src/HOL/Tools/Qelim/cooper.ML
author wenzelm
Mon Sep 29 11:46:47 2008 +0200 (2008-09-29)
changeset 28397 389c5e494605
parent 28290 4cc2b6046258
child 29265 5b4247055bd7
permissions -rw-r--r--
handle _ should be avoided (spurious Interrupt will spoil the game);
     1 (*  Title:      HOL/Tools/Qelim/cooper.ML
     2     ID:         $Id$
     3     Author:     Amine Chaieb, TU Muenchen
     4 *)
     5 
     6 signature COOPER =
     7  sig
     8   val cooper_conv : Proof.context -> conv
     9   exception COOPER of string * exn
    10 end;
    11 
    12 structure Cooper: COOPER =
    13 struct
    14 
    15 open Conv;
    16 open Normalizer;
    17 
    18 exception COOPER of string * exn;
    19 fun simp_thms_conv ctxt =
    20   Simplifier.rewrite (Simplifier.context ctxt HOL_basic_ss addsimps simp_thms);
    21 val FWD = Drule.implies_elim_list;
    22 
    23 val true_tm = @{cterm "True"};
    24 val false_tm = @{cterm "False"};
    25 val zdvd1_eq = @{thm "zdvd1_eq"};
    26 val presburger_ss = @{simpset} addsimps [zdvd1_eq];
    27 val lin_ss = presburger_ss addsimps (@{thm "dvd_eq_mod_eq_0"}::zdvd1_eq::@{thms zadd_ac});
    28 
    29 val iT = HOLogic.intT
    30 val bT = HOLogic.boolT;
    31 val dest_numeral = HOLogic.dest_number #> snd;
    32 
    33 val [miconj, midisj, mieq, mineq, milt, mile, migt, mige, midvd, mindvd, miP] = 
    34     map(instantiate' [SOME @{ctyp "int"}] []) @{thms "minf"};
    35 
    36 val [infDconj, infDdisj, infDdvd,infDndvd,infDP] = 
    37     map(instantiate' [SOME @{ctyp "int"}] []) @{thms "inf_period"};
    38 
    39 val [piconj, pidisj, pieq,pineq,pilt,pile,pigt,pige,pidvd,pindvd,piP] = 
    40     map (instantiate' [SOME @{ctyp "int"}] []) @{thms "pinf"};
    41 
    42 val [miP, piP] = map (instantiate' [SOME @{ctyp "bool"}] []) [miP, piP];
    43 
    44 val infDP = instantiate' (map SOME [@{ctyp "int"}, @{ctyp "bool"}]) [] infDP;
    45 
    46 val [[asetconj, asetdisj, aseteq, asetneq, asetlt, asetle, 
    47       asetgt, asetge, asetdvd, asetndvd,asetP],
    48      [bsetconj, bsetdisj, bseteq, bsetneq, bsetlt, bsetle, 
    49       bsetgt, bsetge, bsetdvd, bsetndvd,bsetP]]  = [@{thms "aset"}, @{thms "bset"}];
    50 
    51 val [miex, cpmi, piex, cppi] = [@{thm "minusinfinity"}, @{thm "cpmi"}, 
    52                                 @{thm "plusinfinity"}, @{thm "cppi"}];
    53 
    54 val unity_coeff_ex = instantiate' [SOME @{ctyp "int"}] [] @{thm "unity_coeff_ex"};
    55 
    56 val [zdvd_mono,simp_from_to,all_not_ex] = 
    57      [@{thm "zdvd_mono"}, @{thm "simp_from_to"}, @{thm "all_not_ex"}];
    58 
    59 val [dvd_uminus, dvd_uminus'] = @{thms "uminus_dvd_conv"};
    60 
    61 val eval_ss = presburger_ss addsimps [simp_from_to] delsimps [insert_iff,bex_triv];
    62 val eval_conv = Simplifier.rewrite eval_ss;
    63 
    64 (* recognising cterm without moving to terms *)
    65 
    66 datatype fm = And of cterm*cterm| Or of cterm*cterm| Eq of cterm | NEq of cterm 
    67             | Lt of cterm | Le of cterm | Gt of cterm | Ge of cterm
    68             | Dvd of cterm*cterm | NDvd of cterm*cterm | Nox
    69 
    70 fun whatis x ct = 
    71 ( case (term_of ct) of 
    72   Const("op &",_)$_$_ => And (Thm.dest_binop ct)
    73 | Const ("op |",_)$_$_ => Or (Thm.dest_binop ct)
    74 | Const ("op =",ty)$y$_ => if term_of x aconv y then Eq (Thm.dest_arg ct) else Nox
    75 | Const (@{const_name Not},_) $ (Const ("op =",_)$y$_) => 
    76   if term_of x aconv y then NEq (funpow 2 Thm.dest_arg ct) else Nox
    77 | Const (@{const_name HOL.less}, _) $ y$ z =>
    78    if term_of x aconv y then Lt (Thm.dest_arg ct) 
    79    else if term_of x aconv z then Gt (Thm.dest_arg1 ct) else Nox
    80 | Const (@{const_name HOL.less_eq}, _) $ y $ z => 
    81    if term_of x aconv y then Le (Thm.dest_arg ct) 
    82    else if term_of x aconv z then Ge (Thm.dest_arg1 ct) else Nox
    83 | Const (@{const_name Ring_and_Field.dvd},_)$_$(Const(@{const_name HOL.plus},_)$y$_) =>
    84    if term_of x aconv y then Dvd (Thm.dest_binop ct ||> Thm.dest_arg) else Nox 
    85 | Const (@{const_name Not},_) $ (Const (@{const_name Ring_and_Field.dvd},_)$_$(Const(@{const_name "HOL.plus"},_)$y$_)) =>
    86    if term_of x aconv y then 
    87    NDvd (Thm.dest_binop (Thm.dest_arg ct) ||> Thm.dest_arg) else Nox 
    88 | _ => Nox)
    89   handle CTERM _ => Nox; 
    90 
    91 fun get_pmi_term t = 
    92   let val (x,eq) = 
    93      (Thm.dest_abs NONE o Thm.dest_arg o snd o Thm.dest_abs NONE o Thm.dest_arg)
    94         (Thm.dest_arg t)
    95 in (Thm.cabs x o Thm.dest_arg o Thm.dest_arg) eq end;
    96 
    97 val get_pmi = get_pmi_term o cprop_of;
    98 
    99 val p_v' = @{cpat "?P' :: int => bool"}; 
   100 val q_v' = @{cpat "?Q' :: int => bool"};
   101 val p_v = @{cpat "?P:: int => bool"};
   102 val q_v = @{cpat "?Q:: int => bool"};
   103 
   104 fun myfwd (th1, th2, th3) p q 
   105       [(th_1,th_2,th_3), (th_1',th_2',th_3')] = 
   106   let  
   107    val (mp', mq') = (get_pmi th_1, get_pmi th_1')
   108    val mi_th = FWD (instantiate ([],[(p_v,p),(q_v,q), (p_v',mp'),(q_v',mq')]) th1) 
   109                    [th_1, th_1']
   110    val infD_th = FWD (instantiate ([],[(p_v,mp'), (q_v, mq')]) th3) [th_3,th_3']
   111    val set_th = FWD (instantiate ([],[(p_v,p), (q_v,q)]) th2) [th_2, th_2']
   112   in (mi_th, set_th, infD_th)
   113   end;
   114 
   115 val inst' = fn cts => instantiate' [] (map SOME cts);
   116 val infDTrue = instantiate' [] [SOME true_tm] infDP;
   117 val infDFalse = instantiate' [] [SOME false_tm] infDP;
   118 
   119 val cadd =  @{cterm "op + :: int => _"}
   120 val cmulC =  @{cterm "op * :: int => _"}
   121 val cminus =  @{cterm "op - :: int => _"}
   122 val cone =  @{cterm "1 :: int"}
   123 val cneg = @{cterm "uminus :: int => _"}
   124 val [addC, mulC, subC, negC] = map term_of [cadd, cmulC, cminus, cneg]
   125 val [zero, one] = [@{term "0 :: int"}, @{term "1 :: int"}];
   126 
   127 val is_numeral = can dest_numeral; 
   128 
   129 fun numeral1 f n = HOLogic.mk_number iT (f (dest_numeral n)); 
   130 fun numeral2 f m n = HOLogic.mk_number iT (f (dest_numeral m) (dest_numeral n));
   131 
   132 val [minus1,plus1] = 
   133     map (fn c => fn t => Thm.capply (Thm.capply c t) cone) [cminus,cadd];
   134 
   135 fun decomp_pinf x dvd inS [aseteq, asetneq, asetlt, asetle, 
   136                            asetgt, asetge,asetdvd,asetndvd,asetP,
   137                            infDdvd, infDndvd, asetconj,
   138                            asetdisj, infDconj, infDdisj] cp =
   139  case (whatis x cp) of
   140   And (p,q) => ([p,q], myfwd (piconj, asetconj, infDconj) (Thm.cabs x p) (Thm.cabs x q))
   141 | Or (p,q) => ([p,q], myfwd (pidisj, asetdisj, infDdisj) (Thm.cabs x p) (Thm.cabs x q))
   142 | Eq t => ([], K (inst' [t] pieq, FWD (inst' [t] aseteq) [inS (plus1 t)], infDFalse))
   143 | NEq t => ([], K (inst' [t] pineq, FWD (inst' [t] asetneq) [inS t], infDTrue))
   144 | Lt t => ([], K (inst' [t] pilt, FWD (inst' [t] asetlt) [inS t], infDFalse))
   145 | Le t => ([], K (inst' [t] pile, FWD (inst' [t] asetle) [inS (plus1 t)], infDFalse))
   146 | Gt t => ([], K (inst' [t] pigt, (inst' [t] asetgt), infDTrue))
   147 | Ge t => ([], K (inst' [t] pige, (inst' [t] asetge), infDTrue))
   148 | Dvd (d,s) => 
   149    ([],let val dd = dvd d
   150 	     in K (inst' [d,s] pidvd, FWD (inst' [d,s] asetdvd) [dd],FWD (inst' [d,s] infDdvd) [dd]) end)
   151 | NDvd(d,s) => ([],let val dd = dvd d
   152 	      in K (inst' [d,s] pindvd, FWD (inst' [d,s] asetndvd) [dd], FWD (inst' [d,s] infDndvd) [dd]) end)
   153 | _ => ([], K (inst' [cp] piP, inst' [cp] asetP, inst' [cp] infDP));
   154 
   155 fun decomp_minf x dvd inS [bseteq,bsetneq,bsetlt, bsetle, bsetgt,
   156                            bsetge,bsetdvd,bsetndvd,bsetP,
   157                            infDdvd, infDndvd, bsetconj,
   158                            bsetdisj, infDconj, infDdisj] cp =
   159  case (whatis x cp) of
   160   And (p,q) => ([p,q], myfwd (miconj, bsetconj, infDconj) (Thm.cabs x p) (Thm.cabs x q))
   161 | Or (p,q) => ([p,q], myfwd (midisj, bsetdisj, infDdisj) (Thm.cabs x p) (Thm.cabs x q))
   162 | Eq t => ([], K (inst' [t] mieq, FWD (inst' [t] bseteq) [inS (minus1 t)], infDFalse))
   163 | NEq t => ([], K (inst' [t] mineq, FWD (inst' [t] bsetneq) [inS t], infDTrue))
   164 | Lt t => ([], K (inst' [t] milt, (inst' [t] bsetlt), infDTrue))
   165 | Le t => ([], K (inst' [t] mile, (inst' [t] bsetle), infDTrue))
   166 | Gt t => ([], K (inst' [t] migt, FWD (inst' [t] bsetgt) [inS t], infDFalse))
   167 | Ge t => ([], K (inst' [t] mige,FWD (inst' [t] bsetge) [inS (minus1 t)], infDFalse))
   168 | Dvd (d,s) => ([],let val dd = dvd d
   169 	      in K (inst' [d,s] midvd, FWD (inst' [d,s] bsetdvd) [dd] , FWD (inst' [d,s] infDdvd) [dd]) end)
   170 | NDvd (d,s) => ([],let val dd = dvd d
   171 	      in K (inst' [d,s] mindvd, FWD (inst' [d,s] bsetndvd) [dd], FWD (inst' [d,s] infDndvd) [dd]) end)
   172 | _ => ([], K (inst' [cp] miP, inst' [cp] bsetP, inst' [cp] infDP))
   173 
   174     (* Canonical linear form for terms, formulae etc.. *)
   175 fun provelin ctxt t = Goal.prove ctxt [] [] t 
   176   (fn _ => EVERY [simp_tac lin_ss 1, TRY (simple_arith_tac ctxt 1)]);
   177 fun linear_cmul 0 tm = zero 
   178   | linear_cmul n tm = case tm of  
   179       Const (@{const_name HOL.plus}, _) $ a $ b => addC $ linear_cmul n a $ linear_cmul n b
   180     | Const (@{const_name HOL.times}, _) $ c $ x => mulC $ numeral1 (fn m => n * m) c $ x
   181     | Const (@{const_name HOL.minus}, _) $ a $ b => subC $ linear_cmul n a $ linear_cmul n b
   182     | (m as Const (@{const_name HOL.uminus}, _)) $ a => m $ linear_cmul n a
   183     | _ => numeral1 (fn m => n * m) tm;
   184 fun earlier [] x y = false 
   185 	| earlier (h::t) x y = 
   186     if h aconv y then false else if h aconv x then true else earlier t x y; 
   187 
   188 fun linear_add vars tm1 tm2 = case (tm1, tm2) of 
   189     (Const (@{const_name HOL.plus}, _) $ (Const (@{const_name HOL.times}, _) $ c1 $ x1) $ r1,
   190     Const (@{const_name HOL.plus}, _) $ (Const (@{const_name HOL.times}, _) $ c2 $ x2) $ r2) =>
   191    if x1 = x2 then 
   192      let val c = numeral2 (curry op +) c1 c2
   193       in if c = zero then linear_add vars r1 r2
   194          else addC$(mulC$c$x1)$(linear_add vars r1 r2)
   195      end 
   196      else if earlier vars x1 x2 then addC $ (mulC $ c1 $ x1) $ linear_add vars r1 tm2
   197    else addC $ (mulC $ c2 $ x2) $ linear_add vars tm1 r2
   198  | (Const (@{const_name HOL.plus}, _) $ (Const (@{const_name HOL.times}, _) $ c1 $ x1) $ r1, _) =>
   199       addC $ (mulC $ c1 $ x1) $ linear_add vars r1 tm2
   200  | (_, Const (@{const_name HOL.plus}, _) $ (Const (@{const_name HOL.times}, _) $ c2 $ x2) $ r2) => 
   201       addC $ (mulC $ c2 $ x2) $ linear_add vars tm1 r2
   202  | (_, _) => numeral2 (curry op +) tm1 tm2;
   203  
   204 fun linear_neg tm = linear_cmul ~1 tm; 
   205 fun linear_sub vars tm1 tm2 = linear_add vars tm1 (linear_neg tm2); 
   206 
   207 
   208 fun lint vars tm =  if is_numeral tm then tm  else case tm of 
   209   Const (@{const_name HOL.uminus}, _) $ t => linear_neg (lint vars t)
   210 | Const (@{const_name HOL.plus}, _) $ s $ t => linear_add vars (lint vars s) (lint vars t)
   211 | Const (@{const_name HOL.minus}, _) $ s $ t => linear_sub vars (lint vars s) (lint vars t)
   212 | Const (@{const_name HOL.times}, _) $ s $ t =>
   213   let val s' = lint vars s  
   214       val t' = lint vars t  
   215   in if is_numeral s' then (linear_cmul (dest_numeral s') t') 
   216      else if is_numeral t' then (linear_cmul (dest_numeral t') s') 
   217      else raise COOPER ("Cooper Failed", TERM ("lint: not linear",[tm]))
   218   end 
   219  | _ => addC $ (mulC $ one $ tm) $ zero;
   220 
   221 fun lin (vs as x::_) (Const (@{const_name Not}, _) $ (Const (@{const_name HOL.less}, T) $ s $ t)) = 
   222     lin vs (Const (@{const_name HOL.less_eq}, T) $ t $ s)
   223   | lin (vs as x::_) (Const (@{const_name Not},_) $ (Const(@{const_name HOL.less_eq}, T) $ s $ t)) = 
   224     lin vs (Const (@{const_name HOL.less}, T) $ t $ s)
   225   | lin vs (Const (@{const_name Not},T)$t) = Const (@{const_name Not},T)$ (lin vs t)
   226   | lin (vs as x::_) (Const(@{const_name Ring_and_Field.dvd},_)$d$t) = 
   227     HOLogic.mk_binrel @{const_name Ring_and_Field.dvd} (numeral1 abs d, lint vs t)
   228   | lin (vs as x::_) ((b as Const("op =",_))$s$t) = 
   229      (case lint vs (subC$t$s) of 
   230       (t as a$(m$c$y)$r) => 
   231         if x <> y then b$zero$t
   232         else if dest_numeral c < 0 then b$(m$(numeral1 ~ c)$y)$r
   233         else b$(m$c$y)$(linear_neg r)
   234       | t => b$zero$t)
   235   | lin (vs as x::_) (b$s$t) = 
   236      (case lint vs (subC$t$s) of 
   237       (t as a$(m$c$y)$r) => 
   238         if x <> y then b$zero$t
   239         else if dest_numeral c < 0 then b$(m$(numeral1 ~ c)$y)$r
   240         else b$(linear_neg r)$(m$c$y)
   241       | t => b$zero$t)
   242   | lin vs fm = fm;
   243 
   244 fun lint_conv ctxt vs ct = 
   245 let val t = term_of ct
   246 in (provelin ctxt ((HOLogic.eq_const iT)$t$(lint vs t) |> HOLogic.mk_Trueprop))
   247              RS eq_reflection
   248 end;
   249 
   250 fun is_intrel (b$_$_) = domain_type (fastype_of b) = HOLogic.intT
   251   | is_intrel (@{term "Not"}$(b$_$_)) = domain_type (fastype_of b) = HOLogic.intT
   252   | is_intrel _ = false;
   253  
   254 fun linearize_conv ctxt vs ct = case term_of ct of
   255   Const(@{const_name Ring_and_Field.dvd},_)$d$t => 
   256   let 
   257     val th = binop_conv (lint_conv ctxt vs) ct
   258     val (d',t') = Thm.dest_binop (Thm.rhs_of th)
   259     val (dt',tt') = (term_of d', term_of t')
   260   in if is_numeral dt' andalso is_numeral tt' 
   261      then Conv.fconv_rule (arg_conv (Simplifier.rewrite presburger_ss)) th
   262      else 
   263      let 
   264       val dth = 
   265       ((if dest_numeral (term_of d') < 0 then 
   266           Conv.fconv_rule (arg_conv (arg1_conv (lint_conv ctxt vs)))
   267                            (Thm.transitive th (inst' [d',t'] dvd_uminus))
   268         else th) handle TERM _ => th)
   269       val d'' = Thm.rhs_of dth |> Thm.dest_arg1
   270      in
   271       case tt' of 
   272         Const(@{const_name HOL.plus},_)$(Const(@{const_name HOL.times},_)$c$_)$_ => 
   273         let val x = dest_numeral c
   274         in if x < 0 then Conv.fconv_rule (arg_conv (arg_conv (lint_conv ctxt vs)))
   275                                        (Thm.transitive dth (inst' [d'',t'] dvd_uminus'))
   276         else dth end
   277       | _ => dth
   278      end
   279   end
   280 | Const (@{const_name Not},_)$(Const(@{const_name Ring_and_Field.dvd},_)$_$_) => arg_conv (linearize_conv ctxt vs) ct
   281 | t => if is_intrel t 
   282       then (provelin ctxt ((HOLogic.eq_const bT)$t$(lin vs t) |> HOLogic.mk_Trueprop))
   283        RS eq_reflection
   284       else reflexive ct;
   285 
   286 val dvdc = @{cterm "op dvd :: int => _"};
   287 
   288 fun unify ctxt q = 
   289  let
   290   val (e,(cx,p)) = q |> Thm.dest_comb ||> Thm.dest_abs NONE
   291   val x = term_of cx 
   292   val ins = insert (op = : int * int -> bool)
   293   fun h (acc,dacc) t = 
   294    case (term_of t) of
   295     Const(s,_)$(Const(@{const_name HOL.times},_)$c$y)$ _ => 
   296     if x aconv y andalso member (op =)
   297       ["op =", @{const_name HOL.less}, @{const_name HOL.less_eq}] s
   298     then (ins (dest_numeral c) acc,dacc) else (acc,dacc)
   299   | Const(s,_)$_$(Const(@{const_name HOL.times},_)$c$y) => 
   300     if x aconv y andalso member (op =)
   301        [@{const_name HOL.less}, @{const_name HOL.less_eq}] s 
   302     then (ins (dest_numeral c) acc, dacc) else (acc,dacc)
   303   | Const(@{const_name Ring_and_Field.dvd},_)$_$(Const(@{const_name HOL.plus},_)$(Const(@{const_name HOL.times},_)$c$y)$_) => 
   304     if x aconv y then (acc,ins (dest_numeral c) dacc) else (acc,dacc)
   305   | Const("op &",_)$_$_ => h (h (acc,dacc) (Thm.dest_arg1 t)) (Thm.dest_arg t)
   306   | Const("op |",_)$_$_ => h (h (acc,dacc) (Thm.dest_arg1 t)) (Thm.dest_arg t)
   307   | Const (@{const_name Not},_)$_ => h (acc,dacc) (Thm.dest_arg t)
   308   | _ => (acc, dacc)
   309   val (cs,ds) = h ([],[]) p
   310   val l = Integer.lcms (cs union ds)
   311   fun cv k ct = 
   312     let val (tm as b$s$t) = term_of ct 
   313     in ((HOLogic.eq_const bT)$tm$(b$(linear_cmul k s)$(linear_cmul k t))
   314          |> HOLogic.mk_Trueprop |> provelin ctxt) RS eq_reflection end
   315   fun nzprop x = 
   316    let 
   317     val th = 
   318      Simplifier.rewrite lin_ss 
   319       (Thm.capply @{cterm Trueprop} (Thm.capply @{cterm "Not"} 
   320            (Thm.capply (Thm.capply @{cterm "op = :: int => _"} (Numeral.mk_cnumber @{ctyp "int"} x)) 
   321            @{cterm "0::int"})))
   322    in equal_elim (Thm.symmetric th) TrueI end;
   323   val notz = let val tab = fold Inttab.update 
   324                                (ds ~~ (map (fn x => nzprop (l div x)) ds)) Inttab.empty 
   325             in 
   326              (fn ct => (valOf (Inttab.lookup tab (ct |> term_of |> dest_numeral)) 
   327                 handle Option => (writeln "noz: Theorems-Table contains no entry for"; 
   328                                     Display.print_cterm ct ; raise Option)))
   329            end
   330   fun unit_conv t = 
   331    case (term_of t) of
   332    Const("op &",_)$_$_ => binop_conv unit_conv t
   333   | Const("op |",_)$_$_ => binop_conv unit_conv t
   334   | Const (@{const_name Not},_)$_ => arg_conv unit_conv t
   335   | Const(s,_)$(Const(@{const_name HOL.times},_)$c$y)$ _ => 
   336     if x=y andalso member (op =)
   337       ["op =", @{const_name HOL.less}, @{const_name HOL.less_eq}] s
   338     then cv (l div dest_numeral c) t else Thm.reflexive t
   339   | Const(s,_)$_$(Const(@{const_name HOL.times},_)$c$y) => 
   340     if x=y andalso member (op =)
   341       [@{const_name HOL.less}, @{const_name HOL.less_eq}] s
   342     then cv (l div dest_numeral c) t else Thm.reflexive t
   343   | Const(@{const_name Ring_and_Field.dvd},_)$d$(r as (Const(@{const_name HOL.plus},_)$(Const(@{const_name HOL.times},_)$c$y)$_)) => 
   344     if x=y then 
   345       let 
   346        val k = l div dest_numeral c
   347        val kt = HOLogic.mk_number iT k
   348        val th1 = inst' [Thm.dest_arg1 t, Thm.dest_arg t] 
   349              ((Thm.dest_arg t |> funpow 2 Thm.dest_arg1 |> notz) RS zdvd_mono)
   350        val (d',t') = (mulC$kt$d, mulC$kt$r)
   351        val thc = (provelin ctxt ((HOLogic.eq_const iT)$d'$(lint [] d') |> HOLogic.mk_Trueprop))
   352                    RS eq_reflection
   353        val tht = (provelin ctxt ((HOLogic.eq_const iT)$t'$(linear_cmul k r) |> HOLogic.mk_Trueprop))
   354                  RS eq_reflection
   355       in Thm.transitive th1 (Thm.combination (Drule.arg_cong_rule dvdc thc) tht) end                 
   356     else Thm.reflexive t
   357   | _ => Thm.reflexive t
   358   val uth = unit_conv p
   359   val clt =  Numeral.mk_cnumber @{ctyp "int"} l
   360   val ltx = Thm.capply (Thm.capply cmulC clt) cx
   361   val th = Drule.arg_cong_rule e (Thm.abstract_rule (fst (dest_Free x )) cx uth)
   362   val th' = inst' [Thm.cabs ltx (Thm.rhs_of uth), clt] unity_coeff_ex
   363   val thf = transitive th 
   364       (transitive (symmetric (beta_conversion true (cprop_of th' |> Thm.dest_arg1))) th')
   365   val (lth,rth) = Thm.dest_comb (cprop_of thf) |>> Thm.dest_arg |>> Thm.beta_conversion true
   366                   ||> beta_conversion true |>> Thm.symmetric
   367  in transitive (transitive lth thf) rth end;
   368 
   369 
   370 val emptyIS = @{cterm "{}::int set"};
   371 val insert_tm = @{cterm "insert :: int => _"};
   372 val mem_tm = Const("op :",[iT , HOLogic.mk_setT iT] ---> bT);
   373 fun mkISet cts = fold_rev (Thm.capply insert_tm #> Thm.capply) cts emptyIS;
   374 val cTrp = @{cterm "Trueprop"};
   375 val eqelem_imp_imp = (thm"eqelem_imp_iff") RS iffD1;
   376 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 
   377                                       |> Thm.dest_abs NONE |> snd |> Thm.dest_fun |> Thm.dest_arg)
   378                       [asetP,bsetP];
   379 
   380 val D_tm = @{cpat "?D::int"};
   381 
   382 fun cooperex_conv ctxt vs q = 
   383 let 
   384 
   385  val uth = unify ctxt q
   386  val (x,p) = Thm.dest_abs NONE (Thm.dest_arg (Thm.rhs_of uth))
   387  val ins = insert (op aconvc)
   388  fun h t (bacc,aacc,dacc) = 
   389   case (whatis x t) of
   390     And (p,q) => h q (h p (bacc,aacc,dacc))
   391   | Or (p,q) => h q  (h p (bacc,aacc,dacc))
   392   | Eq t => (ins (minus1 t) bacc, 
   393              ins (plus1 t) aacc,dacc)
   394   | NEq t => (ins t bacc, 
   395               ins t aacc, dacc)
   396   | Lt t => (bacc, ins t aacc, dacc)
   397   | Le t => (bacc, ins (plus1 t) aacc,dacc)
   398   | Gt t => (ins t bacc, aacc,dacc)
   399   | Ge t => (ins (minus1 t) bacc, aacc,dacc)
   400   | Dvd (d,s) => (bacc,aacc,insert (op =) (term_of d |> dest_numeral) dacc)
   401   | NDvd (d,s) => (bacc,aacc,insert (op =) (term_of d|> dest_numeral) dacc)
   402   | _ => (bacc, aacc, dacc)
   403  val (b0,a0,ds) = h p ([],[],[])
   404  val d = Integer.lcms ds
   405  val cd = Numeral.mk_cnumber @{ctyp "int"} d
   406  val dt = term_of cd
   407  fun divprop x = 
   408    let 
   409     val th = 
   410      Simplifier.rewrite lin_ss 
   411       (Thm.capply @{cterm Trueprop} 
   412            (Thm.capply (Thm.capply dvdc (Numeral.mk_cnumber @{ctyp "int"} x)) cd))
   413    in equal_elim (Thm.symmetric th) TrueI end;
   414  val dvd = let val tab = fold Inttab.update
   415                                (ds ~~ (map divprop ds)) Inttab.empty in 
   416            (fn ct => (valOf (Inttab.lookup tab (term_of ct |> dest_numeral)) 
   417                     handle Option => (writeln "dvd: Theorems-Table contains no entry for"; 
   418                                       Display.print_cterm ct ; raise Option)))
   419            end
   420  val dp = 
   421    let val th = Simplifier.rewrite lin_ss 
   422       (Thm.capply @{cterm Trueprop} 
   423            (Thm.capply (Thm.capply @{cterm "op < :: int => _"} @{cterm "0::int"}) cd))
   424    in equal_elim (Thm.symmetric th) TrueI end;
   425     (* A and B set *)
   426    local 
   427      val insI1 = instantiate' [SOME @{ctyp "int"}] [] @{thm "insertI1"}
   428      val insI2 = instantiate' [SOME @{ctyp "int"}] [] @{thm "insertI2"}
   429    in
   430     fun provein x S = 
   431      case term_of S of
   432         Const("{}",_) => error "Unexpected error in Cooper please email Amine Chaieb"
   433       | Const("insert",_)$y$_ => 
   434          let val (cy,S') = Thm.dest_binop S
   435          in if term_of x aconv y then instantiate' [] [SOME x, SOME S'] insI1
   436          else implies_elim (instantiate' [] [SOME x, SOME S', SOME cy] insI2) 
   437                            (provein x S')
   438          end
   439    end
   440  
   441  val al = map (lint vs o term_of) a0
   442  val bl = map (lint vs o term_of) b0
   443  val (sl,s0,f,abths,cpth) = 
   444    if length (distinct (op aconv) bl) <= length (distinct (op aconv) al) 
   445    then  
   446     (bl,b0,decomp_minf,
   447      fn B => (map (fn th => implies_elim (Thm.instantiate ([],[(B_tm,B), (D_tm,cd)]) th) dp) 
   448                      [bseteq,bsetneq,bsetlt, bsetle, bsetgt,bsetge])@
   449                    (map (Thm.instantiate ([],[(B_tm,B), (D_tm,cd)])) 
   450                         [bsetdvd,bsetndvd,bsetP,infDdvd, infDndvd,bsetconj,
   451                          bsetdisj,infDconj, infDdisj]),
   452                        cpmi) 
   453      else (al,a0,decomp_pinf,fn A => 
   454           (map (fn th => implies_elim (Thm.instantiate ([],[(A_tm,A), (D_tm,cd)]) th) dp)
   455                    [aseteq,asetneq,asetlt, asetle, asetgt,asetge])@
   456                    (map (Thm.instantiate ([],[(A_tm,A), (D_tm,cd)])) 
   457                    [asetdvd,asetndvd, asetP, infDdvd, infDndvd,asetconj,
   458                          asetdisj,infDconj, infDdisj]),cppi)
   459  val cpth = 
   460   let
   461    val sths = map (fn (tl,t0) => 
   462                       if tl = term_of t0 
   463                       then instantiate' [SOME @{ctyp "int"}] [SOME t0] refl
   464                       else provelin ctxt ((HOLogic.eq_const iT)$tl$(term_of t0) 
   465                                  |> HOLogic.mk_Trueprop)) 
   466                    (sl ~~ s0)
   467    val csl = distinct (op aconvc) (map (cprop_of #> Thm.dest_arg #> Thm.dest_arg1) sths)
   468    val S = mkISet csl
   469    val inStab = fold (fn ct => fn tab => Termtab.update (term_of ct, provein ct S) tab) 
   470                     csl Termtab.empty
   471    val eqelem_th = instantiate' [SOME @{ctyp "int"}] [NONE,NONE, SOME S] eqelem_imp_imp
   472    val inS = 
   473      let 
   474       fun transmem th0 th1 = 
   475        Thm.equal_elim 
   476         (Drule.arg_cong_rule cTrp (Drule.fun_cong_rule (Drule.arg_cong_rule 
   477                ((Thm.dest_fun o Thm.dest_fun o Thm.dest_arg o cprop_of) th1) th0) S)) th1
   478       val tab = fold Termtab.update
   479         (map (fn eq => 
   480                 let val (s,t) = cprop_of eq |> Thm.dest_arg |> Thm.dest_binop 
   481                     val th = if term_of s = term_of t 
   482                              then valOf(Termtab.lookup inStab (term_of s))
   483                              else FWD (instantiate' [] [SOME s, SOME t] eqelem_th) 
   484                                 [eq, valOf(Termtab.lookup inStab (term_of s))]
   485                  in (term_of t, th) end)
   486                   sths) Termtab.empty
   487         in fn ct => 
   488           (valOf (Termtab.lookup tab (term_of ct))
   489            handle Option => (writeln "inS: No theorem for " ; Display.print_cterm ct ; raise Option))
   490         end
   491        val (inf, nb, pd) = divide_and_conquer (f x dvd inS (abths S)) p
   492    in [dp, inf, nb, pd] MRS cpth
   493    end
   494  val cpth' = Thm.transitive uth (cpth RS eq_reflection)
   495 in Thm.transitive cpth' ((simp_thms_conv ctxt then_conv eval_conv) (Thm.rhs_of cpth'))
   496 end;
   497 
   498 fun literals_conv bops uops env cv = 
   499  let fun h t =
   500   case (term_of t) of 
   501    b$_$_ => if member (op aconv) bops b then binop_conv h t else cv env t
   502  | u$_ => if member (op aconv) uops u then arg_conv h t else cv env t
   503  | _ => cv env t
   504  in h end;
   505 
   506 fun integer_nnf_conv ctxt env =
   507  nnf_conv then_conv literals_conv [HOLogic.conj, HOLogic.disj] [] env (linearize_conv ctxt);
   508 
   509 local
   510  val pcv = Simplifier.rewrite 
   511      (HOL_basic_ss addsimps (simp_thms @ (List.take(ex_simps,4)) 
   512                       @ [not_all,all_not_ex, ex_disj_distrib]))
   513  val postcv = Simplifier.rewrite presburger_ss
   514  fun conv ctxt p = 
   515   let val _ = ()
   516   in
   517    Qelim.gen_qelim_conv pcv postcv pcv (cons o term_of) 
   518       (term_frees (term_of p)) (linearize_conv ctxt) (integer_nnf_conv ctxt) 
   519       (cooperex_conv ctxt) p 
   520   end
   521   handle  CTERM s => raise COOPER ("Cooper Failed", CTERM s)
   522         | THM s => raise COOPER ("Cooper Failed", THM s) 
   523         | TYPE s => raise COOPER ("Cooper Failed", TYPE s) 
   524 in val cooper_conv = conv 
   525 end;
   526 end;
   527 
   528 
   529 
   530 structure Coopereif =
   531 struct
   532 
   533 open GeneratedCooper;
   534 
   535 fun cooper s = raise Cooper.COOPER ("Cooper oracle failed", ERROR s);
   536 fun i_of_term vs t = case t
   537  of Free (xn, xT) => (case AList.lookup (op aconv) vs t
   538      of NONE   => cooper "Variable not found in the list!"
   539       | SOME n => Bound n)
   540   | @{term "0::int"} => C 0
   541   | @{term "1::int"} => C 1
   542   | Term.Bound i => Bound i
   543   | Const(@{const_name HOL.uminus},_)$t' => Neg (i_of_term vs t')
   544   | Const(@{const_name HOL.plus},_)$t1$t2 => Add (i_of_term vs t1,i_of_term vs t2)
   545   | Const(@{const_name HOL.minus},_)$t1$t2 => Sub (i_of_term vs t1,i_of_term vs t2)
   546   | Const(@{const_name HOL.times},_)$t1$t2 => 
   547      (Mul (HOLogic.dest_number t1 |> snd, i_of_term vs t2)
   548     handle TERM _ => 
   549        (Mul (HOLogic.dest_number t2 |> snd, i_of_term vs t1)
   550         handle TERM _ => cooper "Reification: Unsupported kind of multiplication"))
   551   | _ => (C (HOLogic.dest_number t |> snd) 
   552            handle TERM _ => cooper "Reification: unknown term");
   553 
   554 fun qf_of_term ps vs t =  case t
   555  of Const("True",_) => T
   556   | Const("False",_) => F
   557   | Const(@{const_name HOL.less},_)$t1$t2 => Lt (Sub (i_of_term vs t1,i_of_term vs t2))
   558   | Const(@{const_name HOL.less_eq},_)$t1$t2 => Le (Sub(i_of_term vs t1,i_of_term vs t2))
   559   | Const(@{const_name Ring_and_Field.dvd},_)$t1$t2 => 
   560       (Dvd(HOLogic.dest_number t1 |> snd, i_of_term vs t2) handle _ => cooper "Reification: unsupported dvd")  (* FIXME avoid handle _ *)
   561   | @{term "op = :: int => _"}$t1$t2 => Eq (Sub (i_of_term vs t1,i_of_term vs t2))
   562   | @{term "op = :: bool => _ "}$t1$t2 => Iffa(qf_of_term ps vs t1,qf_of_term ps vs t2)
   563   | Const("op &",_)$t1$t2 => And(qf_of_term ps vs t1,qf_of_term ps vs t2)
   564   | Const("op |",_)$t1$t2 => Or(qf_of_term ps vs t1,qf_of_term ps vs t2)
   565   | Const("op -->",_)$t1$t2 => Impa(qf_of_term ps vs t1,qf_of_term ps vs t2)
   566   | Const (@{const_name Not},_)$t' => Nota(qf_of_term ps vs t')
   567   | Const("Ex",_)$Abs(xn,xT,p) => 
   568      let val (xn',p') = variant_abs (xn,xT,p)
   569          val vs' = (Free (xn',xT), 0) :: (map (fn(v,n) => (v,1+ n)) vs)
   570      in E (qf_of_term ps vs' p')
   571      end
   572   | Const("All",_)$Abs(xn,xT,p) => 
   573      let val (xn',p') = variant_abs (xn,xT,p)
   574          val vs' = (Free (xn',xT), 0) :: (map (fn(v,n) => (v,1+ n)) vs)
   575      in A (qf_of_term ps vs' p')
   576      end
   577   | _ =>(case AList.lookup (op aconv) ps t of 
   578            NONE => cooper "Reification: unknown term!"
   579          | SOME n => Closed n);
   580 
   581 local
   582  val ops = [@{term "op &"}, @{term "op |"}, @{term "op -->"}, @{term "op = :: bool => _"},
   583              @{term "op = :: int => _"}, @{term "op < :: int => _"}, 
   584              @{term "op <= :: int => _"}, @{term "Not"}, @{term "All:: (int => _) => _"}, 
   585              @{term "Ex:: (int => _) => _"}, @{term "True"}, @{term "False"}]
   586 fun ty t = Bool.not (fastype_of t = HOLogic.boolT)
   587 in
   588 fun term_bools acc t =
   589 case t of 
   590     (l as f $ a) $ b => if ty t orelse f mem ops then term_bools (term_bools acc l)b 
   591             else insert (op aconv) t acc
   592   | f $ a => if ty t orelse f mem ops then term_bools (term_bools acc f) a  
   593             else insert (op aconv) t acc
   594   | Abs p => term_bools acc (snd (variant_abs p))
   595   | _ => if ty t orelse t mem ops then acc else insert (op aconv) t acc
   596 end;
   597  
   598 fun myassoc2 l v =
   599     case l of
   600 	[] => NONE
   601       | (x,v')::xs => if v = v' then SOME x
   602 		      else myassoc2 xs v;
   603 
   604 fun term_of_i vs t = case t
   605  of C i => HOLogic.mk_number HOLogic.intT i
   606   | Bound n => the (myassoc2 vs n)
   607   | Neg t' => @{term "uminus :: int => _"} $ term_of_i vs t'
   608   | Add (t1, t2) => @{term "op + :: int => _"} $ term_of_i vs t1 $ term_of_i vs t2
   609   | Sub (t1, t2) => @{term "op - :: int => _"} $ term_of_i vs t1 $ term_of_i vs t2
   610   | Mul (i, t2) => @{term "op * :: int => _"} $
   611       HOLogic.mk_number HOLogic.intT i $ term_of_i vs t2
   612   | Cx (i, t') => term_of_i vs (Add (Mul (i, Bound 0), t'));
   613 
   614 fun term_of_qf ps vs t = 
   615  case t of 
   616    T => HOLogic.true_const 
   617  | F => HOLogic.false_const
   618  | Lt t' => @{term "op < :: int => _ "}$ term_of_i vs t'$ @{term "0::int"}
   619  | Le t' => @{term "op <= :: int => _ "}$ term_of_i vs t' $ @{term "0::int"}
   620  | Gt t' => @{term "op < :: int => _ "}$ @{term "0::int"}$ term_of_i vs t'
   621  | Ge t' => @{term "op <= :: int => _ "}$ @{term "0::int"}$ term_of_i vs t'
   622  | Eq t' => @{term "op = :: int => _ "}$ term_of_i vs t'$ @{term "0::int"}
   623  | NEq t' => term_of_qf ps vs (Nota (Eq t'))
   624  | Dvd(i,t') => @{term "op dvd :: int => _ "} $ 
   625     HOLogic.mk_number HOLogic.intT i $ term_of_i vs t'
   626  | NDvd(i,t')=> term_of_qf ps vs (Nota(Dvd(i,t')))
   627  | Nota t' => HOLogic.Not$(term_of_qf ps vs t')
   628  | And(t1,t2) => HOLogic.conj$(term_of_qf ps vs t1)$(term_of_qf ps vs t2)
   629  | Or(t1,t2) => HOLogic.disj$(term_of_qf ps vs t1)$(term_of_qf ps vs t2)
   630  | Impa(t1,t2) => HOLogic.imp$(term_of_qf ps vs t1)$(term_of_qf ps vs t2)
   631  | Iffa(t1,t2) => @{term "op = :: bool => _"} $ term_of_qf ps vs t1 $ term_of_qf ps vs t2
   632  | Closed n => the (myassoc2 ps n)
   633  | NClosed n => term_of_qf ps vs (Nota (Closed n))
   634  | _ => cooper "If this is raised, Isabelle/HOL or generate_code is inconsistent!";
   635 
   636 fun cooper_oracle ct =
   637   let
   638     val thy = Thm.theory_of_cterm ct;
   639     val t = Thm.term_of ct;
   640     val (vs, ps) = pairself (map_index swap) (term_frees t, term_bools [] t);
   641   in
   642     Thm.cterm_of thy (Logic.mk_equals (HOLogic.mk_Trueprop t,
   643       HOLogic.mk_Trueprop (term_of_qf ps vs (pa (qf_of_term ps vs t)))))
   644   end;
   645 
   646 end;