(* Title: HOL/Complex/ex/mireif.ML
ID: $Id$
Author: Amine Chaieb, TU Muenchen
Oracle for Mixed Real-Integer auantifier elimination
based on the verified code in HOL/Complex/ex/MIR.thy.
*)
structure ReflectedMir =
struct
open Mir;
exception MIR;
fun num_of_term vs t =
case t of
Free(xn,xT) => (case AList.lookup (op =) vs t of
NONE => error "Variable not found in the list!"
| SOME n => Bound n)
| Const("RealDef.real",_)$ @{term "0::int"} => C 0
| Const("RealDef.real",_)$ @{term "1::int"} => C 1
| @{term "0::real"} => C 0
| @{term "1::real"} => C 1
| Term.Bound i => Bound (nat i)
| Const(@{const_name "HOL.uminus"},_)$t' => Neg (num_of_term vs t')
| Const (@{const_name "HOL.plus"},_)$t1$t2 => Add (num_of_term vs t1,num_of_term vs t2)
| Const (@{const_name "HOL.minus"},_)$t1$t2 => Sub (num_of_term vs t1,num_of_term vs t2)
| Const (@{const_name "HOL.times"},_)$t1$t2 =>
(case (num_of_term vs t1) of C i =>
Mul (i,num_of_term vs t2)
| _ => error "num_of_term: unsupported Multiplication")
| Const("RealDef.real",_)$ (Const (@{const_name "RComplete.floor"},_)$ t') => Floor (num_of_term vs t')
| Const("RealDef.real",_)$ (Const (@{const_name "RComplete.ceiling"},_)$ t') => Neg(Floor (Neg (num_of_term vs t')))
| Const("RealDef.real",_) $ Const (@{const_name "Int.number_of"},_)$t' => C (HOLogic.dest_numeral t')
| Const (@{const_name "Int.number_of"},_)$t' => C (HOLogic.dest_numeral t')
| _ => error ("num_of_term: unknown term " ^ Syntax.string_of_term_global CPure.thy t);
(* pseudo reification : term -> fm *)
fun fm_of_term vs t =
case t of
Const("True",_) => T
| Const("False",_) => F
| Const(@{const_name HOL.less},_)$t1$t2 => Lt (Sub (num_of_term vs t1,num_of_term vs t2))
| Const(@{const_name HOL.less_eq},_)$t1$t2 => Le (Sub (num_of_term vs t1,num_of_term vs t2))
| Const (@{const_name "MIR.rdvd"},_ )$ (Const("RealDef.real",_) $ (Const(@{const_name "Int.number_of"},_)$t1))$t2 =>
Dvd (HOLogic.dest_numeral t1, num_of_term vs t2)
| Const("op =",eqT)$t1$t2 =>
if (domain_type eqT = @{typ real})
then Eq (Sub (num_of_term vs t1, num_of_term vs t2))
else Iff (fm_of_term vs t1, fm_of_term vs t2)
| Const("op &",_)$t1$t2 => And (fm_of_term vs t1, fm_of_term vs t2)
| Const("op |",_)$t1$t2 => Or (fm_of_term vs t1, fm_of_term vs t2)
| Const("op -->",_)$t1$t2 => Imp (fm_of_term vs t1, fm_of_term vs t2)
| Const("Not",_)$t' => Not (fm_of_term vs t')
| Const("Ex",_)$Abs(xn,xT,p) =>
E (fm_of_term (map (fn (v, n) => (v, Suc n)) vs) p)
| Const("All",_)$Abs(xn,xT,p) =>
A (fm_of_term (map (fn(v, n) => (v, Suc n)) vs) p)
| _ => error ("fm_of_term : unknown term!" ^ Syntax.string_of_term_global CPure.thy t);
fun start_vs t =
let val fs = term_frees t
in fs ~~ map nat (0 upto (length fs - 1))
end ;
(* transform num and fm back to terms *)
fun myassoc2 l v =
case l of
[] => NONE
| (x,v')::xs => if v = v' then SOME x
else myassoc2 xs v;
val realC = @{term "real :: int => _"};
val rzero = @{term "0::real"};
fun term_of_num vs t =
case t of
C i => realC $ (HOLogic.mk_number HOLogic.intT i)
| Bound n => valOf (myassoc2 vs n)
| Neg (Floor (Neg t')) => realC $ (@{term "ceiling"} $ term_of_num vs t')
| Neg t' => @{term "uminus:: real => _"} $ term_of_num vs t'
| Add(t1,t2) => @{term "op +:: real => _"} $ term_of_num vs t1 $ term_of_num vs t2
| Sub(t1,t2) => @{term "op -:: real => _"} $ term_of_num vs t1 $ term_of_num vs t2
| Mul(i,t2) => @{term "op -:: real => _"} $ term_of_num vs (C i) $ term_of_num vs t2
| Floor t => realC $ (@{term "floor"} $ term_of_num vs t)
| Cn(n,i,t) => term_of_num vs (Add(Mul(i,Bound n),t))
| Cf(c,t,s) => term_of_num vs (Add(Mul(c,Floor t),s));
fun term_of_fm vs t =
case t of
T => HOLogic.true_const
| F => HOLogic.false_const
| Lt t => @{term "op <:: real => _"} $ term_of_num vs t $ rzero
| Le t => @{term "op <=:: real => _"} $ term_of_num vs t $ rzero
| Gt t => @{term "op <:: real => _"}$ rzero $ term_of_num vs t
| Ge t => @{term "op <=:: real => _"} $ rzero $ term_of_num vs t
| Eq t => @{term "op = :: real => _"}$ term_of_num vs t $ rzero
| NEq t => term_of_fm vs (Not (Eq t))
| NDvd (i,t) => term_of_fm vs (Not (Dvd (i,t)))
| Dvd (i,t) => @{term "op rdvd"} $ term_of_num vs (C i) $ term_of_num vs t
| Not t' => HOLogic.Not$(term_of_fm vs t')
| And(t1,t2) => HOLogic.conj $ term_of_fm vs t1 $ term_of_fm vs t2
| Or(t1,t2) => HOLogic.disj $ term_of_fm vs t1 $ term_of_fm vs t2
| Imp(t1,t2) => HOLogic.imp $ term_of_fm vs t1 $ term_of_fm vs t2
| Iff(t1,t2) => HOLogic.mk_eq (term_of_fm vs t1, term_of_fm vs t2)
| _ => error "If this is raised, Isabelle/HOL or generate_code is inconsistent!";
(* The oracle *)
fun mircfr_oracle thy t =
let
val vs = start_vs t
in HOLogic.mk_Trueprop (HOLogic.mk_eq(t, term_of_fm vs (mircfrqe (fm_of_term vs t))))
end;
fun mirlfr_oracle thy t =
let
val vs = start_vs t
in HOLogic.mk_Trueprop (HOLogic.mk_eq(t, term_of_fm vs (mirlfrqe (fm_of_term vs t))))
end;
end;