src/HOL/ex/coopereif.ML

Updated reification : CX discontinued for CN

(*  ID:         $Id$
    Author:     Amine Chaieb, TU Muenchen

Reification for the automatically generated oracle for Presburger arithmetic
in HOL/ex/Reflected_Presburger.thy.
*)

structure Coopereif =
struct

open GeneratedCooper;

val nat = GeneratedCooper.nat o Integer.int;

fun i_of_term vs t = case t
 of Free(xn,xT) => (case AList.lookup (op aconv) vs t
   of NONE   => error "Variable not found in the list!"
    | SOME n => Bound n)
    | @{term "0::int"} => C 0
    | @{term "1::int"} => C 1
    | Term.Bound i => Bound (nat i)
    | Const(@{const_name "HOL.uminus"},_)$t' => Neg (i_of_term vs t')
    | Const(@{const_name "HOL.plus"},_)$t1$t2 => Add (i_of_term vs t1,i_of_term vs t2)
    | Const(@{const_name "HOL.minus"},_)$t1$t2 => Sub (i_of_term vs t1,i_of_term vs t2)
    | Const(@{const_name "HOL.times"},_)$t1$t2 => (Mul (HOLogic.dest_number t1 |> snd,i_of_term vs t2)
        handle TERM _ => 
           (Mul (HOLogic.dest_number t2 |> snd,i_of_term vs t1)
            handle TERM _ => error "i_of_term: Unsupported kind of multiplication"))
    | _ => (C (HOLogic.dest_number t |> snd) 
             handle TERM _ => error "i_of_term: unknown term");

fun qf_of_term ps vs t = case t
     of Const("True",_) => T
      | Const("False",_) => F
      | Const(@{const_name HOL.less},_)$t1$t2 => Lt (Sub (i_of_term vs t1,i_of_term vs t2))
      | Const(@{const_name HOL.less_eq},_)$t1$t2 => Le (Sub(i_of_term vs t1,i_of_term vs t2))
      | Const(@{const_name "Divides.dvd"},_)$t1$t2 => 
          (Dvd(HOLogic.dest_number t1 |> snd, i_of_term vs t2) handle _ => error "qf_of_term: unsupported dvd")
      | @{term "op = :: int => _"}$t1$t2 => Eq (Sub (i_of_term vs t1,i_of_term vs t2))
      | @{term "op = :: bool => _ "}$t1$t2 => Iffa(qf_of_term ps vs t1,qf_of_term ps vs t2)
      | Const("op &",_)$t1$t2 => And(qf_of_term ps vs t1,qf_of_term ps vs t2)
      | Const("op |",_)$t1$t2 => Or(qf_of_term ps vs t1,qf_of_term ps vs t2)
      | Const("op -->",_)$t1$t2 => Impa(qf_of_term ps vs t1,qf_of_term ps vs t2)
      | Const("Not",_)$t' => Nota(qf_of_term ps vs t')
      | Const("Ex",_)$Abs(xn,xT,p) => 
         let val (xn',p') = variant_abs (xn,xT,p)
             val vs' = (Free (xn',xT), nat 0) :: (map (fn(v,n) => (v,1 + n)) vs)
         in E (qf_of_term ps vs' p')
         end
      | Const("All",_)$Abs(xn,xT,p) => 
         let val (xn',p') = variant_abs (xn,xT,p)
             val vs' = (Free (xn',xT), nat 0) :: (map (fn(v,n) => (v,1 + n)) vs)
         in A (qf_of_term ps vs' p')
         end
      | _ =>(case AList.lookup (op aconv) ps t of 
               NONE => error "qf_of_term ps : unknown term!"
             | SOME n => Closed n);

local
  val ops = [@{term "op &"}, @{term "op |"}, @{term "op -->"}, @{term "op = :: bool => _"},
    @{term "op = :: int => _"}, @{term "op < :: int => _"},
    @{term "op <= :: int => _"}, @{term "Not"}, @{term "All:: (int => _) => _"},
    @{term "Ex:: (int => _) => _"}, @{term "True"}, @{term "False"}]
  fun ty t = Bool.not (fastype_of t = HOLogic.boolT)
in

fun term_bools acc t = case t
 of (l as f $ a) $ b => if ty t orelse f mem ops then term_bools (term_bools acc l)b 
      else insert (op aconv) t acc
  | f $ a => if ty t orelse f mem ops then term_bools (term_bools acc f) a  
      else insert (op aconv) t acc
  | Abs p => term_bools acc (snd (variant_abs p))
  | _ => if ty t orelse t mem ops then acc else insert (op aconv) t acc

end;

fun start_vs t =
  let
    val fs = term_frees t
    val ps = term_bools [] t
  in
    (fs ~~ (map nat (0 upto  (length fs - 1))),
      ps ~~ (map nat (0 upto  (length ps - 1))))
  end;

fun term_of_i vs t = case t
 of C i => HOLogic.mk_number HOLogic.intT i
  | Bound n => (fst o the) (find_first (fn (_, m) => m = n) vs)
  | Neg t' => @{term "uminus :: int => _"} $ term_of_i vs t'
  | Add (t1, t2) => @{term "op +:: int => _"} $ term_of_i vs t1 $ term_of_i vs t2
  | Sub (t1, t2) => Const (@{const_name "HOL.minus"}, HOLogic.intT --> HOLogic.intT --> HOLogic.intT) $
      term_of_i vs t1 $ term_of_i vs t2
  | Mul (i, t2) => Const (@{const_name "HOL.times"}, HOLogic.intT --> HOLogic.intT --> HOLogic.intT) $
      HOLogic.mk_number HOLogic.intT i $ term_of_i vs t2
  | Cn (n,i, t') => term_of_i vs (Add (Mul (i, Bound (nat n)), t'));

fun term_of_qf ps vs t = case t
 of T => HOLogic.true_const 
  | F => HOLogic.false_const
  | Lt t' => @{term "op < :: int => _ "}$ term_of_i vs t'$ @{term "0::int"}
  | Le t' => @{term "op <= :: int => _ "}$ term_of_i vs t' $ @{term "0::int"}
  | Gt t' => @{term "op < :: int => _ "}$ @{term "0::int"}$ term_of_i vs t'
  | Ge t' => @{term "op <= :: int => _ "}$ @{term "0::int"}$ term_of_i vs t'
  | Eq t' => @{term "op = :: int => _ "}$ term_of_i vs t'$ @{term "0::int"}
  | NEq t' => term_of_qf ps vs (Nota(Eq t'))
  | Dvd(i,t') => @{term "op dvd :: int => _ "}$ 
      (HOLogic.mk_number HOLogic.intT i)$(term_of_i vs t')
  | NDvd(i,t')=> term_of_qf ps vs (Nota(Dvd(i,t')))
  | Nota t' => HOLogic.Not$(term_of_qf ps vs t')
  | And(t1,t2) => HOLogic.conj$(term_of_qf ps vs t1)$(term_of_qf ps vs t2)
  | Or(t1,t2) => HOLogic.disj$(term_of_qf ps vs t1)$(term_of_qf ps vs t2)
  | Impa(t1,t2) => HOLogic.imp$(term_of_qf ps vs t1)$(term_of_qf ps vs t2)
  | Iffa(t1,t2) => HOLogic.eq_const HOLogic.boolT $ term_of_qf ps vs t1 $ term_of_qf ps vs t2
  | Closed n => (fst o the) (find_first (fn (_, m) => m = n) ps)
  | NClosed n => term_of_qf ps vs (Nota (Closed n))
  | _ => error "If this is raised, Isabelle/HOL or generate_code is inconsistent!";

(* The oracle *)
fun cooper_oracle thy t = 
  let
    val (vs, ps) = start_vs t;
  in HOLogic.mk_Trueprop (HOLogic.mk_eq (t, term_of_qf ps vs (pa (qf_of_term ps vs t)))) end;

end;