(* Title: HOL/Tools/Qelim/qelim.ML
Author: Amine Chaieb, TU Muenchen
Generic quantifier elimination conversions for HOL formulae.
*)
signature QELIM =
sig
val gen_qelim_conv: conv -> conv -> conv -> (cterm -> 'a -> 'a) -> 'a
-> ('a -> conv) -> ('a -> conv) -> ('a -> conv) -> conv
val standard_qelim_conv: (cterm list -> conv) -> (cterm list -> conv)
-> (cterm list -> conv) -> conv
end;
structure Qelim: QELIM =
struct
open Conv;
val all_not_ex = mk_meta_eq @{thm "all_not_ex"};
fun gen_qelim_conv precv postcv simpex_conv ins env atcv ncv qcv =
let
fun conv env p =
case (term_of p) of
Const(s,T)$_$_ =>
if domain_type T = HOLogic.boolT
andalso member (op =) [@{const_name "op &"}, @{const_name "op |"},
@{const_name "op -->"}, @{const_name "op ="}] s
then binop_conv (conv env) p
else atcv env p
| Const(@{const_name "Not"},_)$_ => arg_conv (conv env) p
| Const(@{const_name "Ex"},_)$Abs(s,_,_) =>
let
val (e,p0) = Thm.dest_comb p
val (x,p') = Thm.dest_abs (SOME s) p0
val env' = ins x env
val th = Thm.abstract_rule s x ((conv env' then_conv ncv env') p')
|> Drule.arg_cong_rule e
val th' = simpex_conv (Thm.rhs_of th)
val (l,r) = Thm.dest_equals (cprop_of th')
in if Thm.is_reflexive th' then Thm.transitive th (qcv env (Thm.rhs_of th))
else Thm.transitive (Thm.transitive th th') (conv env r) end
| Const(@{const_name "Ex"},_)$ _ => (Thm.eta_long_conversion then_conv conv env) p
| Const(@{const_name "All"},_)$_ =>
let
val p = Thm.dest_arg p
val ([(_,T)],[]) = Thm.match (@{cpat "All"}, Thm.dest_fun p)
val th = instantiate' [SOME T] [SOME p] all_not_ex
in transitive th (conv env (Thm.rhs_of th))
end
| _ => atcv env p
in precv then_conv (conv env) then_conv postcv end
(* Instantiation of some parameter for most common cases *)
local
val pcv = Simplifier.rewrite
(HOL_basic_ss addsimps (simp_thms @ ex_simps @ all_simps)
@ [@{thm "all_not_ex"}, not_all, ex_disj_distrib]);
in fun standard_qelim_conv atcv ncv qcv p =
gen_qelim_conv pcv pcv pcv cons (Thm.add_cterm_frees p []) atcv ncv qcv p
end;
end;