src/Provers/Arith/fast_lin_arith.ML
author nipkow
Thu Jan 14 13:18:09 1999 +0100 (1999-01-14)
changeset 6128 2acc5d36610c
parent 6110 15c2b571225b
child 7551 8e934d1a9ac6
permissions -rw-r--r--
More arith refinements.
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(*  Title:      Provers/Arith/fast_lin_arith.ML
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    ID:         $Id$
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    Author:     Tobias Nipkow
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    Copyright   1998  TU Munich
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A generic linear arithmetic package.
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It provides two tactics
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    lin_arith_tac:         int -> tactic
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cut_lin_arith_tac: thms -> int -> tactic
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and a simplification procedure
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    lin_arith_prover: Sign.sg -> thm list -> term -> thm option
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Only take premises and conclusions into account that are already (negated)
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(in)equations. lin_arith_prover tries to prove or disprove the term.
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*)
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(*** Data needed for setting up the linear arithmetic package ***)
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signature LIN_ARITH_LOGIC =
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sig
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  val conjI:		thm
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  val ccontr:           thm (* (~ P ==> False) ==> P *)
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  val neqE:             thm (* [| m ~= n; m < n ==> P; n < m ==> P |] ==> P *)
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  val notI:             thm (* (P ==> False) ==> ~ P *)
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  val not_lessD:        thm (* ~(m < n) ==> n <= m *)
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  val not_leD:          thm (* ~(m <= n) ==> n < m *)
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  val sym:		thm (* x = y ==> y = x *)
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  val mk_Eq: thm -> thm
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  val mk_Trueprop: term -> term
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  val neg_prop: term -> term
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  val is_False: thm -> bool
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  val is_nat: typ list * term -> bool
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  val mk_nat_thm: Sign.sg -> term -> thm
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end;
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(*
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mk_Eq(~in) = `in == False'
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mk_Eq(in) = `in == True'
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where `in' is an (in)equality.
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neg_prop(t) = neg if t is wrapped up in Trueprop and
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  nt is the (logically) negated version of t, where the negation
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  of a negative term is the term itself (no double negation!);
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is_nat(parameter-types,t) =  t:nat
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mk_nat_thm(t) = "0 <= t"
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*)
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signature LIN_ARITH_DATA =
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sig
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  val add_mono_thms:    thm list ref
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                            (* [| i rel1 j; m rel2 n |] ==> i + m rel3 j + n *)
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  val lessD:            thm list ref (* m < n ==> m+1 <= n *)
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  val decomp:
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    (term -> ((term * int)list * int * string * (term * int)list * int)option)
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    ref
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  val simp: (thm -> thm) ref
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end;
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(*
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decomp(`x Rel y') should yield (p,i,Rel,q,j)
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   where Rel is one of "<", "~<", "<=", "~<=" and "=" and
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         p/q is the decomposition of the sum terms x/y into a list
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         of summand * multiplicity pairs and a constant summand.
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simp must reduce contradictory <= to False.
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   It should also cancel common summands to keep <= reduced;
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   otherwise <= can grow to massive proportions.
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*)
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signature FAST_LIN_ARITH =
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sig
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  val lin_arith_prover: Sign.sg -> thm list -> term -> thm option
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  val     lin_arith_tac:             int -> tactic
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  val cut_lin_arith_tac: thm list -> int -> tactic
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end;
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functor Fast_Lin_Arith(structure LA_Logic:LIN_ARITH_LOGIC 
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                       and       LA_Data:LIN_ARITH_DATA) : FAST_LIN_ARITH =
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struct
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(*** A fast decision procedure ***)
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(*** Code ported from HOL Light ***)
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(* possible optimizations:
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   use (var,coeff) rep or vector rep  tp save space;
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   treat non-negative atoms separately rather than adding 0 <= atom
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*)
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datatype lineq_type = Eq | Le | Lt;
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datatype injust = Asm of int
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                | Nat of int (* index of atom *)
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                | LessD of injust
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                | NotLessD of injust
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                | NotLeD of injust
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                | Multiplied of int * injust
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                | Added of injust * injust;
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datatype lineq = Lineq of int * lineq_type * int list * injust;
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(* ------------------------------------------------------------------------- *)
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(* Calculate new (in)equality type after addition.                           *)
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(* ------------------------------------------------------------------------- *)
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fun find_add_type(Eq,x) = x
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  | find_add_type(x,Eq) = x
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  | find_add_type(_,Lt) = Lt
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  | find_add_type(Lt,_) = Lt
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  | find_add_type(Le,Le) = Le;
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(* ------------------------------------------------------------------------- *)
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(* Multiply out an (in)equation.                                             *)
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(* ------------------------------------------------------------------------- *)
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fun multiply_ineq n (i as Lineq(k,ty,l,just)) =
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  if n = 1 then i
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  else if n = 0 andalso ty = Lt then sys_error "multiply_ineq"
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  else if n < 0 andalso (ty=Le orelse ty=Lt) then sys_error "multiply_ineq"
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  else Lineq(n * k,ty,map (apl(n,op * )) l,Multiplied(n,just));
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(* ------------------------------------------------------------------------- *)
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(* Add together (in)equations.                                               *)
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(* ------------------------------------------------------------------------- *)
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fun add_ineq (i1 as Lineq(k1,ty1,l1,just1)) (i2 as Lineq(k2,ty2,l2,just2)) =
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  let val l = map2 (op +) (l1,l2)
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  in Lineq(k1+k2,find_add_type(ty1,ty2),l,Added(just1,just2)) end;
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(* ------------------------------------------------------------------------- *)
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(* Elimination of variable between a single pair of (in)equations.           *)
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(* If they're both inequalities, 1st coefficient must be +ve, 2nd -ve.       *)
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(* ------------------------------------------------------------------------- *)
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fun gcd x y =
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  let fun gxd x y =
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    if y = 0 then x else gxd y (x mod y)
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  in if x < y then gxd y x else gxd x y end;
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fun lcm x y = (x * y) div gcd x y;
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fun el 0 (h::_) = h
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  | el n (_::t) = el (n - 1) t
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  | el _ _  = sys_error "el";
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fun elim_var v (i1 as Lineq(k1,ty1,l1,just1)) (i2 as Lineq(k2,ty2,l2,just2)) =
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  let val c1 = el v l1 and c2 = el v l2
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      val m = lcm (abs c1) (abs c2)
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      val m1 = m div (abs c1) and m2 = m div (abs c2)
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      val (n1,n2) =
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        if (c1 >= 0) = (c2 >= 0)
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        then if ty1 = Eq then (~m1,m2)
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             else if ty2 = Eq then (m1,~m2)
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                  else sys_error "elim_var"
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        else (m1,m2)
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      val (p1,p2) = if ty1=Eq andalso ty2=Eq andalso (n1 = ~1 orelse n2 = ~1)
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                    then (~n1,~n2) else (n1,n2)
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  in add_ineq (multiply_ineq n1 i1) (multiply_ineq n2 i2) end;
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(* ------------------------------------------------------------------------- *)
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(* The main refutation-finding code.                                         *)
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(* ------------------------------------------------------------------------- *)
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fun is_trivial (Lineq(_,_,l,_)) = forall (fn i => i=0) l;
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fun is_answer (ans as Lineq(k,ty,l,_)) =
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  case ty  of Eq => k <> 0 | Le => k > 0 | Lt => k >= 0;
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fun calc_blowup l =
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  let val (p,n) = partition (apl(0,op<)) (filter (apl(0,op<>)) l)
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  in (length p) * (length n) end;
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(* ------------------------------------------------------------------------- *)
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(* Main elimination code:                                                    *)
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(*                                                                           *)
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(* (1) Looks for immediate solutions (false assertions with no variables).   *)
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(*                                                                           *)
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(* (2) If there are any equations, picks a variable with the lowest absolute *)
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(* coefficient in any of them, and uses it to eliminate.                     *)
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(*                                                                           *)
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(* (3) Otherwise, chooses a variable in the inequality to minimize the       *)
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(* blowup (number of consequences generated) and eliminates it.              *)
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(* ------------------------------------------------------------------------- *)
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fun allpairs f xs ys =
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  flat(map (fn x => map (fn y => f x y) ys) xs);
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fun extract_first p =
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  let fun extract xs (y::ys) = if p y then (Some y,xs@ys)
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                               else extract (y::xs) ys
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        | extract xs []      = (None,xs)
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  in extract [] end;
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(*
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fun print_ineqs ineqs =
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 writeln(cat_lines(""::map (fn Lineq(c,t,l,_) =>
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   string_of_int c ^
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   (case t of Eq => " =  " | Lt=> " <  " | Le => " <= ") ^
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   commas(map string_of_int l)) ineqs));
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*)
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fun elim ineqs =
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  let (*val dummy = print_ineqs ineqs;*)
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      val (triv,nontriv) = partition is_trivial ineqs in
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  if not(null triv)
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  then case find_first is_answer triv of
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         None => elim nontriv | some => some
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  else
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  if null nontriv then None else
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  let val (eqs,noneqs) = partition (fn (Lineq(_,ty,_,_)) => ty=Eq) nontriv in
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  if not(null eqs) then
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     let val clist = foldl (fn (cs,Lineq(_,_,l,_)) => l union cs) ([],eqs)
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         val sclist = sort (fn (x,y) => int_ord(abs(x),abs(y)))
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                           (filter (fn i => i<>0) clist)
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         val c = hd sclist
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         val (Some(eq as Lineq(_,_,ceq,_)),othereqs) =
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               extract_first (fn Lineq(_,_,l,_) => c mem l) eqs
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         val v = find_index (fn k => k=c) ceq
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         val (ioth,roth) = partition (fn (Lineq(_,_,l,_)) => el v l = 0)
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                                     (othereqs @ noneqs)
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         val others = map (elim_var v eq) roth @ ioth
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     in elim others end
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  else
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  let val lists = map (fn (Lineq(_,_,l,_)) => l) noneqs
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      val numlist = 0 upto (length(hd lists) - 1)
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      val coeffs = map (fn i => map (el i) lists) numlist
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      val blows = map calc_blowup coeffs
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      val iblows = blows ~~ numlist
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      val nziblows = filter (fn (i,_) => i<>0) iblows
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  in if null nziblows then None else
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     let val (c,v) = hd(sort (fn (x,y) => int_ord(fst(x),fst(y))) nziblows)
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         val (no,yes) = partition (fn (Lineq(_,_,l,_)) => el v l = 0) ineqs
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         val (pos,neg) = partition(fn (Lineq(_,_,l,_)) => el v l > 0) yes
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     in elim (no @ allpairs (elim_var v) pos neg) end
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  end
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  end
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  end;
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(* ------------------------------------------------------------------------- *)
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(* Translate back a proof.                                                   *)
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(* ------------------------------------------------------------------------- *)
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(* FIXME OPTIMIZE!!!!
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   Addition/Multiplication need i*t representation rather than t+t+...
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Simplification may detect a contradiction 'prematurely' due to type
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information: n+1 <= 0 is simplified to False and does not need to be crossed
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with 0 <= n.
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*)
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local
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 exception FalseE of thm
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in
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fun mkthm sg asms just =
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  let val atoms = foldl (fn (ats,(lhs,_,_,rhs,_)) =>
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                            map fst lhs  union  (map fst rhs  union  ats))
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                        ([], mapfilter (!LA_Data.decomp o concl_of) asms)
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      fun addthms thm1 thm2 =
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        let val conj = thm1 RS (thm2 RS LA_Logic.conjI)
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        in the(get_first (fn th => Some(conj RS th) handle _ => None)
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                         (!LA_Data.add_mono_thms))
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        end;
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      fun multn(n,thm) =
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        let fun mul(i,th) = if i=1 then th else mul(i-1, addthms thm th)
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        in if n < 0 then mul(~n,thm) RS LA_Logic.sym else mul(n,thm) end;
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      fun simp thm =
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        let val thm' = !LA_Data.simp thm
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        in if LA_Logic.is_False thm' then raise FalseE thm' else thm' end
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      fun mk(Asm i) = nth_elem(i,asms)
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        | mk(Nat(i)) = LA_Logic.mk_nat_thm sg (nth_elem(i,atoms))
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        | mk(LessD(j)) = hd([mk j] RL !LA_Data.lessD)
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        | mk(NotLeD(j)) = hd([mk j RS LA_Logic.not_leD] RL !LA_Data.lessD)
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        | mk(NotLessD(j)) = mk j RS LA_Logic.not_lessD
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        | mk(Added(j1,j2)) = simp(addthms (mk j1) (mk j2))
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        | mk(Multiplied(n,j)) = multn(n,mk j)
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  in !LA_Data.simp(mk just) handle FalseE thm => thm end
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end;
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fun coeff poly atom = case assoc(poly,atom) of None => 0 | Some i => i;
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fun mklineq atoms =
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  let val n = length atoms in
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    fn ((lhs,i,rel,rhs,j),k) =>
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    let val lhsa = map (coeff lhs) atoms
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        and rhsa = map (coeff rhs) atoms
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        val diff = map2 (op -) (rhsa,lhsa)
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        val c = i-j
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        val just = Asm k
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    in case rel of
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        "<="   => Some(Lineq(c,Le,diff,just))
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       | "~<=" => Some(Lineq(1-c,Le,map (op ~) diff,NotLeD(just)))
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       | "<"   => Some(Lineq(c+1,Le,diff,LessD(just)))
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       | "~<"  => Some(Lineq(~c,Le,map (op~) diff,NotLessD(just)))
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       | "="   => Some(Lineq(c,Eq,diff,just))
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       | "~="  => None
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       | _     => sys_error("mklineq" ^ rel)   
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    end
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  end;
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fun mknat pTs ixs (atom,i) =
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  if LA_Logic.is_nat(pTs,atom)
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  then let val l = map (fn j => if j=i then 1 else 0) ixs
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       in Some(Lineq(0,Le,l,Nat(i))) end
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  else None
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fun abstract pTs items =
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  let val atoms = foldl (fn (ats,((lhs,_,_,rhs,_),_)) =>
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                            (map fst lhs) union ((map fst rhs) union ats))
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                        ([],items)
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      val ixs = 0 upto (length(atoms)-1)
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      val iatoms = atoms ~~ ixs
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  in mapfilter (mklineq atoms) items @ mapfilter (mknat pTs ixs) iatoms end;
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(* Ordinary refutation *)
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fun refute1(pTs,items) =
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  (case elim (abstract pTs items) of
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       None => []
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     | Some(Lineq(_,_,_,j)) => [j]);
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fun refute1_tac(i,just) =
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  fn state =>
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    let val sg = #sign(rep_thm state)
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    in resolve_tac [LA_Logic.notI,LA_Logic.ccontr] i THEN
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       METAHYPS (fn asms => rtac (mkthm sg asms just) 1) i
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    end
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    state;
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(* Double refutation caused by equality in conclusion *)
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fun refute2(pTs,items, (rhs,i,_,lhs,j), nHs) =
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  (case elim (abstract pTs (items@[((rhs,i,"<",lhs,j),nHs)])) of
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    None => []
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  | Some(Lineq(_,_,_,j1)) =>
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      (case elim (abstract pTs (items@[((lhs,j,"<",rhs,i),nHs)])) of
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        None => []
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      | Some(Lineq(_,_,_,j2)) => [j1,j2]));
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fun refute2_tac(i,just1,just2) =
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  fn state => 
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    let val sg = #sign(rep_thm state)
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    in rtac LA_Logic.ccontr i THEN rotate_tac ~1 i THEN etac LA_Logic.neqE i THEN
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       METAHYPS (fn asms => rtac (mkthm sg asms just1) 1) i THEN
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       METAHYPS (fn asms => rtac (mkthm sg asms just2) 1) i
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    end
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    state;
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fun prove(pTs,Hs,concl) =
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let val nHs = length Hs
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    val ixHs = Hs ~~ (0 upto (nHs-1))
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    val Hitems = mapfilter (fn (h,i) => case !LA_Data.decomp h of
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                                 None => None | Some(it) => Some(it,i)) ixHs
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in case !LA_Data.decomp concl of
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     None => if null Hitems then [] else refute1(pTs,Hitems)
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   | Some(citem as (r,i,rel,l,j)) =>
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       if rel = "="
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       then refute2(pTs,Hitems,citem,nHs)
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       else let val neg::rel0 = explode rel
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                val nrel = if neg = "~" then implode rel0 else "~"^rel
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            in refute1(pTs, Hitems@[((r,i,nrel,l,j),nHs)]) end
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end;
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(*
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Fast but very incomplete decider. Only premises and conclusions
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that are already (negated) (in)equations are taken into account.
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*)
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val lin_arith_tac = SUBGOAL (fn (A,n) =>
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  let val pTs = rev(map snd (Logic.strip_params A))
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      val Hs = Logic.strip_assums_hyp A
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      val concl = Logic.strip_assums_concl A
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  in case prove(pTs,Hs,concl) of
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       [j] => refute1_tac(n,j)
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     | [j1,j2] => refute2_tac(n,j1,j2)
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     | _ => no_tac
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  end);
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fun cut_lin_arith_tac thms i = cut_facts_tac thms i THEN lin_arith_tac i;
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fun prover1(just,sg,thms,concl,pos) =
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let val nconcl = LA_Logic.neg_prop concl
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    val cnconcl = cterm_of sg nconcl
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    val Fthm = mkthm sg (thms @ [assume cnconcl]) just
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    val contr = if pos then LA_Logic.ccontr else LA_Logic.notI
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in Some(LA_Logic.mk_Eq ((implies_intr cnconcl Fthm) COMP contr)) end
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handle _ => None;
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(* handle thm with equality conclusion *)
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fun prover2(just1,just2,sg,thms,concl) =
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let val nconcl = LA_Logic.neg_prop concl (* m ~= n *)
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    val cnconcl = cterm_of sg nconcl
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    val neqthm = assume cnconcl
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    val casethm = neqthm COMP LA_Logic.neqE (* [|m<n ==> R; n<m ==> R|] ==> R *)
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    val [lessimp1,lessimp2] = prems_of casethm
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    val less1 = fst(Logic.dest_implies lessimp1) (* m<n *)
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    and less2 = fst(Logic.dest_implies lessimp2) (* n<m *)
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    val cless1 = cterm_of sg less1 and cless2 = cterm_of sg less2
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    val thm1 = mkthm sg (thms @ [assume cless1]) just1
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    and thm2 = mkthm sg (thms @ [assume cless2]) just2
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    val dthm1 = implies_intr cless1 thm1 and dthm2 = implies_intr cless2 thm2
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    val thm = dthm2 COMP (dthm1 COMP casethm)
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in Some(LA_Logic.mk_Eq ((implies_intr cnconcl thm) COMP LA_Logic.ccontr)) end
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   404
handle _ => None;
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   405
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(* PRE: concl is not negated! *)
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fun lin_arith_prover sg thms concl =
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let val Hs = map (#prop o rep_thm) thms
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    val Tconcl = LA_Logic.mk_Trueprop concl
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in case prove([],Hs,Tconcl) of
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     [j] => prover1(j,sg,thms,Tconcl,true)
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   | [j1,j2] => prover2(j1,j2,sg,thms,Tconcl)
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   | _ => let val nTconcl = LA_Logic.neg_prop Tconcl
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          in case prove([],Hs,nTconcl) of
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               [j] => prover1(j,sg,thms,nTconcl,false)
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               (* [_,_] impossible because of negation *)
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             | _ => None
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          end
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end;
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   421
end;