number iT k
   351        val th1 = inst' [Thm.dest_arg1 t, Thm.dest_arg t]
   352              ((Thm.dest_arg t |> funpow 2 Thm.dest_arg1 |> notz) RS zdvd_mono)
   353        val (d',t') = (mulC$kt$d, mulC$kt$r)
   354        val thc = (provelin ctxt ((HOLogic.eq_const iT)$d'$(lint [] d') |> HOLogic.mk_Trueprop))
   355                    RS eq_reflection
   356        val tht = (provelin ctxt ((HOLogic.eq_const iT)$t'$(linear_cmul k r) |> HOLogic.mk_Trueprop))
   357                  RS eq_reflection
   358       in Thm.transitive th1 (Thm.combination (Drule.arg_cong_rule dvdc thc) tht) end
   359     else Thm.reflexive t
   360   | _ => Thm.reflexive t
   361   val uth = unit_conv p
   362   val clt =  Numeral.mk_cnumber @{ctyp "int"} l
   363   val ltx = Thm.capply (Thm.capply cmulC clt) cx
   364   val th = Drule.arg_cong_rule e (Thm.abstract_rule (fst (dest_Free x )) cx uth)
   365   val th' = inst' [Thm.cabs ltx (Thm.rhs_of uth), clt] unity_coeff_ex
   366   val thf = transitive th
   367       (transitive (symmetric (beta_conversion true (cprop_of th' |> Thm.dest_arg1))) th')
   368   val (lth,rth) = Thm.dest_comb (cprop_of thf) |>> Thm.dest_arg |>> Thm.beta_conversion true
   369                   ||> beta_conversion true |>> Thm.symmetric
   370  in transitive (transitive lth thf) rth end;
   371 
   372 
   373 val emptyIS = @{cterm "{}::int set"};
   374 val insert_tm = @{cterm "insert :: int => _"};
   375 val mem_tm = Const("op :",[iT , HOLogic.mk_setT iT] ---> bT);
   376 fun mkISet cts = fold_rev (Thm.capply insert_tm #> Thm.capply) cts emptyIS;
   377 val cTrp = @{cterm "Trueprop"};
   378 val eqelem_imp_imp = (thm"eqelem_imp_iff") RS iffD1;
   379 val [A_tm,B_tm] = map (fn th => cprop_of th |> funpow 2 Thm.dest_arg |> Thm.dest_abs NONE |> snd |> Thm.dest_arg1 |> Thm.dest_arg
   380                                       |> Thm.dest_abs NONE |> snd |> Thm.dest_fun |> Thm.dest_arg)
   381                       [asetP,bsetP];
   382 
   383 val D_tm = @{cpat "?D::int"};
   384 
   385 fun cooperex_conv ctxt vs q =
   386 let
   387 
   388  val uth = unify ctxt q
   389  val (x,p) = Thm.dest_abs NONE (Thm.dest_arg (Thm.rhs_of uth))
   390  val ins = insert (op aconvc)
   391  fun h t (bacc,aacc,dacc) =
   392   case (whatis x t) of
   393     And (p,q) => h q (h p (bacc,aacc,dacc))
   394   | Or (p,q) => h q  (h p (bacc,aacc,dacc))
   395   | Eq t => (ins (minus1 t) bacc,
   396              ins (plus1 t) aacc,dacc)
   397   | NEq t => (ins t bacc,
   398               ins t aacc, dacc)
   399   | Lt t => (bacc, ins t aacc, dacc)
   400   | Le t => (bacc, ins (plus1 t) aacc,dacc)
   401   | Gt t => (ins t bacc, aacc,dacc)
   402   | Ge t => (ins (minus1 t) bacc, aacc,dacc)
   403   | Dvd (d,s) => (bacc,aacc,insert (op =) (term_of d |> dest_numeral) dacc)
   404   | NDvd (d,s) => (bacc,aacc,insert (op =) (term_of d|> dest_numeral) dacc)
   405   | _ => (bacc, aacc, dacc)
   406  val (b0,a0,ds) = h p ([],[],[])
   407  val d = Integer.lcms ds
   408  val cd = Numeral.mk_cnumber @{ctyp "int"} d
   409  val dt = term_of cd
   410  fun divprop x =
   411    let
   412     val th =
   413      Simplifier.rewrite lin_ss
   414       (Thm.capply @{cterm Trueprop}
   415            (Thm.capply (Thm.capply dvdc (Numeral.mk_cnumber @{ctyp "int"} x)) cd))
   416    in equal_elim (Thm.symmetric th) TrueI end;
   417  val dvd =
   418    let val tab = fold Inttab.update (ds ~~ (map divprop ds)) Inttab.empty in
   419      fn ct => the (Inttab.lookup tab (term_of ct |> dest_numeral))
   420        handle Option =>
   421         (writeln ("dvd: Theorems-Table contains no entry for" ^
   422             Syntax.string_of_term ctxt (Thm.term_of ct)); raise Option)
   423    end
   424  val dp =
   425    let val th = Simplifier.rewrite lin_ss
   426       (Thm.capply @{cterm Trueprop}
   427            (Thm.capply (Thm.capply @{cterm "op < :: int => _"} @{cterm "0::int"}) cd))
   428    in equal_elim (Thm.symmetric th) TrueI end;
   429     (* A and B set *)
   430    local
   431      val insI1 = instantiate' [SOME @{ctyp "int"}] [] @{thm "insertI1"}
   432      val insI2 = instantiate' [SOME @{ctyp "int"}] [] @{thm "insertI2"}
   433    in
   434     fun provein x S =
   435      case term_of S of
   436         Const(@{const_name Orderings.bot}, _) => error "Unexpected error in Cooper, please email Amine Chaieb"
   437       | Const(@{const_name insert}, _) $ y $ _ =>
   438          let val (cy,S') = Thm.dest_binop S
   439          in if term_of x aconv y then instantiate' [] [SOME x, SOME S'] insI1
   440          else implies_elim (instantiate' [] [SOME x, SOME S', SOME cy] insI2)
   441                            (provein x S')
   442          end
   443    end
   444 
   445  val al = map (lint vs o term_of) a0
   446  val bl = map (lint vs o term_of) b0
   447  val (sl,s0,f,abths,cpth) =
   448    if length (distinct (op aconv) bl) <= length (distinct (op aconv) al)
   449    then
   450     (bl,b0,decomp_minf,
   451      fn B => (map (fn th => implies_elim (Thm.instantiate ([],[(B_tm,B), (D_tm,cd)]) th) dp)
   452                      [bseteq,bsetneq,bsetlt, bsetle, bsetgt,bsetge])@
   453                    (map (Thm.instantiate ([],[(B_tm,B), (D_tm,cd)]))
   454                         [bsetdvd,bsetndvd,bsetP,infDdvd, infDndvd,bsetconj,
   455                          bsetdisj,infDconj, infDdisj]),
   456                        cpmi)
   457      else (al,a0,decomp_pinf,fn A =>
   458           (map (fn th => implies_elim (Thm.instantiate ([],[(A_tm,A), (D_tm,cd)]) th) dp)
   459                    [aseteq,asetneq,asetlt, asetle, asetgt,asetge])@
   460                    (map (Thm.instantiate ([],[(A_tm,A), (D_tm,cd)]))
   461                    [asetdvd,asetndvd, asetP, infDdvd, infDndvd,asetconj,
   462                          asetdisj,infDconj, infDdisj]),cppi)
   463  val cpth =
   464   let
   465    val sths = map (fn (tl,t0) =>
   466                       if tl = term_of t0
   467                       then instantiate' [SOME @{ctyp "int"}] [SOME t0] refl
   468                       else provelin ctxt ((HOLogic.eq_const iT)$tl$(term_of t0)
   469                                  |> HOLogic.mk_Trueprop))
   470                    (sl ~~ s0)
   471    val csl = distinct (op aconvc) (map (cprop_of #> Thm.dest_arg #> Thm.dest_arg1) sths)
   472    val S = mkISet csl
   473    val inStab = fold (fn ct => fn tab => Termtab.update (term_of ct, provein ct S) tab)
   474                     csl Termtab.empty
   475    val eqelem_th = instantiate' [SOME @{ctyp "int"}] [NONE,NONE, SOME S] eqelem_imp_imp
   476    val inS =
   477      let
   478       fun transmem th0 th1 =
   479        Thm.equal_elim
   480         (Drule.arg_cong_rule cTrp (Drule.fun_cong_rule (Drule.arg_cong_rule
   481                ((Thm.dest_fun o Thm.dest_fun o Thm.dest_arg o cprop_of) th1) th0) S)) th1
   482       val tab = fold Termtab.update
   483         (map (fn eq =>
   484                 let val (s,t) = cprop_of eq |> Thm.dest_arg |> Thm.dest_binop
   485                     val th = if term_of s = term_of t
   486                              then the (Termtab.lookup inStab (term_of s))
   487                              else FWD (instantiate' [] [SOME s, SOME t] eqelem_th)
   488                                 [eq, the (Termtab.lookup inStab (term_of s))]
   489                  in (term_of t, th) end)
   490                   sths) Termtab.empty
   491         in
   492           fn ct => the (Termtab.lookup tab (term_of ct))
   493             handle Option =>
   494               (writeln ("inS: No theorem for " ^ Syntax.string_of_term ctxt (Thm.term_of ct));
   495                 raise Option)
   496         end
   497        val (inf, nb, pd) = divide_and_conquer (f x dvd inS (abths S)) p
   498    in [dp, inf, nb, pd] MRS cpth
   499    end
   500  val cpth' = Thm.transitive uth (cpth RS eq_reflection)
   501 in Thm.transitive cpth' ((simp_thms_conv ctxt then_conv eval_conv) (Thm.rhs_of cpth'))
   502 end;
   503 
   504 fun literals_conv bops uops env cv =
   505  let fun h t =
   506   case (term_of t) of
   507    b$_$_ => if member (op aconv) bops b then binop_conv h t else cv env t
   508  | u$_ => if member (op aconv) uops u then arg_conv h t else cv env t
   509  | _ => cv env t
   510  in h end;
   511 
   512 fun integer_nnf_conv ctxt env =
   513  nnf_conv then_conv literals_conv [HOLogic.conj, HOLogic.disj] [] env (linearize_conv ctxt);
   514 
   515 local
   516  val pcv = Simplifier.rewrite
   517      (HOL_basic_ss addsimps (@{thms simp_thms} @ List.take(@{thms ex_simps}, 4)
   518                       @ [not_all, all_not_ex, @{thm ex_disj_distrib}]))
   519  val postcv = Simplifier.rewrite presburger_ss
   520  fun conv ctxt p =
   521   let val _ = ()
   522   in
   523    Qelim.gen_qelim_conv pcv postcv pcv (cons o term_of)
   524       (OldTerm.term_frees (term_of p)) (linearize_conv ctxt) (integer_nnf_conv ctxt)
   525       (cooperex_conv ctxt) p
   526   end
   527   handle  CTERM s => raise COOPER ("Cooper Failed", CTERM s)
   528         | THM s => raise COOPER ("Cooper Failed", THM s)
   529         | TYPE s => raise COOPER ("Cooper Failed", TYPE s)
   530 in val cooper_conv = conv
   531 end;
   532 end;
   533 
   534 
   535 
   536 structure Coopereif =
   537 struct
   538 
   539 open Generated_Cooper;
   540 
   541 fun member eq = Library.member eq;
   542 
   543 fun cooper s = raise Cooper.COOPER ("Cooper oracle failed", ERROR s);
   544 fun i_of_term vs t = case t
   545  of Free (xn, xT) => (case AList.lookup (op aconv) vs t
   546      of NONE   => cooper "Variable not found in the list!"
   547       | SOME n => Bound n)
   548   | @{term "0::int"} => C 0
   549   | @{term "1::int"} => C 1
   550   | Term.Bound i => Bound i
   551   | Const(@{const_name Groups.uminus},_)$t' => Neg (i_of_term vs t')
   552   | Const(@{const_name Groups.plus},_)$t1$t2 => Add (i_of_term vs t1,i_of_term vs t2)
   553   | Const(@{const_name Groups.minus},_)$t1$t2 => Sub (i_of_term vs t1,i_of_term vs t2)
   554   | Const(@{const_name Groups.times},_)$t1$t2 =>
   555      (Mul (HOLogic.dest_number t1 |> snd, i_of_term vs t2)
   556     handle TERM _ =>
   557        (Mul (HOLogic.dest_number t2 |> snd, i_of_term vs t1)
   558         handle TERM _ => cooper "Reification: Unsupported kind of multiplication"))
   559   | _ => (C (HOLogic.dest_number t |> snd)
   560            handle TERM _ => cooper "Reification: unknown term");
   561 
   562 fun qf_of_term ps vs t =  case t
   563  of Const("True",_) => T
   564   | Const("False",_) => F
   565   | Const(@{const_name Orderings.less},_)$t1$t2 => Lt (Sub (i_of_term vs t1,i_of_term vs t2))
   566   | Const(@{const_name Orderings.less_eq},_)$t1$t2 => Le (Sub(i_of_term vs t1,i_of_term vs t2))
   567   | Const(@{const_name Rings.dvd},_)$t1$t2 =>
   568       (Dvd(HOLogic.dest_number t1 |> snd, i_of_term vs t2) handle _ => cooper "Reification: unsupported dvd")  (* FIXME avoid handle _ *)
   569   | @{term "op = :: int => _"}$t1$t2 => Eq (Sub (i_of_term vs t1,i_of_term vs t2))
   570   | @{term "op = :: bool => _ "}$t1$t2 => Iff(qf_of_term ps vs t1,qf_of_term ps vs t2)
   571   | Const("op &",_)$t1$t2 => And(qf_of_term ps vs t1,qf_of_term ps vs t2)
   572   | Const("op |",_)$t1$t2 => Or(qf_of_term ps vs t1,qf_of_term ps vs t2)
   573   | Const("op -->",_)$t1$t2 => Imp(qf_of_term ps vs t1,qf_of_term ps vs t2)
   574   | Const (@{const_name Not},_)$t' => Not(qf_of_term ps vs t')
   575   | Const("Ex",_)$Abs(xn,xT,p) =>
   576      let val (xn',p') = variant_abs (xn,xT,p)
   577          val vs' = (Free (xn',xT), 0) :: (map (fn(v,n) => (v,1+ n)) vs)
   578      in E (qf_of_term ps vs' p')
   579      end
   580   | Const("All",_)$Abs(xn,xT,p) =>
   581      let val (xn',p') = variant_abs (xn,xT,p)
   582          val vs' = (Free (xn',xT), 0) :: (map (fn(v,n) => (v,1+ n)) vs)
   583      in A (qf_of_term ps vs' p')
   584      end
   585   | _ =>(case AList.lookup (op aconv) ps t of
   586            NONE => cooper "Reification: unknown term!"
   587          | SOME n => Closed n);
   588 
   589 local
   590  val ops = [@{term "op &"}, @{term "op |"}, @{term "op -->"}, @{term "op = :: bool => _"},
   591              @{term "op = :: int => _"}, @{term "op < :: int => _"},
   592              @{term "op <= :: int => _"}, @{term "Not"}, @{term "All:: (int => _) => _"},
   593              @{term "Ex:: (int => _) => _"}, @{term "True"}, @{term "False"}]
   594 fun ty t = Bool.not (fastype_of t = HOLogic.boolT)
   595 in
   596 fun term_bools acc t =
   597 case t of
   598     (l as f $ a) $ b => if ty t orelse member (op =) ops f then term_bools (term_bools acc l)b
   599             else insert (op aconv) t acc
   600   | f $ a => if ty t orelse member (op =) ops f then term_bools (term_bools acc f) a
   601             else insert (op aconv) t acc
   602   | Abs p => term_bools acc (snd (variant_abs p))
   603   | _ => if ty t orelse member (op =) ops t then acc else insert (op aconv) t acc
   604 end;
   605 
   606 fun myassoc2 l v =
   607     case l of
   608   [] => NONE
   609       | (x,v')::xs => if v = v' then SOME x
   610           else myassoc2 xs v;
   611 
   612 fun term_of_i vs t = case t
   613  of C i => HOLogic.mk_number HOLogic.intT i
   614   | Bound n => the (myassoc2 vs n)
   615   | Neg t' => @{term "uminus :: int => _"} $ term_of_i vs t'
   616   | Add (t1, t2) => @{term "op + :: int => _"} $ term_of_i vs t1 $ term_of_i vs t2
   617   | Sub (t1, t2) => @{term "op - :: int => _"} $ term_of_i vs t1 $ term_of_i vs t2
   618   | Mul (i, t2) => @{term "op * :: int => _"} $
   619       HOLogic.mk_number HOLogic.intT i $ term_of_i vs t2
   620   | Cn (n, i, t') => term_of_i vs (Add (Mul (i, Bound n), t'));
   621 
   622 fun term_of_qf ps vs t =
   623  case t of
   624    T => HOLogic.true_const
   625  | F => HOLogic.false_const
   626  | Lt t' => @{term "op < :: int => _ "}$ term_of_i vs t'$ @{term "0::int"}
   627  | Le t' => @{term "op <= :: int => _ "}$ term_of_i vs t' $ @{term "0::int"}
   628  | Gt t' => @{term "op < :: int => _ "}$ @{term "0::int"}$ term_of_i vs t'
   629  | Ge t' => @{term "op <= :: int => _ "}$ @{term "0::int"}$ term_of_i vs t'
   630  | Eq t' => @{term "op = :: int => _ "}$ term_of_i vs t'$ @{term "0::int"}
   631  | NEq t' => term_of_qf ps vs (Not (Eq t'))
   632  | Dvd(i,t') => @{term "op dvd :: int => _ "} $
   633     HOLogic.mk_number HOLogic.intT i $ term_of_i vs t'
   634  | NDvd(i,t')=> term_of_qf ps vs (Not(Dvd(i,t')))
   635  | Not t' => HOLogic.Not$(term_of_qf ps vs t')
   636  | And(t1,t2) => HOLogic.conj$(term_of_qf ps vs t1)$(term_of_qf ps vs t2)
   637  | Or(t1,t2) => HOLogic.disj$(term_of_qf ps vs t1)$(term_of_qf ps vs t2)
   638  | Imp(t1,t2) => HOLogic.imp$(term_of_qf ps vs t1)$(term_of_qf ps vs t2)
   639  | Iff(t1,t2) => @{term "op = :: bool => _"} $ term_of_qf ps vs t1 $ term_of_qf ps vs t2
   640  | Closed n => the (myassoc2 ps n)
   641  | NClosed n => term_of_qf ps vs (Not (Closed n))
   642  | _ => cooper "If this is raised, Isabelle/HOL or code generator is inconsistent!";
   643 
   644 fun cooper_oracle ct =
   645   let
   646     val thy = Thm.theory_of_cterm ct;
   647     val t = Thm.term_of ct;
   648     val (vs, ps) = pairself (map_index swap) (OldTerm.term_frees t, term_bools [] t);
   649   in
   650     Thm.cterm_of thy (Logic.mk_equals (HOLogic.mk_Trueprop t,
   651       HOLogic.mk_Trueprop (term_of_qf ps vs (pa (qf_of_term ps vs t)))))
   652   end;
   653 
   654 end;