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|
(**************************************************************************)
(* *)
(* The Why platform for program certification *)
(* Copyright (C) 2002-2008 *)
(* Romain BARDOU *)
(* Jean-Franois COUCHOT *)
(* Mehdi DOGGUY *)
(* Jean-Christophe FILLITRE *)
(* Thierry HUBERT *)
(* Claude MARCH *)
(* Yannick MOY *)
(* Christine PAULIN *)
(* Yann RGIS-GIANAS *)
(* Nicolas ROUSSET *)
(* Xavier URBAIN *)
(* *)
(* This software is free software; you can redistribute it and/or *)
(* modify it under the terms of the GNU General Public *)
(* License version 2, as published by the Free Software Foundation. *)
(* *)
(* This software is distributed in the hope that it will be useful, *)
(* but WITHOUT ANY WARRANTY; without even the implied warranty of *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. *)
(* *)
(* See the GNU General Public License version 2 for more details *)
(* (enclosed in the file GPL). *)
(* *)
(**************************************************************************)
(* ========================================================================= *)
(* Cooper's algorithm for Presburger arithmetic. *)
(* *)
(* Copyright (c) 2003, John Harrison. (See "LICENSE.txt" for details.) *)
(* ========================================================================= *)
(* ------------------------------------------------------------------------- *)
(* Lift operations up to numerals. *)
(* ------------------------------------------------------------------------- *)
let mk_numeral n = Fn(string_of_num n,[]);;
let dest_numeral =
function (Fn(ns,[])) -> num_of_string ns
| _ -> failwith "dest_numeral";;
let is_numeral = can dest_numeral;;
let numeral1 fn n = mk_numeral(fn(dest_numeral n));;
let numeral2 fn m n = mk_numeral(fn (dest_numeral m) (dest_numeral n));;
(* ------------------------------------------------------------------------- *)
(* Operations on canonical linear terms c1 * x1 + ... + cn * xn + k *)
(* *)
(* Note that we're quite strict: the ci must be present even if 1 *)
(* (but if 0 we expect the monomial to be omitted) and k must be there *)
(* even if it's zero. Thus, it's a constant iff not an addition term. *)
(* ------------------------------------------------------------------------- *)
let rec linear_cmul n tm =
if n =/ Int 0 then Fn("0",[]) else
match tm with
Fn("+",[Fn("*",[c1; x1]); rest]) ->
Fn("+",[Fn("*",[numeral1 (( */ ) n) c1; x1]);
linear_cmul n rest])
| k -> numeral1 (( */ ) n) k;;
let earlierv vars (Var x) (Var y) = earlier vars x y;;
let rec linear_add vars tm1 tm2 =
match (tm1,tm2) with
(Fn("+",[Fn("*",[c1; x1]); rest1]),
Fn("+",[Fn("*",[c2; x2]); rest2])) ->
if x1 = x2 then
let c = numeral2 (+/) c1 c2 in
if c = Fn("0",[]) then linear_add vars rest1 rest2
else Fn("+",[Fn("*",[c; x1]); linear_add vars rest1 rest2])
else if earlierv vars x1 x2 then
Fn("+",[Fn("*",[c1; x1]); linear_add vars rest1 tm2])
else
Fn("+",[Fn("*",[c2; x2]); linear_add vars tm1 rest2])
| (Fn("+",[Fn("*",[c1; x1]); rest1]),_) ->
Fn("+",[Fn("*",[c1; x1]); linear_add vars rest1 tm2])
| (_,Fn("+",[Fn("*",[c2; x2]); rest2])) ->
Fn("+",[Fn("*",[c2; x2]); linear_add vars tm1 rest2])
| _ -> numeral2 (+/) tm1 tm2;;
let linear_neg tm = linear_cmul (Int(-1)) tm;;
let linear_sub vars tm1 tm2 = linear_add vars tm1 (linear_neg tm2);;
(* ------------------------------------------------------------------------- *)
(* Linearize a term. *)
(* ------------------------------------------------------------------------- *)
let rec lint vars tm =
match tm with
Var x -> Fn("+",[Fn("*",[Fn("1",[]); tm]); Fn("0",[])])
| Fn("-",[t]) -> linear_neg (lint vars t)
| Fn("+",[s;t]) -> linear_add vars (lint vars s) (lint vars t)
| Fn("-",[s;t]) -> linear_sub vars (lint vars s) (lint vars t)
| Fn("*",[s;t]) ->
let s' = lint vars s and t' = lint vars t in
if is_numeral s' then linear_cmul (dest_numeral s') t'
else if is_numeral t' then linear_cmul (dest_numeral t') s'
else failwith "lint: apparent nonlinearity"
| _ -> if is_numeral tm then tm else failwith "lint: unknown term";;
(* ------------------------------------------------------------------------- *)
(* Linearize the atoms in a formula, and eliminate non-strict inequalities. *)
(* ------------------------------------------------------------------------- *)
let mkatom vars p t = Atom(R(p,[Fn("0",[]);lint vars t]));;
let linform vars fm =
match fm with
Atom(R("divides",[c;t])) ->
let c' = mk_numeral(abs_num(dest_numeral c)) in
Atom(R("divides",[c';lint vars t]))
| Atom(R("=",[s;t])) -> mkatom vars "=" (Fn("-",[t;s]))
| Atom(R("<",[s;t])) -> mkatom vars "<" (Fn("-",[t;s]))
| Atom(R(">",[s;t])) -> mkatom vars "<" (Fn("-",[s;t]))
| Atom(R("<=",[s;t])) ->
mkatom vars "<" (Fn("-",[Fn("+",[t;Fn("1",[])]);s]))
| Atom(R(">=",[s;t])) ->
mkatom vars "<" (Fn("-",[Fn("+",[s;Fn("1",[])]);t]))
| _ -> fm;;
(* ------------------------------------------------------------------------- *)
(* Post-NNF transformation eliminating negated inequalities. *)
(* ------------------------------------------------------------------------- *)
let rec posineq fm =
match fm with
| Not(Atom(R("<",[Fn("0",[]); t]))) ->
Atom(R("<",[Fn("0",[]); linear_sub [] (Fn("1",[])) t]))
| _ -> fm;;
(* ------------------------------------------------------------------------- *)
(* Find the LCM of the coefficients of x. *)
(* ------------------------------------------------------------------------- *)
let rec formlcm x fm =
match fm with
Atom(R(p,[_;Fn("+",[Fn("*",[c;y]);z])])) when y = x ->
abs_num(dest_numeral c)
| Not(p) -> formlcm x p
| And(p,q) -> lcm_num (formlcm x p) (formlcm x q)
| Or(p,q) -> lcm_num (formlcm x p) (formlcm x q)
| _ -> Int 1;;
(* ------------------------------------------------------------------------- *)
(* Adjust all coefficients of x in formula; fold in reduction to +/- 1. *)
(* ------------------------------------------------------------------------- *)
let rec adjustcoeff x l fm =
match fm with
Atom(R(p,[d; Fn("+",[Fn("*",[c;y]);z])])) when y = x ->
let m = l // dest_numeral c in
let n = if p = "<" then abs_num(m) else m in
let xtm = Fn("*",[mk_numeral(m // n); x]) in
Atom(R(p,[linear_cmul (abs_num m) d;
Fn("+",[xtm; linear_cmul n z])]))
| Not(p) -> Not(adjustcoeff x l p)
| And(p,q) -> And(adjustcoeff x l p,adjustcoeff x l q)
| Or(p,q) -> Or(adjustcoeff x l p,adjustcoeff x l q)
| _ -> fm;;
(* ------------------------------------------------------------------------- *)
(* Hence make coefficient of x one in existential formula. *)
(* ------------------------------------------------------------------------- *)
let unitycoeff x fm =
let l = formlcm x fm in
let fm' = adjustcoeff x l fm in
if l =/ Int 1 then fm' else
let xp = Fn("+",[Fn("*",[Fn("1",[]);x]); Fn("0",[])]) in
And(Atom(R("divides",[mk_numeral l; xp])),adjustcoeff x l fm);;
(* ------------------------------------------------------------------------- *)
(* The "minus infinity" version. *)
(* ------------------------------------------------------------------------- *)
let rec minusinf x fm =
match fm with
Atom(R("=",[Fn("0",[]); Fn("+",[Fn("*",[Fn("1",[]);y]);z])]))
when y = x -> False
| Atom(R("<",[Fn("0",[]); Fn("+",[Fn("*",[pm1;y]);z])])) when y = x ->
if pm1 = Fn("1",[]) then False else True
| Not(p) -> Not(minusinf x p)
| And(p,q) -> And(minusinf x p,minusinf x q)
| Or(p,q) -> Or(minusinf x p,minusinf x q)
| _ -> fm;;
(* ------------------------------------------------------------------------- *)
(* The LCM of all the divisors that involve x. *)
(* ------------------------------------------------------------------------- *)
let rec divlcm x fm =
match fm with
Atom(R("divides",[d;Fn("+",[Fn("*",[c;y]);z])])) when y = x ->
dest_numeral d
| Not(p) -> divlcm x p
| And(p,q) -> lcm_num (divlcm x p) (divlcm x q)
| Or(p,q) -> lcm_num (divlcm x p) (divlcm x q)
| _ -> Int 1;;
(* ------------------------------------------------------------------------- *)
(* Construct the B-set. *)
(* ------------------------------------------------------------------------- *)
let rec bset x fm =
match fm with
Not(Atom(R("=",[Fn("0",[]); Fn("+",[Fn("*",[Fn("1",[]);y]);a])])))
when y = x -> [linear_neg a]
| Atom(R("=",[Fn("0",[]); Fn("+",[Fn("*",[Fn("1",[]);y]);a])]))
when y = x -> [linear_neg(linear_add [] a (Fn("1",[])))]
| Atom(R("<",[Fn("0",[]); Fn("+",[Fn("*",[Fn("1",[]);y]);a])]))
when y = x -> [linear_neg a]
| Not(p) -> bset x p
| And(p,q) -> union (bset x p) (bset x q)
| Or(p,q) -> union (bset x p) (bset x q)
| _ -> [];;
(* ------------------------------------------------------------------------- *)
(* Replace top variable with another linear form, retaining canonicality. *)
(* ------------------------------------------------------------------------- *)
let rec linrep vars x t fm =
match fm with
Atom(R(p,[d; Fn("+",[Fn("*",[c;y]);z])])) when y = x ->
let ct = linear_cmul (dest_numeral c) t in
Atom(R(p,[d; linear_add vars ct z]))
| Not(p) -> Not(linrep vars x t p)
| And(p,q) -> And(linrep vars x t p,linrep vars x t q)
| Or(p,q) -> Or(linrep vars x t p,linrep vars x t q)
| _ -> fm;;
(* ------------------------------------------------------------------------- *)
(* Evaluation of constant expressions. *)
(* ------------------------------------------------------------------------- *)
let operations =
["=",(=/); "<",(</); ">",(>/); "<=",(<=/); ">=",(>=/);
"divides",(fun x y -> mod_num y x =/ Int 0)];;
let evalc_atom at =
match at with
R(p,[s;t]) ->
(try if assoc p operations (dest_numeral s) (dest_numeral t)
then True else False
with Failure _ -> Atom at)
| _ -> Atom at;;
let evalc = onatoms evalc_atom;;
(* ------------------------------------------------------------------------- *)
(* Hence the core quantifier elimination procedure. *)
(* ------------------------------------------------------------------------- *)
let cooper vars fm =
match fm with
Exists(x0,p0) ->
let x = Var x0 in
let p = unitycoeff x p0 in
let p_inf = simplify(minusinf x p) and bs = bset x p
and js = Int 1 --- divlcm x p in
let p_element j b =
linrep vars x (linear_add vars b (mk_numeral j)) p in
let stage j = list_disj
(linrep vars x (mk_numeral j) p_inf ::
map (p_element j) bs) in
list_disj (map stage js)
| _ -> failwith "cooper: not an existential formula";;
(* ------------------------------------------------------------------------- *)
(* Overall function. *)
(* ------------------------------------------------------------------------- *)
let integer_qelim =
simplify ** evalc **
lift_qelim linform (cnnf posineq ** evalc) cooper;;
(* ------------------------------------------------------------------------- *)
(* Examples. *)
(* ------------------------------------------------------------------------- *)
START_INTERACTIVE;;
integer_qelim <<forall x y. x < y ==> 2 * x + 1 < 2 * y>>;;
integer_qelim <<forall x y. ~(2 * x + 1 = 2 * y)>>;;
integer_qelim <<exists x y. x > 0 /\ y >= 0 /\ 3 * x - 5 * y = 1>>;;
integer_qelim <<exists x y z. 4 * x - 6 * y = 1>>;;
integer_qelim <<forall x. b < x ==> a <= x>>;;
integer_qelim <<forall x. a < 3 * x ==> b < 3 * x>>;;
time integer_qelim <<forall x y. x <= y ==> 2 * x + 1 < 2 * y>>;;
time integer_qelim <<(exists d. y = 65 * d) ==> (exists d. y = 5 * d)>>;;
time integer_qelim
<<forall y. (exists d. y = 65 * d) ==> (exists d. y = 5 * d)>>;;
time integer_qelim <<forall x y. ~(2 * x + 1 = 2 * y)>>;;
time integer_qelim
<<forall x y z. (2 * x + 1 = 2 * y) ==> x + y + z > 129>>;;
time integer_qelim <<forall x. a < x ==> b < x>>;;
time integer_qelim <<forall x. a <= x ==> b < x>>;;
(* ------------------------------------------------------------------------- *)
(* Formula examples from Cooper's paper. *)
(* ------------------------------------------------------------------------- *)
time integer_qelim <<forall a b. exists x. a < 20 * x /\ 20 * x < b>>;;
time integer_qelim <<exists x. a < 20 * x /\ 20 * x < b>>;;
time integer_qelim <<forall b. exists x. a < 20 * x /\ 20 * x < b>>;;
time integer_qelim
<<forall a. exists b. a < 4 * b + 3 * a \/ (~(a < b) /\ a > b + 1)>>;;
time integer_qelim
<<exists y. forall x. x + 5 * y > 1 /\ 13 * x - y > 1 /\ x + 2 < 0>>;;
(* ------------------------------------------------------------------------- *)
(* Some more. *)
(* ------------------------------------------------------------------------- *)
time integer_qelim <<forall x y. x >= 0 /\ y >= 0
==> 12 * x - 8 * y < 0 \/ 12 * x - 8 * y > 2>>;;
time integer_qelim <<exists x y. 5 * x + 3 * y = 1>>;;
time integer_qelim <<exists x y. 5 * x + 10 * y = 1>>;;
time integer_qelim <<exists x y. x >= 0 /\ y >= 0 /\ 5 * x - 6 * y = 1>>;;
time integer_qelim <<exists w x y z. 2 * w + 3 * x + 4 * y + 5 * z = 1>>;;
time integer_qelim <<exists x y. x >= 0 /\ y >= 0 /\ 5 * x - 3 * y = 1>>;;
time integer_qelim <<exists x y. x >= 0 /\ y >= 0 /\ 3 * x - 5 * y = 1>>;;
time integer_qelim <<exists x y. x >= 0 /\ y >= 0 /\ 6 * x - 3 * y = 1>>;;
time integer_qelim
<<forall x y. ~(x = 0) ==> 5 * y < 6 * x \/ 5 * y > 6 * x>>;;
time integer_qelim
<<forall x y. ~divides(5,x) /\ ~divides(6,y) ==> ~(6 * x = 5 * y)>>;;
time integer_qelim <<forall x y. ~divides(5,x) ==> ~(6 * x = 5 * y)>>;;
time integer_qelim <<forall x y. ~(6 * x = 5 * y)>>;;
time integer_qelim <<forall x y. 6 * x = 5 * y ==> exists d. y = 3 * d>>;;
time integer_qelim <<6 * x = 5 * y ==> exists d. y = 3 * d>>;;
(* ------------------------------------------------------------------------- *)
(* Positive variant of the Bezout theorem. *)
(* ------------------------------------------------------------------------- *)
time integer_qelim
<<forall z. z > 7 ==> exists x y. x >= 0 /\ y >= 0 /\ 3 * x + 5 * y = z>>;;
time integer_qelim
<<forall z. z > 2 ==> exists x y. x >= 0 /\ y >= 0 /\ 3 * x + 5 * y = z>>;;
time integer_qelim
<<forall z.
z <= 7
==> ((exists x y. x >= 0 /\ y >= 0 /\ 3 * x + 5 * y = z) <=>
~(exists x y. x >= 0 /\ y >= 0 /\ 3 * x + 5 * y = 7 - z))>>;;
(* ------------------------------------------------------------------------- *)
(* Basic result about congruences. *)
(* ------------------------------------------------------------------------- *)
time integer_qelim
<<forall x. ~divides(2,x) /\ divides(3,x-1) <=>
divides(12,x-1) \/ divides(12,x-7)>>;;
time integer_qelim
<<forall x. ~(exists m. x = 2 * m) /\ (exists m. x = 3 * m + 1) <=>
(exists m. x = 12 * m + 1) \/ (exists m. x = 12 * m + 7)>>;;
(* ------------------------------------------------------------------------- *)
(* Something else. *)
(* ------------------------------------------------------------------------- *)
time integer_qelim
<<forall x.
~(divides(2,x))
==> divides(4,x-1) \/
divides(8,x-1) \/
divides(8,x-3) \/
divides(6,x-1) \/
divides(14,x-1) \/
divides(14,x-9) \/
divides(14,x-11) \/
divides(24,x-5) \/
divides(24,x-11)>>;;
(* ------------------------------------------------------------------------- *)
(* Testing fix for an earlier version. *)
(* ------------------------------------------------------------------------- *)
(integer_qelim ** generalize)
<<a + 2 = b /\ v_3 = b - a + 1 /\ v_2 = b - 2 /\ v_1 = 3 ==> false>>;;
(* ------------------------------------------------------------------------- *)
(* Inspired by the Collatz conjecture. *)
(* ------------------------------------------------------------------------- *)
integer_qelim
<<exists a b. ~(a = 1) /\ ((2 * b = a) \/ (2 * b = 3 * a + 1)) /\ (a = b)>>;;
integer_qelim
<<exists a b. a > 1 /\ b > 1 /\
((2 * b = a) \/ (2 * b = 3 * a + 1)) /\
(a = b)>>;;
END_INTERACTIVE;;
|
