(* 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: Proof.context -> conv -> conv -> conv -> (cterm -> 'a -> 'a) -> 'a ->
('a -> conv) -> ('a -> conv) -> ('a -> conv) -> conv
val standard_qelim_conv: Proof.context ->
(cterm list -> conv) -> (cterm list -> conv) ->
(cterm list -> conv) -> conv
end;
structure Qelim: QELIM =
struct
fun gen_qelim_conv ctxt precv postcv simpex_conv ins env atcv ncv qcv =
let
fun conv env p =
(case Thm.term_of p of
\<^Const_>\<open>conj for _ _\<close> => Conv.binop_conv (conv env) p
| \<^Const_>\<open>disj for _ _\<close> => Conv.binop_conv (conv env) p
| \<^Const_>\<open>implies for _ _\<close> => Conv.binop_conv (conv env) p
| \<^Const_>\<open>HOL.eq _ for _ _\<close> => Conv.binop_conv (conv env) p
| \<^Const_>\<open>Not for _\<close> => Conv.arg_conv (conv env) p
| \<^Const_>\<open>Ex _ for \<open>Abs (s, _, _)\<close>\<close> =>
let
val (e,p0) = Thm.dest_comb p
val (x,p') = Thm.dest_abs_global 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 (_, r) = Thm.dest_equals (Thm.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_>\<open>Ex _ for _\<close> => (Thm.eta_long_conversion then_conv conv env) p
| \<^Const_>\<open>All _ for _\<close> =>
let
val allT = Thm.ctyp_of_cterm (Thm.dest_fun p)
val T = Thm.dest_ctyp0 (Thm.dest_ctyp0 allT)
val P = Thm.dest_arg p
val th = \<^instantiate>\<open>'a = T and P in lemma "\<forall>x::'a. P x \<equiv> \<nexists>x. \<not> P x" by simp\<close>
in Thm.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 ss =
simpset_of
(put_simpset HOL_basic_ss \<^context>
addsimps @{thms simp_thms ex_simps all_simps all_not_ex not_all ex_disj_distrib});
fun pcv ctxt = Simplifier.rewrite (put_simpset ss ctxt)
in
fun standard_qelim_conv ctxt atcv ncv qcv p =
let
val pcv = pcv ctxt
val env = Cterms.list_set_rev (Cterms.build (Drule.add_frees_cterm p))
in gen_qelim_conv ctxt pcv pcv pcv cons env atcv ncv qcv p end
end;
end;