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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(* *)
(* Micromega: A reflexive tactic using the Positivstellensatz *)
(* *)
(* Frédéric Besson (Irisa/Inria) 2006-2008 *)
(* *)
(************************************************************************)
Require Import List.
Require Import Refl.
Require Import Bool.
Set Implicit Arguments.
Inductive BFormula (A:Type) : Type :=
| TT : BFormula A
| FF : BFormula A
| X : Prop -> BFormula A
| A : A -> BFormula A
| Cj : BFormula A -> BFormula A -> BFormula A
| D : BFormula A-> BFormula A -> BFormula A
| N : BFormula A -> BFormula A
| I : BFormula A-> BFormula A-> BFormula A.
Fixpoint eval_f (A:Type) (ev:A -> Prop ) (f:BFormula A) {struct f}: Prop :=
match f with
| TT => True
| FF => False
| A a => ev a
| X p => p
| Cj e1 e2 => (eval_f ev e1) /\ (eval_f ev e2)
| D e1 e2 => (eval_f ev e1) \/ (eval_f ev e2)
| N e => ~ (eval_f ev e)
| I f1 f2 => (eval_f ev f1) -> (eval_f ev f2)
end.
Lemma map_simpl : forall A B f l, @map A B f l = match l with
| nil => nil
| a :: l=> (f a) :: (@map A B f l)
end.
Proof.
destruct l ; reflexivity.
Qed.
Section S.
Variable Env : Type.
Variable Term : Type.
Variable eval : Env -> Term -> Prop.
Variable Term' : Type.
Variable eval' : Env -> Term' -> Prop.
Variable no_middle_eval' : forall env d, (eval' env d) \/ ~ (eval' env d).
Definition clause := list Term'.
Definition cnf := list clause.
Variable normalise : Term -> cnf.
Variable negate : Term -> cnf.
Definition tt : cnf := @nil clause.
Definition ff : cnf := cons (@nil Term') nil.
Definition or_clause_cnf (t:clause) (f:cnf) : cnf :=
List.map (fun x => (t++x)) f.
Fixpoint or_cnf (f : cnf) (f' : cnf) {struct f}: cnf :=
match f with
| nil => tt
| e :: rst => (or_cnf rst f') ++ (or_clause_cnf e f')
end.
Definition and_cnf (f1 : cnf) (f2 : cnf) : cnf :=
f1 ++ f2.
Fixpoint xcnf (pol : bool) (f : BFormula Term) {struct f}: cnf :=
match f with
| TT => if pol then tt else ff
| FF => if pol then ff else tt
| X p => if pol then ff else ff (* This is not complete - cannot negate any proposition *)
| A x => if pol then normalise x else negate x
| N e => xcnf (negb pol) e
| Cj e1 e2 =>
(if pol then and_cnf else or_cnf) (xcnf pol e1) (xcnf pol e2)
| D e1 e2 => (if pol then or_cnf else and_cnf) (xcnf pol e1) (xcnf pol e2)
| I e1 e2 => (if pol then or_cnf else and_cnf) (xcnf (negb pol) e1) (xcnf pol e2)
end.
Definition eval_cnf (env : Term' -> Prop) (f:cnf) := make_conj (fun cl => ~ make_conj env cl) f.
Lemma eval_cnf_app : forall env x y, eval_cnf (eval' env) (x++y) -> eval_cnf (eval' env) x /\ eval_cnf (eval' env) y.
Proof.
unfold eval_cnf.
intros.
rewrite make_conj_app in H ; auto.
Qed.
Lemma or_clause_correct : forall env t f, eval_cnf (eval' env) (or_clause_cnf t f) -> (~ make_conj (eval' env) t) \/ (eval_cnf (eval' env) f).
Proof.
unfold eval_cnf.
unfold or_clause_cnf.
induction f.
simpl.
intros ; right;auto.
(**)
rewrite map_simpl.
intros.
rewrite make_conj_cons in H.
destruct H as [HH1 HH2].
generalize (IHf HH2) ; clear IHf ; intro.
destruct H.
left ; auto.
rewrite make_conj_cons.
destruct (not_make_conj_app _ _ _ (no_middle_eval' env) HH1).
tauto.
tauto.
Qed.
Lemma eval_cnf_cons : forall env a f, (~ make_conj (eval' env) a) -> eval_cnf (eval' env) f -> eval_cnf (eval' env) (a::f).
Proof.
intros.
unfold eval_cnf in *.
rewrite make_conj_cons ; eauto.
Qed.
Lemma or_cnf_correct : forall env f f', eval_cnf (eval' env) (or_cnf f f') -> (eval_cnf (eval' env) f) \/ (eval_cnf (eval' env) f').
Proof.
induction f.
unfold eval_cnf.
simpl.
tauto.
(**)
intros.
simpl in H.
destruct (eval_cnf_app _ _ _ H).
clear H.
destruct (IHf _ H0).
destruct (or_clause_correct _ _ _ H1).
left.
apply eval_cnf_cons ; auto.
right ; auto.
right ; auto.
Qed.
Variable normalise_correct : forall env t, eval_cnf (eval' env) (normalise t) -> eval env t.
Variable negate_correct : forall env t, eval_cnf (eval' env) (negate t) -> ~ eval env t.
Lemma xcnf_correct : forall f pol env, eval_cnf (eval' env) (xcnf pol f) -> eval_f (eval env) (if pol then f else N f).
Proof.
induction f.
(* TT *)
unfold eval_cnf.
simpl.
destruct pol ; simpl ; auto.
(* FF *)
unfold eval_cnf.
destruct pol; simpl ; auto.
(* P *)
simpl.
destruct pol ; intros ;simpl.
unfold eval_cnf in H.
(* Here I have to drop the proposition *)
simpl in H.
tauto.
(* Here, I could store P in the clause *)
unfold eval_cnf in H;simpl in H.
tauto.
(* A *)
simpl.
destruct pol ; simpl.
intros.
apply normalise_correct ; auto.
(* A 2 *)
intros.
apply negate_correct ; auto.
auto.
(* Cj *)
destruct pol ; simpl.
(* pol = true *)
intros.
unfold and_cnf in H.
destruct (eval_cnf_app _ _ _ H).
clear H.
split.
apply (IHf1 _ _ H0).
apply (IHf2 _ _ H1).
(* pol = false *)
intros.
destruct (or_cnf_correct _ _ _ H).
generalize (IHf1 false env H0).
simpl.
tauto.
generalize (IHf2 false env H0).
simpl.
tauto.
(* D *)
simpl.
destruct pol.
(* pol = true *)
intros.
destruct (or_cnf_correct _ _ _ H).
generalize (IHf1 _ env H0).
simpl.
tauto.
generalize (IHf2 _ env H0).
simpl.
tauto.
(* pol = true *)
unfold and_cnf.
intros.
destruct (eval_cnf_app _ _ _ H).
clear H.
simpl.
generalize (IHf1 _ _ H0).
generalize (IHf2 _ _ H1).
simpl.
tauto.
(**)
simpl.
destruct pol ; simpl.
intros.
apply (IHf false) ; auto.
intros.
generalize (IHf _ _ H).
tauto.
(* I *)
simpl; intros.
destruct pol.
simpl.
intro.
destruct (or_cnf_correct _ _ _ H).
generalize (IHf1 _ _ H1).
simpl in *.
tauto.
generalize (IHf2 _ _ H1).
auto.
(* pol = false *)
unfold and_cnf in H.
simpl in H.
destruct (eval_cnf_app _ _ _ H).
generalize (IHf1 _ _ H0).
generalize (IHf2 _ _ H1).
simpl.
tauto.
Qed.
Variable Witness : Type.
Variable checker : list Term' -> Witness -> bool.
Variable checker_sound : forall t w, checker t w = true -> forall env, make_impl (eval' env) t False.
Fixpoint cnf_checker (f : cnf) (l : list Witness) {struct f}: bool :=
match f with
| nil => true
| e::f => match l with
| nil => false
| c::l => match checker e c with
| true => cnf_checker f l
| _ => false
end
end
end.
Lemma cnf_checker_sound : forall t w, cnf_checker t w = true -> forall env, eval_cnf (eval' env) t.
Proof.
unfold eval_cnf.
induction t.
(* bc *)
simpl.
auto.
(* ic *)
simpl.
destruct w.
intros ; discriminate.
case_eq (checker a w) ; intros ; try discriminate.
generalize (@checker_sound _ _ H env).
generalize (IHt _ H0 env) ; intros.
destruct t.
red ; intro.
rewrite <- make_conj_impl in H2.
tauto.
rewrite <- make_conj_impl in H2.
tauto.
Qed.
Definition tauto_checker (f:BFormula Term) (w:list Witness) : bool :=
cnf_checker (xcnf true f) w.
Lemma tauto_checker_sound : forall t w, tauto_checker t w = true -> forall env, eval_f (eval env) t.
Proof.
unfold tauto_checker.
intros.
change (eval_f (eval env) t) with (eval_f (eval env) (if true then t else TT Term)).
apply (xcnf_correct t true).
eapply cnf_checker_sound ; eauto.
Qed.
End S.
(* Local Variables: *)
(* coding: utf-8 *)
(* End: *)
|
