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|
(***********************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *)
(* \VV/ *************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
(*i $Id$ i*)
open Util
open Names
open General
open Dpctypes
open Subst
open Unif
open Graph
(* Herbrand form *************)
let h_fun v ex f =
let lt = List.map (fun x -> VAR(x)) ex
in subst v (APP (GLOB (SK v)::lt)) f
let free_her fv f =
let rec fh_rec f = function
v::ll -> let f0 = subst v (APP [GLOB (SK v)]) f
in fh_rec f0 ll
| [] -> f
in fh_rec f fv
let herbrand f =
let rec h_rec ex sign f = match f with
ForAll(v,f1) -> if sign then (h_rec ex sign (h_fun v ex f1))
else (h_rec (v::ex) sign f1)
| Exists(v,f1) -> if sign then (h_rec (v::ex) sign f1)
else (h_rec ex sign (h_fun v ex f1))
| Neg(f1) -> let (e,h) = (h_rec ex ((not (sign))) f1)
in (e , Neg h)
| Atom(s,lt) -> (ex,Atom(s,lt))
| Imp(f1,f2) -> (let (ex1,h1) = (h_rec ex ((not (sign))) f1)
and (ex2,h2) = (h_rec ex sign f2)
in (union ex1 ex2,Imp(h1,h2)))
| Conj(f1,f2) -> (let (ex1,h1) = (h_rec ex sign f1)
and (ex2,h2) = (h_rec ex sign f2)
in (union ex1 ex2,Conj(h1,h2)))
| Disj(f1,f2) -> (let (ex1,h1) = (h_rec ex sign f1)
and (ex2,h2) = (h_rec ex sign f2)
in (union ex1 ex2,Disj(h1,h2)))
in let (ex,f0) = h_rec [] true f
in let fv = free_var_formula f
in (ex,free_her fv f0)
let separate f =
let n = ref 0
in let rec sep_rec f = match f with
Atom((s,0),lt) -> n := !n+1;
Atom((s,!n),lt)
| Atom((s,_),_) -> anomaly "separate"
| Neg(f1) -> Neg(sep_rec f1)
| ForAll(s,f1) -> ForAll(s,sep_rec f1)
| Exists(s,f1) -> Exists(s,sep_rec f1)
| Imp(f1,f2)-> Imp(sep_rec f1,sep_rec f2)
| Conj(f1,f2) -> Conj(sep_rec f1,sep_rec f2)
| Disj(f1,f2) -> Disj(sep_rec f1,sep_rec f2)
in sep_rec f
(* Searching for paths... *********)
let lit_conj f =
let rec lc_rec sign = function
Atom(_,_) as a -> if sign then ([a],[],[]) else ([],[a],[])
| Neg(f1) -> lc_rec ((not (sign))) f1
| Conj(f1,f2) -> let (p1,n1,c1) = lc_rec sign f1
and (p2,n2,c2) = lc_rec sign f2
in let c = if sign then [((f1,f2),(p1@n1,p2@n2))]
else []
in (p1@p2,n1@n2,c@c1@c2)
| Disj(f1,f2) -> let (p1,n1,c1) = lc_rec sign f1
and (p2,n2,c2) = lc_rec sign f2
in let c = if sign then []
else [((f1,f2),(p1@n1,p2@n2))]
in (p1@p2,n1@n2,c@c1@c2)
| Imp(f1,f2) -> let (p1,n1,c1) = lc_rec ((not (sign))) f1
and (p2,n2,c2) = lc_rec sign f2
in let c = if sign then []
else [((f1,f2),(p1@n1,p2@n2))]
in (p1@p2,n1@n2,c@c1@c2)
| _ -> raise Impossible_case
in lc_rec true f
(* Finding all matches ********)
let mb p q =
let rec mb_rec = function
(_,(lf1,lf2))::ll -> ((not(( ((List.mem p lf1) & (List.mem q lf2))
or ((List.mem q lf1) & (List.mem p lf2))))) )
& (mb_rec ll)
| [] -> true
in mb_rec
let all_matches pos neg lcj =
let rec all_matches_p (x,y) = function
((Atom (x_0,x_1)) as q::neg1) ->
let lm = try [((Atom (x,y)),q,unif_atoms (x,y) (x_0,x_1))]
with Not_unifiable -> []
in if mb (Atom (x,y)) q lcj then lm@(all_matches_p (x,y) neg1)
else (all_matches_p (x,y) neg1)
| _ -> []
and all_matches_rec = function
((Atom (x_0,x_1))::pos1) -> (all_matches_p (x_0,x_1) neg)@(all_matches_rec pos1)
| _ -> []
in all_matches_rec pos
(* combine_path P L : returns the least path containing all the matches
in P and L. If no such path exists, it raises
CP_error *********)
exception CP_error
let rec lit_unicity = function
(p::ll) -> if List.mem p ll then false else lit_unicity ll
| [] -> true
let combine_path (m1,u1) (m2,u2) =
let lm = union m1 m2
in let llt = List.map (fun (p,q,_) -> [p;q]) lm
in let lt = List.flatten llt
in if lit_unicity lt
then let u = try up u1 u2 with Up_error -> raise CP_error
in (lm , u)
else raise CP_error
(* lit_in_matches : formula list -> (match list) -> bool *****)
let lit_in_matches lF ml =
let rec lim_rec = function
((p,q,_)::mmll) -> if (List.mem p lF) or (List.mem q lF)
then true
else lim_rec mmll
| [] -> false
in lim_rec ml
(* 16.III.94 *********)
(* Simple Depth First Search (p.138) *******)
(* NB : path = match list * unifier ********)
let rec matches_pos p = function
((p1,q,u) as m::mM) -> if p=p1
then [([m] , u)]@(matches_pos p mM)
else matches_pos p mM
| [] -> []
let rec matches_neg q = function
((p,q1,u) as m::mM) -> if q=q1
then [([m] , u)]@(matches_neg q mM)
else matches_neg q mM
| [] -> []
let uncovered (ml,_) lcj =
let rec uncov_rec = function
(((f1,f2),(lf1,lf2))::llccjj) ->
let sat1 = lit_in_matches lf1 ml
in let sat2 = lit_in_matches lf2 ml
in if sat1 & ((not (sat2)))
then (f2,lf2)::(uncov_rec llccjj)
else if sat2 & ((not (sat1)))
then (f1,lf1)::(uncov_rec llccjj)
else (uncov_rec llccjj)
| [] -> []
in uncov_rec lcj
let paths pos neg m lcj =
let rec paths_rec = function
((_,_,u) as m_0::mM) -> (extend ([m_0],u))@(paths_rec mM)
| [] -> []
and extension p (f1,lf1) =
let lm = glue (List.map (fun x -> if List.mem x pos
then matches_pos x m
else matches_neg x m) lf1)
in let rec ext_rec = function
(l::ll) ->( try (combine_path p l)::(ext_rec ll)
with CP_error -> ext_rec ll
)
| [] -> []
in ext_rec lm
and extend p =
let uc = uncovered p lcj
in if uc = []
then [p]
else glue (List.map (fun x -> (glue (List.map extend (extension p x)))) uc)
in paths_rec m
(* Dealing with cycles... ******)
let conj_graph lm lc lt =
let rec nb_of_x = function
(p::lpp) -> (try let q = List.assoc p lm in q::(nb_of_x lpp)
with Not_found -> nb_of_x lpp)
| [] -> []
and nb_rec x = function
((l1,l2)::lcc) ->
if List.mem x l1
then (nb_of_x l2)@(nb_rec x lcc)
else if List.mem x l2
then (nb_of_x l1)@(nb_rec x lcc)
else nb_rec x lcc
| [] -> []
and conj_graph_rec = function
(p::ll) -> (p,nb_rec p lc)::(conj_graph_rec ll)
| [] -> []
in conj_graph_rec lt
let is_cyclic_path lcj (lmu,u) =
let lm = List.flatten (List.map (fun (p,q,_) -> [(p,q);(q,p)]) lmu)
in let lc = List.map (fun (_,pl) -> pl) lcj
in let lt = List.map (fun (p,_) -> p) lm
in let g = conj_graph lm lc lt
in have_cycle g
let good_paths lcj lp =
let rec good_paths_rec = function
(p::pp) -> if is_cyclic_path lcj p
then good_paths_rec pp
else p::(good_paths_rec pp)
| [] -> []
in good_paths_rec lp
|
