The Coq Proof Assistant

# Library Coq.micromega.Tauto

Require Import List.
Require Import Refl.
Require Import Bool.
Require Import Relation_Definitions Setoid.
Set Implicit Arguments.
Formulae are either interpreted over Prop or bool.
Inductive kind : Type :=
|isProp
|isBool.
Register isProp as micromega.kind.isProp.
Register isBool as micromega.kind.isBool.
Inductive Trace (A : Type) :=
| null : Trace A
| push : A -> Trace A -> Trace A
| merge : Trace A -> Trace A -> Trace A
.
Section S.
Context {TA : Type}. Context {TX : kind -> Type}. Context {AA : Type}. Context {AF : Type}.
Inductive GFormula : kind -> Type :=
| TT : forall (k: kind), GFormula k
| FF : forall (k: kind), GFormula k
| X : forall (k: kind), TX k -> GFormula k
| A : forall (k: kind), TA -> AA -> GFormula k
| AND : forall (k: kind), GFormula k -> GFormula k -> GFormula k
| OR : forall (k: kind), GFormula k -> GFormula k -> GFormula k
| NOT : forall (k: kind), GFormula k -> GFormula k
| IMPL : forall (k: kind), GFormula k -> option AF -> GFormula k -> GFormula k
| IFF : forall (k: kind), GFormula k -> GFormula k -> GFormula k
| EQ : GFormula isBool -> GFormula isBool -> GFormula isProp.
Register TT as micromega.GFormula.TT.
Register FF as micromega.GFormula.FF.
Register X as micromega.GFormula.X.
Register A as micromega.GFormula.A.
Register AND as micromega.GFormula.AND.
Register OR as micromega.GFormula.OR.
Register NOT as micromega.GFormula.NOT.
Register IMPL as micromega.GFormula.IMPL.
Register IFF as micromega.GFormula.IFF.
Register EQ as micromega.GFormula.EQ.
Section MAPX.
Variable F : forall k, TX k -> TX k.
Fixpoint mapX (k:kind) (f : GFormula k) : GFormula k :=
match f with
| TT k => TT k
| FF k => FF k
| X x => X (F x)
| A k a an => A k a an
| AND f1 f2 => AND (mapX f1) (mapX f2)
| OR f1 f2 => OR (mapX f1) (mapX f2)
| NOT f => NOT (mapX f)
| IMPL f1 o f2 => IMPL (mapX f1) o (mapX f2)
| IFF f1 f2 => IFF (mapX f1) (mapX f2)
| EQ f1 f2 => EQ (mapX f1) (mapX f2)
end.
End MAPX.
Section FOLDANNOT.
Variable ACC : Type.
Variable F : ACC -> AA -> ACC.
Fixpoint foldA (k: kind) (f : GFormula k) (acc : ACC) : ACC :=
match f with
| TT _ => acc
| FF _ => acc
| X x => acc
| A _ a an => F acc an
| AND f1 f2
| OR f1 f2
| IFF f1 f2
| IMPL f1 _ f2 | EQ f1 f2 => foldA f1 (foldA f2 acc)
| NOT f => foldA f acc
end.
End FOLDANNOT.
Definition cons_id (id : option AF) (l : list AF) :=
match id with
| None => l
| Some id => id :: l
end.
Fixpoint ids_of_formula (k: kind) (f:GFormula k) :=
match f with
| IMPL f id f' => cons_id id (ids_of_formula f')
| _ => nil
end.
Fixpoint collect_annot (k: kind) (f : GFormula k) : list AA :=
match f with
| TT _ | FF _ | X _ => nil
| A _ _ a => a ::nil
| AND f1 f2
| OR f1 f2
| IFF f1 f2 | EQ f1 f2
| IMPL f1 _ f2 => collect_annot f1 ++ collect_annot f2
| NOT f => collect_annot f
end.
Definition rtyp (k: kind) : Type := if k then Prop else bool.
Variable ex : forall (k: kind), TX k -> rtyp k.
Section EVAL.
Variable ea : forall (k: kind), TA -> rtyp k.
Definition eTT (k: kind) : rtyp k :=
if k as k' return rtyp k' then True else true.
Definition eFF (k: kind) : rtyp k :=
if k as k' return rtyp k' then False else false.
Definition eAND (k: kind) : rtyp k -> rtyp k -> rtyp k :=
if k as k' return rtyp k' -> rtyp k' -> rtyp k'
then and else andb.
Definition eOR (k: kind) : rtyp k -> rtyp k -> rtyp k :=
if k as k' return rtyp k' -> rtyp k' -> rtyp k'
then or else orb.
Definition eIMPL (k: kind) : rtyp k -> rtyp k -> rtyp k :=
if k as k' return rtyp k' -> rtyp k' -> rtyp k'
then (fun x y => x -> y) else implb.
Definition eIFF (k: kind) : rtyp k -> rtyp k -> rtyp k :=
if k as k' return rtyp k' -> rtyp k' -> rtyp k'
then iff else eqb.
Definition eNOT (k: kind) : rtyp k -> rtyp k :=
if k as k' return rtyp k' -> rtyp k'
then not else negb.
Fixpoint eval_f (k: kind) (f:GFormula k) {struct f}: rtyp k :=
match f in GFormula k' return rtyp k' with
| TT tk => eTT tk
| FF tk => eFF tk
| A k a _ => ea k a
| X p => ex p
| @AND k e1 e2 => eAND k (eval_f e1) (eval_f e2)
| @OR k e1 e2 => eOR k (eval_f e1) (eval_f e2)
| @NOT k e => eNOT k (eval_f e)
| @IMPL k f1 _ f2 => eIMPL k (eval_f f1) (eval_f f2)
| @IFF k f1 f2 => eIFF k (eval_f f1) (eval_f f2)
| EQ f1 f2 => (eval_f f1) = (eval_f f2)
end.
Lemma eval_f_rew : forall k (f:GFormula k),
eval_f f =
match f in GFormula k' return rtyp k' with
| TT tk => eTT tk
| FF tk => eFF tk
| A k a _ => ea k a
| X p => ex p
| @AND k e1 e2 => eAND k (eval_f e1) (eval_f e2)
| @OR k e1 e2 => eOR k (eval_f e1) (eval_f e2)
| @NOT k e => eNOT k (eval_f e)
| @IMPL k f1 _ f2 => eIMPL k (eval_f f1) (eval_f f2)
| @IFF k f1 f2 => eIFF k (eval_f f1) (eval_f f2)
| EQ f1 f2 => (eval_f f1) = (eval_f f2)
end.
End EVAL.
Definition hold (k: kind) : rtyp k -> Prop :=
if k as k0 return (rtyp k0 -> Prop) then fun x => x else is_true.
Definition eiff (k: kind) : rtyp k -> rtyp k -> Prop :=
if k as k' return rtyp k' -> rtyp k' -> Prop then iff else @eq bool.
Lemma eiff_refl (k: kind) (x : rtyp k) :
eiff k x x.
Lemma eiff_sym k (x y : rtyp k) : eiff k x y -> eiff k y x.
Lemma eiff_trans k (x y z : rtyp k) : eiff k x y -> eiff k y z -> eiff k x z.
Lemma hold_eiff (k: kind) (x y : rtyp k) :
(hold k x <-> hold k y) <-> eiff k x y.
Instance eiff_eq (k: kind) : Equivalence (eiff k).
Add Parametric Morphism (k: kind) : (@eAND k) with signature eiff k ==> eiff k ==> eiff k as eAnd_morph.
Add Parametric Morphism (k: kind) : (@eOR k) with signature eiff k ==> eiff k ==> eiff k as eOR_morph.
Add Parametric Morphism (k: kind) : (@eIMPL k) with signature eiff k ==> eiff k ==> eiff k as eIMPL_morph.
Add Parametric Morphism (k: kind) : (@eIFF k) with signature eiff k ==> eiff k ==> eiff k as eIFF_morph.
Add Parametric Morphism (k: kind) : (@eNOT k) with signature eiff k ==> eiff k as eNOT_morph.
Lemma eval_f_morph :
forall (ev ev' : forall (k: kind), TA -> rtyp k),
(forall k a, eiff k (ev k a) (ev' k a)) ->
forall (k: kind)(f : GFormula k),
(eiff k (eval_f ev f) (eval_f ev' f)).
End S.
Typical boolean formulae
Definition eKind (k: kind) := if k then Prop else bool.
Register eKind as micromega.eKind.
Definition BFormula (A : Type) := @GFormula A eKind unit unit.
Register BFormula as micromega.BFormula.type.
Section MAPATOMS.
Context {TA TA':Type}.
Context {TX : kind -> Type}.
Context {AA : Type}.
Context {AF : Type}.
Fixpoint map_bformula (k: kind)(fct : TA -> TA') (f : @GFormula TA TX AA AF k) : @GFormula TA' TX AA AF k:=
match f with
| TT k => TT k
| FF k => FF k
| X k p => X k p
| A k a t => A k (fct a) t
| AND f1 f2 => AND (map_bformula fct f1) (map_bformula fct f2)
| OR f1 f2 => OR (map_bformula fct f1) (map_bformula fct f2)
| NOT f => NOT (map_bformula fct f)
| IMPL f1 a f2 => IMPL (map_bformula fct f1) a (map_bformula fct f2)
| IFF f1 f2 => IFF (map_bformula fct f1) (map_bformula fct f2)
| EQ f1 f2 => EQ (map_bformula fct f1) (map_bformula fct f2)
end.
End MAPATOMS.
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.
Section S.
A cnf tracking annotations of atoms.
Type parameters
Variable Env : Type.
Variable Term : Type.
Variable Term' : Type.
Variable Annot : Type.
Local Notation Trace := (Trace Annot).
Variable unsat : Term' -> bool. Variable deduce : Term' -> Term' -> option Term'.
Local Notation null := (@null Annot).
Local Notation push := (@push Annot).
Local Notation merge := (@merge Annot).
Definition clause := list (Term' * Annot).
Definition cnf := list clause.
Variable normalise : Term -> Annot -> cnf.
Variable negate : Term -> Annot -> cnf.
Definition cnf_tt : cnf := @nil clause.
Definition cnf_ff : cnf := cons (@nil (Term' * Annot)) nil.
Our cnf is optimised and detects contradictions on the fly.
Fixpoint add_term (t: Term' * Annot) (cl : clause) : option clause :=
match cl with
| nil =>
match deduce (fst t) (fst t) with
| None => Some (t ::nil)
| Some u => if unsat u then None else Some (t::nil)
end
| t'::cl =>
match deduce (fst t) (fst t') with
| None =>
match add_term t cl with
| None => None
| Some cl' => Some (t' :: cl')
end
| Some u =>
if unsat u then None else
match add_term t cl with
| None => None
| Some cl' => Some (t' :: cl')
end
end
end.
Fixpoint or_clause (cl1 cl2 : clause) : option clause :=
match cl1 with
| nil => Some cl2
| t::cl => match add_term t cl2 with
| None => None
| Some cl' => or_clause cl cl'
end
end.
Definition xor_clause_cnf (t:clause) (f:cnf) : cnf :=
List.fold_left (fun acc e =>
match or_clause t e with
| None => acc
| Some cl => cl :: acc
end) f nil .
Definition or_clause_cnf (t: clause) (f:cnf) : cnf :=
match t with
| nil => f
| _ => xor_clause_cnf t f
end.
Fixpoint or_cnf (f : cnf) (f' : cnf) {struct f}: cnf :=
match f with
| nil => cnf_tt
| e :: rst => (or_cnf rst f') +++ (or_clause_cnf e f')
end.
Definition and_cnf (f1 : cnf) (f2 : cnf) : cnf :=
f1 +++ f2.
TX is Prop in Coq and EConstr.constr in Ocaml. AF is unit in Coq and Names.Id.t in Ocaml
Definition TFormula (TX: kind -> Type) (AF: Type) := @GFormula Term TX Annot AF.
Definition is_cnf_tt (c : cnf) : bool :=
match c with
| nil => true
| _ => false
end.
Definition is_cnf_ff (c : cnf) : bool :=
match c with
| nil::nil => true
| _ => false
end.
Definition and_cnf_opt (f1 : cnf) (f2 : cnf) : cnf :=
if is_cnf_ff f1 || is_cnf_ff f2
then cnf_ff
else
if is_cnf_tt f2
then f1
else and_cnf f1 f2.
Definition or_cnf_opt (f1 : cnf) (f2 : cnf) : cnf :=
if is_cnf_tt f1 || is_cnf_tt f2
then cnf_tt
else if is_cnf_ff f2
then f1 else or_cnf f1 f2.
Section REC.
Context {TX : kind -> Type}.
Context {AF : Type}.
Variable REC : forall (pol : bool) (k: kind) (f : TFormula TX AF k), cnf.
Definition mk_and (k: kind) (pol:bool) (f1 f2 : TFormula TX AF k):=
(if pol then and_cnf_opt else or_cnf_opt) (REC pol f1) (REC pol f2).
Definition mk_or (k: kind) (pol:bool) (f1 f2 : TFormula TX AF k):=
(if pol then or_cnf_opt else and_cnf_opt) (REC pol f1) (REC pol f2).
Definition mk_impl (k: kind) (pol:bool) (f1 f2 : TFormula TX AF k):=
(if pol then or_cnf_opt else and_cnf_opt) (REC (negb pol) f1) (REC pol f2).
Definition mk_iff (k: kind) (pol:bool) (f1 f2: TFormula TX AF k):=
or_cnf_opt (and_cnf_opt (REC (negb pol) f1) (REC false f2))
(and_cnf_opt (REC pol f1) (REC true f2)).
End REC.
Definition is_bool {TX : kind -> Type} {AF: Type} (k: kind) (f : TFormula TX AF k) :=
match f with
| TT _ => Some true
| FF _ => Some false
| _ => None
end.
Lemma is_bool_inv : forall {TX : kind -> Type} {AF: Type} (k: kind) (f : TFormula TX AF k) res,
is_bool f = Some res -> f = if res then TT _ else FF _.
Fixpoint xcnf {TX : kind -> Type} {AF: Type} (pol : bool) (k: kind) (f : TFormula TX AF k) {struct f}: cnf :=
match f with
| TT _ => if pol then cnf_tt else cnf_ff
| FF _ => if pol then cnf_ff else cnf_tt
| X _ p => if pol then cnf_ff else cnf_ff
| A _ x t => if pol then normalise x t else negate x t
| NOT e => xcnf (negb pol) e
| AND e1 e2 => mk_and xcnf pol e1 e2
| OR e1 e2 => mk_or xcnf pol e1 e2
| IMPL e1 _ e2 => mk_impl xcnf pol e1 e2
| IFF e1 e2 => match is_bool e2 with
| Some isb => xcnf (if isb then pol else negb pol) e1
| None => mk_iff xcnf pol e1 e2
end
| EQ e1 e2 =>
match is_bool e2 with
| Some isb => xcnf (if isb then pol else negb pol) e1
| None => mk_iff xcnf pol e1 e2
end
end.
Section CNFAnnot.
Records annotations used to optimise the cnf. Those need to be kept when pruning the formula. For efficiency, this is a separate function.
Fixpoint radd_term (t : Term' * Annot) (cl : clause) : clause + Trace :=
match cl with
| nil =>
match deduce (fst t) (fst t) with
| Some u => if unsat u then inr (push (snd t) null) else inl (t::nil)
| None => inl (t::nil)
end
| t'::cl =>
match deduce (fst t) (fst t') with
| Some u => if unsat u then inr (push (snd t) (push (snd t') null))
else match radd_term t cl with
| inl cl' => inl (t'::cl')
| inr l => inr l
end
| None => match radd_term t cl with
| inl cl' => inl (t'::cl')
| inr l => inr l
end
end
end.
Fixpoint ror_clause cl1 cl2 :=
match cl1 with
| nil => inl cl2
| t::cl => match radd_term t cl2 with
| inl cl' => ror_clause cl cl'
| inr l => inr l
end
end.
Definition xror_clause_cnf t f :=
List.fold_left (fun '(acc,tg) e =>
match ror_clause t e with
| inl cl => (cl :: acc,tg)
| inr l => (acc,merge tg l)
end) f (nil, null).
Definition ror_clause_cnf t f :=
match t with
| nil => (f, null)
| _ => xror_clause_cnf t f
end.
Fixpoint ror_cnf (f f':list clause) :=
match f with
| nil => (cnf_tt, null)
| e :: rst =>
let (rst_f',t) := ror_cnf rst f' in
let (e_f', t') := ror_clause_cnf e f' in
(rst_f' +++ e_f', merge t t')
end.
Definition annot_of_clause (l : clause) : list Annot :=
List.map snd l.
Definition annot_of_cnf (f : cnf) : list Annot :=
List.fold_left (fun acc e => annot_of_clause e +++ acc ) f nil.
Definition ror_cnf_opt f1 f2 :=
if is_cnf_tt f1
then (cnf_tt, null)
else if is_cnf_tt f2
then (cnf_tt, null)
else if is_cnf_ff f2
then (f1, null)
else ror_cnf f1 f2.
Definition ocons {A : Type} (o : option A) (l : list A) : list A :=
match o with
| None => l
| Some e => e ::l
end.
Definition ratom (c : cnf) (a : Annot) : cnf * Trace :=
if is_cnf_ff c || is_cnf_tt c
then (c,push a null)
else (c,null).
Section REC.
Context {TX : kind -> Type} {AF : Type}.
Variable RXCNF : forall (polarity: bool) (k: kind) (f: TFormula TX AF k) , cnf * Trace.
Definition rxcnf_and (polarity:bool) (k: kind) (e1 e2 : TFormula TX AF k) :=
let '(e1,t1) := RXCNF polarity e1 in
let '(e2,t2) := RXCNF polarity e2 in
if polarity
then (and_cnf_opt e1 e2, merge t1 t2)
else let (f',t') := ror_cnf_opt e1 e2 in
(f', merge t1 (merge t2 t')).
Definition rxcnf_or (polarity:bool) (k: kind) (e1 e2 : TFormula TX AF k) :=
let '(e1,t1) := RXCNF polarity e1 in
let '(e2,t2) := RXCNF polarity e2 in
if polarity
then let (f',t') := ror_cnf_opt e1 e2 in
(f', merge t1 (merge t2 t'))
else (and_cnf_opt e1 e2, merge t1 t2).
Definition rxcnf_impl (polarity:bool) (k: kind) (e1 e2 : TFormula TX AF k) :=
let '(e1 , t1) := (RXCNF (negb polarity) e1) in
if polarity
then
if is_cnf_tt e1
then (e1,t1)
else if is_cnf_ff e1
then
RXCNF polarity e2
else
let '(e2 , t2) := (RXCNF polarity e2) in
let (f',t') := ror_cnf_opt e1 e2 in
(f', merge t1 (merge t2 t'))
else
let '(e2 , t2) := (RXCNF polarity e2) in
(and_cnf_opt e1 e2, merge t1 t2).
Definition rxcnf_iff (polarity:bool) (k: kind) (e1 e2 : TFormula TX AF k) :=
let '(c1,t1) := RXCNF (negb polarity) e1 in
let '(c2,t2) := RXCNF false e2 in
let '(c3,t3) := RXCNF polarity e1 in
let '(c4,t4) := RXCNF true e2 in
let (f',t') := ror_cnf_opt (and_cnf_opt c1 c2) (and_cnf_opt c3 c4) in
(f', merge t1 (merge t2 (merge t3 (merge t4 t'))))
.
End REC.
Fixpoint rxcnf {TX : kind -> Type} {AF: Type}(polarity : bool) (k: kind) (f : TFormula TX AF k) : cnf * Trace :=
match f with
| TT _ => if polarity then (cnf_tt, null) else (cnf_ff, null)
| FF _ => if polarity then (cnf_ff, null) else (cnf_tt, null)
| X b p => if polarity then (cnf_ff, null) else (cnf_ff, null)
| A _ x t => ratom (if polarity then normalise x t else negate x t) t
| NOT e => rxcnf (negb polarity) e
| AND e1 e2 => rxcnf_and rxcnf polarity e1 e2
| OR e1 e2 => rxcnf_or rxcnf polarity e1 e2
| IMPL e1 a e2 => rxcnf_impl rxcnf polarity e1 e2
| IFF e1 e2 => rxcnf_iff rxcnf polarity e1 e2
| EQ e1 e2 => rxcnf_iff rxcnf polarity e1 e2
end.
Section Abstraction.
Variable TX : kind -> Type.
Variable AF : Type.
Class to_constrT : Type :=
{
mkTT : forall (k: kind), TX k;
mkFF : forall (k: kind), TX k;
mkA : forall (k: kind), Term -> Annot -> TX k;
mkAND : forall (k: kind), TX k -> TX k -> TX k;
mkOR : forall (k: kind), TX k -> TX k -> TX k;
mkIMPL : forall (k: kind), TX k -> TX k -> TX k;
mkIFF : forall (k: kind), TX k -> TX k -> TX k;
mkNOT : forall (k: kind), TX k -> TX k;
mkEQ : TX isBool -> TX isBool -> TX isProp
}.
Context {to_constr : to_constrT}.
Fixpoint aformula (k: kind) (f : TFormula TX AF k) : TX k :=
match f with
| TT b => mkTT b
| FF b => mkFF b
| X b p => p
| A b x t => mkA b x t
| AND f1 f2 => mkAND (aformula f1) (aformula f2)
| OR f1 f2 => mkOR (aformula f1) (aformula f2)
| IMPL f1 o f2 => mkIMPL (aformula f1) (aformula f2)
| IFF f1 f2 => mkIFF (aformula f1) (aformula f2)
| NOT f => mkNOT (aformula f)
| EQ f1 f2 => mkEQ (aformula f1) (aformula f2)
end.
Definition is_X (k: kind) (f : TFormula TX AF k) : option (TX k) :=
match f with
| X _ p => Some p
| _ => None
end.
Definition is_X_inv : forall (k: kind) (f: TFormula TX AF k) x,
is_X f = Some x -> f = X k x.
Variable needA : Annot -> bool.
Definition abs_and (k: kind) (f1 f2 : TFormula TX AF k)
(c : forall (k: kind), TFormula TX AF k -> TFormula TX AF k -> TFormula TX AF k) :=
match is_X f1 , is_X f2 with
| Some _ , _ | _ , Some _ => X k (aformula (c k f1 f2))
| _ , _ => c k f1 f2
end.
Definition abs_or (k: kind) (f1 f2 : TFormula TX AF k)
(c : forall (k: kind), TFormula TX AF k -> TFormula TX AF k -> TFormula TX AF k) :=
match is_X f1 , is_X f2 with
| Some _ , Some _ => X k (aformula (c k f1 f2))
| _ , _ => c k f1 f2
end.
Definition abs_not (k: kind) (f1 : TFormula TX AF k)
(c : forall (k: kind), TFormula TX AF k -> TFormula TX AF k) :=
match is_X f1 with
| Some _ => X k (aformula (c k f1 ))
| _ => c k f1
end.
Definition mk_arrow (o : option AF) (k: kind) (f1 f2: TFormula TX AF k) :=
match o with
| None => IMPL f1 None f2
| Some _ => if is_X f1 then f2 else IMPL f1 o f2
end.
Fixpoint abst_simpl (k: kind) (f : TFormula TX AF k) : TFormula TX AF k:=
match f with
| TT k => TT k
| FF k => FF k
| X k p => X k p
| A k x t => if needA t then A k x t else X k (mkA k x t)
| AND f1 f2 => AND (abst_simpl f1) (abst_simpl f2)
| OR f1 f2 => OR (abst_simpl f1) (abst_simpl f2)
| IMPL f1 o f2 => IMPL (abst_simpl f1) o (abst_simpl f2)
| NOT f => NOT (abst_simpl f)
| IFF f1 f2 => IFF (abst_simpl f1) (abst_simpl f2)
| EQ f1 f2 => EQ (abst_simpl f1) (abst_simpl f2)
end.
Section REC.
Variable REC : forall (pol : bool) (k: kind) (f : TFormula TX AF k), TFormula TX AF k.
Definition abst_and (pol : bool) (k: kind) (f1 f2:TFormula TX AF k) : TFormula TX AF k:=
(if pol then abs_and else abs_or) k (REC pol f1) (REC pol f2) AND.
Definition abst_or (pol : bool) (k: kind) (f1 f2:TFormula TX AF k) : TFormula TX AF k:=
(if pol then abs_or else abs_and) k (REC pol f1) (REC pol f2) OR.
Definition abst_impl (pol : bool) (o :option AF) (k: kind) (f1 f2:TFormula TX AF k) : TFormula TX AF k:=
(if pol then abs_or else abs_and) k (REC (negb pol) f1) (REC pol f2) (mk_arrow o).
Definition or_is_X (k: kind) (f1 f2: TFormula TX AF k) : bool :=
match is_X f1 , is_X f2 with
| Some _ , _
| _ , Some _ => true
| _ , _ => false
end.
Definition abs_iff (k: kind) (nf1 ff2 f1 tf2 : TFormula TX AF k) (r: kind) (def : TFormula TX AF r) : TFormula TX AF r :=
if andb (or_is_X nf1 ff2) (or_is_X f1 tf2)
then X r (aformula def)
else def.
Definition abst_iff (pol : bool) (k: kind) (f1 f2: TFormula TX AF k) : TFormula TX AF k :=
abs_iff (REC (negb pol) f1) (REC false f2) (REC pol f1) (REC true f2) (IFF (abst_simpl f1) (abst_simpl f2)).
Definition abst_eq (pol : bool) (f1 f2: TFormula TX AF isBool) : TFormula TX AF isProp :=
abs_iff (REC (negb pol) f1) (REC false f2) (REC pol f1) (REC true f2) (EQ (abst_simpl f1) (abst_simpl f2)).
End REC.
Fixpoint abst_form (pol : bool) (k: kind) (f : TFormula TX AF k) : TFormula TX AF k:=
match f with
| TT k => if pol then TT k else X k (mkTT k)
| FF k => if pol then X k (mkFF k) else FF k
| X k p => X k p
| A k x t => if needA t then A k x t else X k (mkA k x t)
| AND f1 f2 => abst_and abst_form pol f1 f2
| OR f1 f2 => abst_or abst_form pol f1 f2
| IMPL f1 o f2 => abst_impl abst_form pol o f1 f2
| NOT f => abs_not (abst_form (negb pol) f) NOT
| IFF f1 f2 => abst_iff abst_form pol f1 f2
| EQ f1 f2 => abst_eq abst_form pol f1 f2
end.
Lemma if_same : forall {A: Type} (b: bool) (t:A),
(if b then t else t) = t.
Lemma is_cnf_tt_cnf_ff :
is_cnf_tt cnf_ff = false.
Lemma is_cnf_ff_cnf_ff :
is_cnf_ff cnf_ff = true.
Lemma is_cnf_tt_inv : forall f1,
is_cnf_tt f1 = true -> f1 = cnf_tt.
Lemma is_cnf_ff_inv : forall f1,
is_cnf_ff f1 = true -> f1 = cnf_ff.
Lemma if_cnf_tt : forall f, (if is_cnf_tt f then cnf_tt else f) = f.
Lemma or_cnf_opt_cnf_ff : forall f,
or_cnf_opt cnf_ff f = f.
Lemma abs_and_pol : forall (k: kind) (f1 f2: TFormula TX AF k) pol,
and_cnf_opt (xcnf pol f1) (xcnf pol f2) =
xcnf pol (abs_and f1 f2 (if pol then AND else OR)).
Lemma abs_or_pol : forall (k: kind) (f1 f2:TFormula TX AF k) pol,
or_cnf_opt (xcnf pol f1) (xcnf pol f2) =
xcnf pol (abs_or f1 f2 (if pol then OR else AND)).
Variable needA_all : forall a, needA a = true.
Lemma xcnf_true_mk_arrow_l : forall b o t (f:TFormula TX AF b),
xcnf true (mk_arrow o (X b t) f) = xcnf true f.
Lemma or_cnf_opt_cnf_ff_r : forall f,
or_cnf_opt f cnf_ff = f.
Lemma xcnf_true_mk_arrow_r : forall b o t (f:TFormula TX AF b),
xcnf true (mk_arrow o f (X b t)) = xcnf false f.
Lemma and_cnf_opt_cnf_ff_r : forall f,
and_cnf_opt f cnf_ff = cnf_ff.
Lemma and_cnf_opt_cnf_ff : forall f,
and_cnf_opt cnf_ff f = cnf_ff.
Lemma and_cnf_opt_cnf_tt : forall f,
and_cnf_opt f cnf_tt = f.
Lemma is_bool_abst_simpl : forall b (f:TFormula TX AF b),
is_bool (abst_simpl f) = is_bool f.
Lemma abst_simpl_correct : forall b (f:TFormula TX AF b) pol,
xcnf pol f = xcnf pol (abst_simpl f).
Ltac is_X t :=
match goal with
| |-context[is_X ?X] =>
let f := fresh "EQ" in
destruct (is_X X) as [t|] eqn:f ;
[apply is_X_inv in f|]
end.
Ltac cnf_simpl :=
repeat match goal with
| |- context[and_cnf_opt cnf_ff _ ] => rewrite and_cnf_opt_cnf_ff
| |- context[and_cnf_opt _ cnf_ff] => rewrite and_cnf_opt_cnf_ff_r
| |- context[and_cnf_opt _ cnf_tt] => rewrite and_cnf_opt_cnf_tt
| |- context[or_cnf_opt cnf_ff _] => rewrite or_cnf_opt_cnf_ff
| |- context[or_cnf_opt _ cnf_ff] => rewrite or_cnf_opt_cnf_ff_r
end.
Lemma or_is_X_inv : forall (k: kind) (f1 f2 : TFormula TX AF k),
or_is_X f1 f2 = true ->
exists k1, is_X f1 = Some k1 \/ is_X f2 = Some k1.
Lemma mk_iff_is_bool : forall (k: kind) (f1 f2:TFormula TX AF k) pol,
match is_bool f2 with
| Some isb => xcnf (if isb then pol else negb pol) f1
| None => mk_iff xcnf pol f1 f2
end = mk_iff xcnf pol f1 f2.
Lemma abst_iff_correct : forall
(k: kind)
(f1 f2 : GFormula k)
(IHf1 : forall pol : bool, xcnf pol f1 = xcnf pol (abst_form pol f1))
(IHf2 : forall pol : bool, xcnf pol f2 = xcnf pol (abst_form pol f2))
(pol : bool),
xcnf pol (IFF f1 f2) = xcnf pol (abst_iff abst_form pol f1 f2).
Lemma abst_eq_correct : forall
(f1 f2 : GFormula isBool)
(IHf1 : forall pol : bool, xcnf pol f1 = xcnf pol (abst_form pol f1))
(IHf2 : forall pol : bool, xcnf pol f2 = xcnf pol (abst_form pol f2))
(pol : bool),
xcnf pol (EQ f1 f2) = xcnf pol (abst_form pol (EQ f1 f2)).
Lemma abst_form_correct : forall b (f:TFormula TX AF b) pol,
xcnf pol f = xcnf pol (abst_form pol f).
End Abstraction.
End CNFAnnot.
Lemma radd_term_term : forall a' a cl, radd_term a a' = inl cl -> add_term a a' = Some cl.
Lemma radd_term_term' : forall a' a cl, add_term a a' = Some cl -> radd_term a a' = inl cl.
Lemma xror_clause_clause : forall a f,
fst (xror_clause_cnf a f) = xor_clause_cnf a f.
Lemma ror_clause_clause : forall a f,
fst (ror_clause_cnf a f) = or_clause_cnf a f.
Lemma ror_cnf_cnf : forall f1 f2, fst (ror_cnf f1 f2) = or_cnf f1 f2.
Lemma ror_opt_cnf_cnf : forall f1 f2, fst (ror_cnf_opt f1 f2) = or_cnf_opt f1 f2.
Lemma ratom_cnf : forall f a,
fst (ratom f a) = f.
Lemma rxcnf_and_xcnf : forall {TX : kind -> Type} {AF:Type} (k: kind) (f1 f2:TFormula TX AF k)
(IHf1 : forall pol : bool, fst (rxcnf pol f1) = xcnf pol f1)
(IHf2 : forall pol : bool, fst (rxcnf pol f2) = xcnf pol f2),
forall pol : bool, fst (rxcnf_and rxcnf pol f1 f2) = mk_and xcnf pol f1 f2.
Lemma rxcnf_or_xcnf :
forall {TX : kind -> Type} {AF:Type} (k: kind) (f1 f2:TFormula TX AF k)
(IHf1 : forall pol : bool, fst (rxcnf pol f1) = xcnf pol f1)
(IHf2 : forall pol : bool, fst (rxcnf pol f2) = xcnf pol f2),
forall pol : bool, fst (rxcnf_or rxcnf pol f1 f2) = mk_or xcnf pol f1 f2.
Lemma rxcnf_impl_xcnf :
forall {TX : kind -> Type} {AF:Type} (k: kind) (f1 f2:TFormula TX AF k)
(IHf1 : forall pol : bool, fst (rxcnf pol f1) = xcnf pol f1)
(IHf2 : forall pol : bool, fst (rxcnf pol f2) = xcnf pol f2),
forall pol : bool, fst (rxcnf_impl rxcnf pol f1 f2) = mk_impl xcnf pol f1 f2.
Lemma rxcnf_iff_xcnf :
forall {TX : kind -> Type} {AF:Type} (k: kind) (f1 f2:TFormula TX AF k)
(IHf1 : forall pol : bool, fst (rxcnf pol f1) = xcnf pol f1)
(IHf2 : forall pol : bool, fst (rxcnf pol f2) = xcnf pol f2),
forall pol : bool, fst (rxcnf_iff rxcnf pol f1 f2) = mk_iff xcnf pol f1 f2.
Lemma rxcnf_xcnf : forall {TX : kind -> Type} {AF:Type} (k: kind) (f:TFormula TX AF k) pol,
fst (rxcnf pol f) = xcnf pol f.
Variable eval' : Env -> Term' -> Prop.
Variable no_middle_eval' : forall env d, (eval' env d) \/ ~ (eval' env d).
Variable unsat_prop : forall t, unsat t = true ->
forall env, eval' env t -> False.
Variable deduce_prop : forall t t' u,
deduce t t' = Some u -> forall env,
eval' env t -> eval' env t' -> eval' env u.
Definition eval_tt (env : Env) (tt : Term' * Annot) := eval' env (fst tt).
Definition eval_clause (env : Env) (cl : clause) := ~ make_conj (eval_tt env) cl.
Definition eval_cnf (env : Env) (f:cnf) := make_conj (eval_clause env) f.
Lemma eval_cnf_app : forall env x y, eval_cnf env (x+++y) <-> eval_cnf env x /\ eval_cnf env y.
Lemma eval_cnf_ff : forall env, eval_cnf env cnf_ff <-> False.
Lemma eval_cnf_tt : forall env, eval_cnf env cnf_tt <-> True.
Lemma eval_cnf_and_opt : forall env x y, eval_cnf env (and_cnf_opt x y) <-> eval_cnf env (and_cnf x y).
Definition eval_opt_clause (env : Env) (cl: option clause) :=
match cl with
| None => True
| Some cl => eval_clause env cl
end.
Lemma add_term_correct : forall env t cl , eval_opt_clause env (add_term t cl) <-> eval_clause env (t::cl).
Lemma no_middle_eval_tt : forall env a,
eval_tt env a \/ ~ eval_tt env a.
#[local]
Hint Resolve no_middle_eval_tt : tauto.
Lemma or_clause_correct : forall cl cl' env, eval_opt_clause env (or_clause cl cl') <-> eval_clause env cl \/ eval_clause env cl'.
Lemma or_clause_cnf_correct : forall env t f, eval_cnf env (or_clause_cnf t f) <-> (eval_clause env t) \/ (eval_cnf env f).
Lemma eval_cnf_cons : forall env a f, (~ make_conj (eval_tt env) a /\ eval_cnf env f) <-> eval_cnf env (a::f).
Lemma eval_cnf_cons_iff : forall env a f, ((~ make_conj (eval_tt env) a) /\ eval_cnf env f) <-> eval_cnf env (a::f).
Lemma or_cnf_correct : forall env f f', eval_cnf env (or_cnf f f') <-> (eval_cnf env f) \/ (eval_cnf env f').
Lemma or_cnf_opt_correct : forall env f f', eval_cnf env (or_cnf_opt f f') <-> eval_cnf env (or_cnf f f').
Variable eval : Env -> forall (k: kind), Term -> rtyp k.
Variable normalise_correct : forall env b t tg, eval_cnf env (normalise t tg) -> hold b (eval env b t).
Variable negate_correct : forall env b t tg, eval_cnf env (negate t tg) -> hold b (eNOT b (eval env b t)).
Definition e_rtyp (k: kind) (x : rtyp k) : rtyp k := x.
Lemma hold_eTT : forall k, hold k (eTT k).
#[local]
Hint Resolve hold_eTT : tauto.
Lemma hold_eFF : forall k,
hold k (eNOT k (eFF k)).
#[local]
Hint Resolve hold_eFF : tauto.
Lemma hold_eAND : forall k r1 r2,
hold k (eAND k r1 r2) <-> (hold k r1 /\ hold k r2).
Lemma hold_eOR : forall k r1 r2,
hold k (eOR k r1 r2) <-> (hold k r1 \/ hold k r2).
Lemma hold_eNOT : forall k e,
hold k (eNOT k e) <-> not (hold k e).
Lemma hold_eIMPL : forall k e1 e2,
hold k (eIMPL k e1 e2) <-> (hold k e1 -> hold k e2).
Lemma hold_eIFF : forall k e1 e2,
hold k (eIFF k e1 e2) <-> (hold k e1 <-> hold k e2).
Lemma xcnf_impl :
forall
(k: kind)
(f1 : GFormula k)
(o : option unit)
(f2 : GFormula k)
(IHf1 : forall (pol : bool) (env : Env),
eval_cnf env (xcnf pol f1) ->
hold k (eval_f e_rtyp (eval env) (if pol then f1 else NOT f1)))
(IHf2 : forall (pol : bool) (env : Env),
eval_cnf env (xcnf pol f2) ->
hold k (eval_f e_rtyp (eval env) (if pol then f2 else NOT f2))),
forall (pol : bool) (env : Env),
eval_cnf env (xcnf pol (IMPL f1 o f2)) ->
hold k (eval_f e_rtyp (eval env) (if pol then IMPL f1 o f2 else NOT (IMPL f1 o f2))).
Lemma hold_eIFF_IMPL : forall k e1 e2,
hold k (eIFF k e1 e2) <-> (hold k (eAND k (eIMPL k e1 e2) (eIMPL k e2 e1))).
Lemma hold_eEQ : forall e1 e2,
hold isBool (eIFF isBool e1 e2) <-> e1 = e2.
Lemma xcnf_iff : forall
(k : kind)
(f1 f2 : @GFormula Term rtyp Annot unit k)
(IHf1 : forall (pol : bool) (env : Env),
eval_cnf env (xcnf pol f1) ->
hold k (eval_f e_rtyp (eval env) (if pol then f1 else NOT f1)))
(IHf2 : forall (pol : bool) (env : Env),
eval_cnf env (xcnf pol f2) ->
hold k (eval_f e_rtyp (eval env) (if pol then f2 else NOT f2))),
forall (pol : bool) (env : Env),
eval_cnf env (xcnf pol (IFF f1 f2)) ->
hold k (eval_f e_rtyp (eval env) (if pol then IFF f1 f2 else NOT (IFF f1 f2))).
Lemma xcnf_correct : forall (k: kind) (f : @GFormula Term rtyp Annot unit k) pol env,
eval_cnf env (xcnf pol f) -> hold k (eval_f e_rtyp (eval env) (if pol then f else NOT f)).
Variable Witness : Type.
Variable checker : list (Term'*Annot) -> Witness -> bool.
Variable checker_sound : forall t w, checker t w = true -> forall env, make_impl (eval_tt 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 env t.
Definition tauto_checker (f:@GFormula Term rtyp Annot unit isProp) (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 e_rtyp (eval env) t.
Definition eval_bf {A : Type} (ea : forall (k: kind), A -> rtyp k) (k: kind) (f: BFormula A k) := eval_f e_rtyp ea f.
Lemma eval_bf_map : forall T U (fct: T-> U) env (k: kind) (f:BFormula T k) ,
eval_bf env (map_bformula fct f) = eval_bf (fun b x => env b (fct x)) f.
End S.