(* Title: HOL/Complex/ex/mireif.ML ID: $Id: mireif.ML,v 1.5 2007/08/24 12:14:22 haftmann Exp $ 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 "Numeral.number_of"},_)$t' => C (HOLogic.dest_numeral t') | Const (@{const_name "Numeral.number_of"},_)$t' => C (HOLogic.dest_numeral t') | _ => error ("num_of_term: unknown term " ^ (Display.raw_string_of_term 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 "Numeral.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!" ^ Display.raw_string_of_term 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;