--- a/src/HOL/Integ/cooper_dec.ML Thu May 31 11:00:06 2007 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,920 +0,0 @@
-(* Title: HOL/Integ/cooper_dec.ML
- ID: $Id$
- Author: Amine Chaieb and Tobias Nipkow, TU Muenchen
-
-File containing the implementation of Cooper Algorithm
-decision procedure (intensively inspired from J.Harrison)
-*)
-
-
-signature COOPER_DEC =
-sig
- exception COOPER
- val mk_number : IntInf.int -> term
- val zero : term
- val one : term
- val dest_number : term -> IntInf.int
- val is_number : term -> bool
- val is_arith_rel : term -> bool
- val linear_cmul : IntInf.int -> term -> term
- val linear_add : string list -> term -> term -> term
- val linear_sub : string list -> term -> term -> term
- val linear_neg : term -> term
- val lint : string list -> term -> term
- val linform : string list -> term -> term
- val formlcm : term -> term -> IntInf.int
- val adjustcoeff : term -> IntInf.int -> term -> term
- val unitycoeff : term -> term -> term
- val divlcm : term -> term -> IntInf.int
- val bset : term -> term -> term list
- val aset : term -> term -> term list
- val linrep : string list -> term -> term -> term -> term
- val list_disj : term list -> term
- val list_conj : term list -> term
- val simpl : term -> term
- val fv : term -> string list
- val negate : term -> term
- val operations : (string * (IntInf.int * IntInf.int -> bool)) list
- val conjuncts : term -> term list
- val disjuncts : term -> term list
- val has_bound : term -> bool
- val minusinf : term -> term -> term
- val plusinf : term -> term -> term
- val onatoms : (term -> term) -> term -> term
- val evalc : term -> term
- val cooper_w : string list -> term -> (term option * term)
- val integer_qelim : Term.term -> Term.term
-end;
-
-structure CooperDec : COOPER_DEC =
-struct
-
-(* ========================================================================= *)
-(* Cooper's algorithm for Presburger arithmetic. *)
-(* ========================================================================= *)
-exception COOPER;
-
-
-(* ------------------------------------------------------------------------- *)
-(* Lift operations up to numerals. *)
-(* ------------------------------------------------------------------------- *)
-
-(*Assumption : The construction of atomar formulas in linearl arithmetic is based on
-relation operations of Type : [IntInf.int,IntInf.int]---> bool *)
-
-(* ------------------------------------------------------------------------- *)
-
-(*Function is_arith_rel returns true if and only if the term is an atomar presburger
-formula *)
-fun is_arith_rel tm = case tm
- of Const(p, Type ("fun", [Type ("IntDef.int", []), Type ("fun", [Type ("IntDef.int", []),
- Type ("bool", [])])])) $ _ $_ => true
- | _ => false;
-
-(*Function is_arith_rel returns true if and only if the term is an operation of the
-form [int,int]---> int*)
-
-val mk_number = HOLogic.mk_number HOLogic.intT;
-val zero = mk_number 0;
-val one = mk_number 1;
-fun dest_number t = let
- val (T, n) = HOLogic.dest_number t
- in if T = HOLogic.intT then n else error ("bad typ: " ^ Display.raw_string_of_typ T) end;
-val is_number = can dest_number;
-
-(*maps a unary natural function on a term containing an natural number*)
-fun numeral1 f n = mk_number (f (dest_number n));
-
-(*maps a binary natural function on 2 term containing natural numbers*)
-fun numeral2 f m n = mk_number (f (dest_number m) (dest_number 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. *)
-(* ------------------------------------------------------------------------- *)
-
-
-fun linear_cmul n tm = if n = 0 then zero else let fun times n k = n*k in
- ( case tm of
- (Const(@{const_name HOL.plus},T) $ (Const (@{const_name HOL.times},T1 ) $c1 $ x1) $ rest) =>
- Const(@{const_name HOL.plus},T) $ ((Const(@{const_name HOL.times},T1) $ (numeral1 (times n) c1) $ x1)) $ (linear_cmul n rest)
- |_ => numeral1 (times n) tm)
- end ;
-
-
-
-
-(* Whether the first of two items comes earlier in the list *)
-fun earlier [] x y = false
- |earlier (h::t) x y =if h = y then false
- else if h = x then true
- else earlier t x y ;
-
-fun earlierv vars (Bound i) (Bound j) = i < j
- |earlierv vars (Bound _) _ = true
- |earlierv vars _ (Bound _) = false
- |earlierv vars (Free (x,_)) (Free (y,_)) = earlier vars x y;
-
-
-fun linear_add vars tm1 tm2 =
- let fun addwith x y = x + y in
- (case (tm1,tm2) of
- ((Const (@{const_name HOL.plus},T1) $ ( Const(@{const_name HOL.times},T2) $ c1 $ x1) $ rest1),(Const
- (@{const_name HOL.plus},T3)$( Const(@{const_name HOL.times},T4) $ c2 $ x2) $ rest2)) =>
- if x1 = x2 then
- let val c = (numeral2 (addwith) c1 c2)
- in
- if c = zero then (linear_add vars rest1 rest2)
- else (Const(@{const_name HOL.plus},T1) $ (Const(@{const_name HOL.times},T2) $ c $ x1) $ (linear_add vars rest1 rest2))
- end
- else
- if earlierv vars x1 x2 then (Const(@{const_name HOL.plus},T1) $
- (Const(@{const_name HOL.times},T2)$ c1 $ x1) $ (linear_add vars rest1 tm2))
- else (Const(@{const_name HOL.plus},T1) $ (Const(@{const_name HOL.times},T2) $ c2 $ x2) $ (linear_add vars tm1 rest2))
- |((Const(@{const_name HOL.plus},T1) $ (Const(@{const_name HOL.times},T2) $ c1 $ x1) $ rest1) ,_) =>
- (Const(@{const_name HOL.plus},T1)$ (Const(@{const_name HOL.times},T2) $ c1 $ x1) $ (linear_add vars
- rest1 tm2))
- |(_, (Const(@{const_name HOL.plus},T1) $(Const(@{const_name HOL.times},T2) $ c2 $ x2) $ rest2)) =>
- (Const(@{const_name HOL.plus},T1) $ (Const(@{const_name HOL.times},T2) $ c2 $ x2) $ (linear_add vars tm1
- rest2))
- | (_,_) => numeral2 (addwith) tm1 tm2)
-
- end;
-
-(*To obtain the unary - applyed on a formula*)
-
-fun linear_neg tm = linear_cmul (0 - 1) tm;
-
-(*Substraction of two terms *)
-
-fun linear_sub vars tm1 tm2 = linear_add vars tm1 (linear_neg tm2);
-
-
-(* ------------------------------------------------------------------------- *)
-(* Linearize a term. *)
-(* ------------------------------------------------------------------------- *)
-
-(* linearises a term from the point of view of Variable Free (x,T).
-After this fuction the all expressions containig ths variable will have the form
- c*Free(x,T) + t where c is a constant ant t is a Term which is not containing
- Free(x,T)*)
-
-fun lint vars tm = if is_number tm then tm else case tm of
- (Free (x,T)) => (HOLogic.mk_binop @{const_name HOL.plus} ((HOLogic.mk_binop @{const_name HOL.times} ((mk_number 1),Free (x,T))), zero))
- |(Bound i) => (Const(@{const_name HOL.plus},HOLogic.intT -->HOLogic.intT -->HOLogic.intT) $
- (Const(@{const_name HOL.times},HOLogic.intT -->HOLogic.intT -->HOLogic.intT) $ (mk_number 1) $ (Bound i)) $ zero)
- |(Const(@{const_name HOL.uminus},_) $ t ) => (linear_neg (lint vars t))
- |(Const(@{const_name HOL.plus},_) $ s $ t) => (linear_add vars (lint vars s) (lint vars t))
- |(Const(@{const_name HOL.minus},_) $ s $ t) => (linear_sub vars (lint vars s) (lint vars t))
- |(Const (@{const_name HOL.times},_) $ s $ t) =>
- let val s' = lint vars s
- val t' = lint vars t
- in
- if is_number s' then (linear_cmul (dest_number s') t')
- else if is_number t' then (linear_cmul (dest_number t') s')
-
- else raise COOPER
- end
- |_ => raise COOPER;
-
-
-
-(* ------------------------------------------------------------------------- *)
-(* Linearize the atoms in a formula, and eliminate non-strict inequalities. *)
-(* ------------------------------------------------------------------------- *)
-
-fun mkatom vars p t = Const(p,HOLogic.intT --> HOLogic.intT --> HOLogic.boolT) $ zero $ (lint vars t);
-
-fun linform vars (Const ("Divides.dvd",_) $ c $ t) =
- if is_number c then
- let val c' = (mk_number(abs(dest_number c)))
- in (HOLogic.mk_binrel "Divides.dvd" (c,lint vars t))
- end
- else (warning "Nonlinear term --- Non numeral leftside at dvd"
- ;raise COOPER)
- |linform vars (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ s $ t ) = (mkatom vars "op =" (Const (@{const_name HOL.minus},HOLogic.intT --> HOLogic.intT --> HOLogic.intT) $ t $ s) )
- |linform vars (Const(@{const_name Orderings.less},_)$ s $t ) = (mkatom vars @{const_name Orderings.less} (Const (@{const_name HOL.minus},HOLogic.intT --> HOLogic.intT --> HOLogic.intT) $ t $ s))
- |linform vars (Const("op >",_) $ s $ t ) = (mkatom vars @{const_name Orderings.less} (Const (@{const_name HOL.minus},HOLogic.intT --> HOLogic.intT --> HOLogic.intT) $ s $ t))
- |linform vars (Const(@{const_name Orderings.less_eq},_)$ s $ t ) =
- (mkatom vars @{const_name Orderings.less} (Const (@{const_name HOL.minus},HOLogic.intT --> HOLogic.intT --> HOLogic.intT) $ (Const(@{const_name HOL.plus},HOLogic.intT --> HOLogic.intT --> HOLogic.intT) $t $(mk_number 1)) $ s))
- |linform vars (Const("op >=",_)$ s $ t ) =
- (mkatom vars @{const_name Orderings.less} (Const (@{const_name HOL.minus},HOLogic.intT --> HOLogic.intT -->
- HOLogic.intT) $ (Const(@{const_name HOL.plus},HOLogic.intT --> HOLogic.intT -->
- HOLogic.intT) $s $(mk_number 1)) $ t))
-
- |linform vars fm = fm;
-
-(* ------------------------------------------------------------------------- *)
-(* Post-NNF transformation eliminating negated inequalities. *)
-(* ------------------------------------------------------------------------- *)
-
-fun posineq fm = case fm of
- (Const ("Not",_)$(Const(@{const_name Orderings.less},_)$ c $ t)) =>
- (HOLogic.mk_binrel @{const_name Orderings.less} (zero , (linear_sub [] (mk_number 1) (linear_add [] c t ) )))
- | ( Const ("op &",_) $ p $ q) => HOLogic.mk_conj (posineq p,posineq q)
- | ( Const ("op |",_) $ p $ q ) => HOLogic.mk_disj (posineq p,posineq q)
- | _ => fm;
-
-
-(* ------------------------------------------------------------------------- *)
-(* Find the LCM of the coefficients of x. *)
-(* ------------------------------------------------------------------------- *)
-(*gcd calculates gcd (a,b) and helps lcm_num calculating lcm (a,b)*)
-
-(*BEWARE: replaces Library.gcd!! There is also Library.lcm!*)
-fun gcd (a:IntInf.int) b = if a=0 then b else gcd (b mod a) a ;
-fun lcm_num a b = (abs a*b) div (gcd (abs a) (abs b));
-
-fun formlcm x fm = case fm of
- (Const (p,_)$ _ $(Const (@{const_name HOL.plus}, _)$(Const (@{const_name HOL.times},_)$ c $ y ) $z ) ) => if
- (is_arith_rel fm) andalso (x = y) then (abs(dest_number c)) else 1
- | ( Const ("Not", _) $p) => formlcm x p
- | ( Const ("op &",_) $ p $ q) => lcm_num (formlcm x p) (formlcm x q)
- | ( Const ("op |",_) $ p $ q )=> lcm_num (formlcm x p) (formlcm x q)
- | _ => 1;
-
-(* ------------------------------------------------------------------------- *)
-(* Adjust all coefficients of x in formula; fold in reduction to +/- 1. *)
-(* ------------------------------------------------------------------------- *)
-
-fun adjustcoeff x l fm =
- case fm of
- (Const(p,_) $d $( Const (@{const_name HOL.plus}, _)$(Const (@{const_name HOL.times},_) $
- c $ y ) $z )) => if (is_arith_rel fm) andalso (x = y) then
- let val m = l div (dest_number c)
- val n = (if p = @{const_name Orderings.less} then abs(m) else m)
- val xtm = HOLogic.mk_binop @{const_name HOL.times} ((mk_number (m div n)), x)
- in
- (HOLogic.mk_binrel p ((linear_cmul n d),(HOLogic.mk_binop @{const_name HOL.plus} ( xtm ,( linear_cmul n z) ))))
- end
- else fm
- |( Const ("Not", _) $ p) => HOLogic.Not $ (adjustcoeff x l p)
- |( Const ("op &",_) $ p $ q) => HOLogic.conj$(adjustcoeff x l p) $(adjustcoeff x l q)
- |( Const ("op |",_) $ p $ q) => HOLogic.disj $(adjustcoeff x l p)$ (adjustcoeff x l q)
- |_ => fm;
-
-(* ------------------------------------------------------------------------- *)
-(* Hence make coefficient of x one in existential formula. *)
-(* ------------------------------------------------------------------------- *)
-
-fun unitycoeff x fm =
- let val l = formlcm x fm
- val fm' = adjustcoeff x l fm in
- if l = 1 then fm'
- else
- let val xp = (HOLogic.mk_binop @{const_name HOL.plus}
- ((HOLogic.mk_binop @{const_name HOL.times} ((mk_number 1), x )), zero))
- in
- HOLogic.conj $(HOLogic.mk_binrel "Divides.dvd" ((mk_number l) , xp )) $ (adjustcoeff x l fm)
- end
- end;
-
-(* adjustcoeffeq l fm adjusts the coeffitients c_i of x overall in fm to l*)
-(* Here l must be a multiple of all c_i otherwise the obtained formula is not equivalent*)
-(*
-fun adjustcoeffeq x l fm =
- case fm of
- (Const(p,_) $d $( Const (@{const_name HOL.plus}, _)$(Const (@{const_name HOL.times},_) $
- c $ y ) $z )) => if (is_arith_rel fm) andalso (x = y) then
- let val m = l div (dest_number c)
- val n = (if p = @{const_name Orderings.less} then abs(m) else m)
- val xtm = (HOLogic.mk_binop @{const_name HOL.times} ((mk_number ((m div n)*l) ), x))
- in (HOLogic.mk_binrel p ((linear_cmul n d),(HOLogic.mk_binop @{const_name HOL.plus} ( xtm ,( linear_cmul n z) ))))
- end
- else fm
- |( Const ("Not", _) $ p) => HOLogic.Not $ (adjustcoeffeq x l p)
- |( Const ("op &",_) $ p $ q) => HOLogic.conj$(adjustcoeffeq x l p) $(adjustcoeffeq x l q)
- |( Const ("op |",_) $ p $ q) => HOLogic.disj $(adjustcoeffeq x l p)$ (adjustcoeffeq x l q)
- |_ => fm;
-
-
-*)
-
-(* ------------------------------------------------------------------------- *)
-(* The "minus infinity" version. *)
-(* ------------------------------------------------------------------------- *)
-
-fun minusinf x fm = case fm of
- (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ (c1 ) $(Const (@{const_name HOL.plus}, _) $(Const (@{const_name HOL.times},_) $ c2 $ y) $z)) =>
- if (is_arith_rel fm) andalso (x=y) andalso (c2 = one) andalso (c1 =zero) then HOLogic.false_const
- else fm
-
- |(Const(@{const_name Orderings.less},_) $ c $(Const (@{const_name HOL.plus}, _) $(Const (@{const_name HOL.times},_) $ pm1 $ y ) $ z
- )) => if (x = y)
- then if (pm1 = one) andalso (c = zero) then HOLogic.false_const
- else if (dest_number pm1 = ~1) andalso (c = zero) then HOLogic.true_const
- else error "minusinf : term not in normal form!!!"
- else fm
-
- |(Const ("Not", _) $ p) => HOLogic.Not $ (minusinf x p)
- |(Const ("op &",_) $ p $ q) => HOLogic.conj $ (minusinf x p) $ (minusinf x q)
- |(Const ("op |",_) $ p $ q) => HOLogic.disj $ (minusinf x p) $ (minusinf x q)
- |_ => fm;
-
-(* ------------------------------------------------------------------------- *)
-(* The "Plus infinity" version. *)
-(* ------------------------------------------------------------------------- *)
-
-fun plusinf x fm = case fm of
- (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ (c1 ) $(Const (@{const_name HOL.plus}, _) $(Const (@{const_name HOL.times},_) $ c2 $ y) $z)) =>
- if (is_arith_rel fm) andalso (x=y) andalso (c2 = one) andalso (c1 =zero) then HOLogic.false_const
- else fm
-
- |(Const(@{const_name Orderings.less},_) $ c $(Const (@{const_name HOL.plus}, _) $(Const (@{const_name HOL.times},_) $ pm1 $ y ) $ z
- )) => if (x = y)
- then if (pm1 = one) andalso (c = zero) then HOLogic.true_const
- else if (dest_number pm1 = ~1) andalso (c = zero) then HOLogic.false_const
- else error "plusinf : term not in normal form!!!"
- else fm
-
- |(Const ("Not", _) $ p) => HOLogic.Not $ (plusinf x p)
- |(Const ("op &",_) $ p $ q) => HOLogic.conj $ (plusinf x p) $ (plusinf x q)
- |(Const ("op |",_) $ p $ q) => HOLogic.disj $ (plusinf x p) $ (plusinf x q)
- |_ => fm;
-
-(* ------------------------------------------------------------------------- *)
-(* The LCM of all the divisors that involve x. *)
-(* ------------------------------------------------------------------------- *)
-
-fun divlcm x (Const("Divides.dvd",_)$ d $ (Const (@{const_name HOL.plus},_) $ (Const (@{const_name HOL.times},_) $ c $ y ) $ z ) ) =
- if x = y then abs(dest_number d) else 1
- |divlcm x ( Const ("Not", _) $ p) = divlcm x p
- |divlcm x ( Const ("op &",_) $ p $ q) = lcm_num (divlcm x p) (divlcm x q)
- |divlcm x ( Const ("op |",_) $ p $ q ) = lcm_num (divlcm x p) (divlcm x q)
- |divlcm x _ = 1;
-
-(* ------------------------------------------------------------------------- *)
-(* Construct the B-set. *)
-(* ------------------------------------------------------------------------- *)
-
-fun bset x fm = case fm of
- (Const ("Not", _) $ p) => if (is_arith_rel p) then
- (case p of
- (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const (@{const_name HOL.plus}, _) $(Const (@{const_name HOL.times},_) $c2 $y) $a ) )
- => if (is_arith_rel p) andalso (x= y) andalso (c2 = one) andalso (c1 = zero)
- then [linear_neg a]
- else bset x p
- |_ =>[])
-
- else bset x p
- |(Const ("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const (@{const_name HOL.plus},_) $ (Const (@{const_name HOL.times},_) $c2 $ x) $ a)) => if (c1 =zero) andalso (c2 = one) then [linear_neg(linear_add [] a (mk_number 1))] else []
- |(Const (@{const_name Orderings.less},_) $ c1$ (Const (@{const_name HOL.plus},_) $(Const (@{const_name HOL.times},_)$ c2 $ x) $ a)) => if (c1 =zero) andalso (c2 = one) then [linear_neg a] else []
- |(Const ("op &",_) $ p $ q) => (bset x p) union (bset x q)
- |(Const ("op |",_) $ p $ q) => (bset x p) union (bset x q)
- |_ => [];
-
-(* ------------------------------------------------------------------------- *)
-(* Construct the A-set. *)
-(* ------------------------------------------------------------------------- *)
-
-fun aset x fm = case fm of
- (Const ("Not", _) $ p) => if (is_arith_rel p) then
- (case p of
- (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const (@{const_name HOL.plus}, _) $(Const (@{const_name HOL.times},_) $c2 $y) $a ) )
- => if (x= y) andalso (c2 = one) andalso (c1 = zero)
- then [linear_neg a]
- else []
- |_ =>[])
-
- else aset x p
- |(Const ("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const (@{const_name HOL.plus},_) $ (Const (@{const_name HOL.times},_) $c2 $ x) $ a)) => if (c1 =zero) andalso (c2 = one) then [linear_sub [] (mk_number 1) a] else []
- |(Const (@{const_name Orderings.less},_) $ c1$ (Const (@{const_name HOL.plus},_) $(Const (@{const_name HOL.times},_)$ c2 $ x) $ a)) => if (c1 =zero) andalso (c2 = (mk_number (~1))) then [a] else []
- |(Const ("op &",_) $ p $ q) => (aset x p) union (aset x q)
- |(Const ("op |",_) $ p $ q) => (aset x p) union (aset x q)
- |_ => [];
-
-
-(* ------------------------------------------------------------------------- *)
-(* Replace top variable with another linear form, retaining canonicality. *)
-(* ------------------------------------------------------------------------- *)
-
-fun linrep vars x t fm = case fm of
- ((Const(p,_)$ d $ (Const(@{const_name HOL.plus},_)$(Const(@{const_name HOL.times},_)$ c $ y) $ z))) =>
- if (x = y) andalso (is_arith_rel fm)
- then
- let val ct = linear_cmul (dest_number c) t
- in (HOLogic.mk_binrel p (d, linear_add vars ct z))
- end
- else fm
- |(Const ("Not", _) $ p) => HOLogic.Not $ (linrep vars x t p)
- |(Const ("op &",_) $ p $ q) => HOLogic.conj $ (linrep vars x t p) $ (linrep vars x t q)
- |(Const ("op |",_) $ p $ q) => HOLogic.disj $ (linrep vars x t p) $ (linrep vars x t q)
- |_ => fm;
-
-(* ------------------------------------------------------------------------- *)
-(* Evaluation of constant expressions. *)
-(* ------------------------------------------------------------------------- *)
-
-(* An other implementation of divides, that covers more cases*)
-
-exception DVD_UNKNOWN
-
-fun dvd_op (d, t) =
- if not(is_number d) then raise DVD_UNKNOWN
- else let
- val dn = dest_number d
- fun coeffs_of x = case x of
- Const(p,_) $ tl $ tr =>
- if p = @{const_name HOL.plus} then (coeffs_of tl) union (coeffs_of tr)
- else if p = @{const_name HOL.times}
- then if (is_number tr)
- then [(dest_number tr) * (dest_number tl)]
- else [dest_number tl]
- else []
- |_ => if (is_number t) then [dest_number t] else []
- val ts = coeffs_of t
- in case ts of
- [] => raise DVD_UNKNOWN
- |_ => fold_rev (fn k => fn r => r andalso (k mod dn = 0)) ts true
- end;
-
-
-val operations =
- [("op =",op=), (@{const_name Orderings.less},IntInf.<), ("op >",IntInf.>), (@{const_name Orderings.less_eq},IntInf.<=) ,
- ("op >=",IntInf.>=),
- ("Divides.dvd",fn (x,y) =>((IntInf.mod(y, x)) = 0))];
-
-fun applyoperation (SOME f) (a,b) = f (a, b)
- |applyoperation _ (_, _) = false;
-
-(*Evaluation of constant atomic formulas*)
- (*FIXME : This is an optimation but still incorrect !! *)
-(*
-fun evalc_atom at = case at of
- (Const (p,_) $ s $ t) =>
- (if p="Divides.dvd" then
- ((if dvd_op(s,t) then HOLogic.true_const
- else HOLogic.false_const)
- handle _ => at)
- else
- case AList.lookup (op =) operations p of
- SOME f => ((if (f ((dest_number s),(dest_number t))) then HOLogic.true_const else HOLogic.false_const)
- handle _ => at)
- | _ => at)
- |Const("Not",_)$(Const (p,_) $ s $ t) =>(
- case AList.lookup (op =) operations p of
- SOME f => ((if (f ((dest_number s),(dest_number t))) then
- HOLogic.false_const else HOLogic.true_const)
- handle _ => at)
- | _ => at)
- | _ => at;
-
-*)
-
-fun evalc_atom at = case at of
- (Const (p,_) $ s $ t) =>
- ( case AList.lookup (op =) operations p of
- SOME f => ((if (f ((dest_number s),(dest_number t))) then HOLogic.true_const
- else HOLogic.false_const)
- handle _ => at)
- | _ => at)
- |Const("Not",_)$(Const (p,_) $ s $ t) =>(
- case AList.lookup (op =) operations p of
- SOME f => ((if (f ((dest_number s),(dest_number t)))
- then HOLogic.false_const else HOLogic.true_const)
- handle _ => at)
- | _ => at)
- | _ => at;
-
- (*Function onatoms apllys function f on the atomic formulas involved in a.*)
-
-fun onatoms f a = if (is_arith_rel a) then f a else case a of
-
- (Const ("Not",_) $ p) => if is_arith_rel p then HOLogic.Not $ (f p)
-
- else HOLogic.Not $ (onatoms f p)
- |(Const ("op &",_) $ p $ q) => HOLogic.conj $ (onatoms f p) $ (onatoms f q)
- |(Const ("op |",_) $ p $ q) => HOLogic.disj $ (onatoms f p) $ (onatoms f q)
- |(Const ("op -->",_) $ p $ q) => HOLogic.imp $ (onatoms f p) $ (onatoms f q)
- |((Const ("op =", Type ("fun",[Type ("bool", []),_]))) $ p $ q) => (Const ("op =", [HOLogic.boolT, HOLogic.boolT] ---> HOLogic.boolT)) $ (onatoms f p) $ (onatoms f q)
- |(Const("All",_) $ Abs(x,T,p)) => Const("All", [HOLogic.intT -->
- HOLogic.boolT] ---> HOLogic.boolT)$ Abs (x ,T, (onatoms f p))
- |(Const("Ex",_) $ Abs(x,T,p)) => Const("Ex", [HOLogic.intT --> HOLogic.boolT]---> HOLogic.boolT) $ Abs( x ,T, (onatoms f p))
- |_ => a;
-
-val evalc = onatoms evalc_atom;
-
-(* ------------------------------------------------------------------------- *)
-(* Hence overall quantifier elimination. *)
-(* ------------------------------------------------------------------------- *)
-
-
-(*list_disj[conj] makes a disj[conj] of a given list. used with conjucts or disjuncts
-it liearises iterated conj[disj]unctions. *)
-
-fun list_disj [] = HOLogic.false_const
- | list_disj ps = foldr1 (fn (p, q) => HOLogic.disj $ p $ q) ps;
-
-fun list_conj [] = HOLogic.true_const
- | list_conj ps = foldr1 (fn (p, q) => HOLogic.conj $ p $ q) ps;
-
-
-(*Simplification of Formulas *)
-
-(*Function q_bnd_chk checks if a quantified Formula makes sens : Means if in
-the body of the existential quantifier there are bound variables to the
-existential quantifier.*)
-
-fun has_bound fm =let fun has_boundh fm i = case fm of
- Bound n => (i = n)
- |Abs (_,_,p) => has_boundh p (i+1)
- |t1 $ t2 => (has_boundh t1 i) orelse (has_boundh t2 i)
- |_ =>false
-
-in case fm of
- Bound _ => true
- |Abs (_,_,p) => has_boundh p 0
- |t1 $ t2 => (has_bound t1 ) orelse (has_bound t2 )
- |_ =>false
-end;
-
-(*has_sub_abs checks if in a given Formula there are subformulas which are quantifed
-too. Is no used no more.*)
-
-fun has_sub_abs fm = case fm of
- Abs (_,_,_) => true
- |t1 $ t2 => (has_bound t1 ) orelse (has_bound t2 )
- |_ =>false ;
-
-(*update_bounds called with i=0 udates the numeration of bounded variables because the
-formula will not be quantified any more.*)
-
-fun update_bounds fm i = case fm of
- Bound n => if n >= i then Bound (n-1) else fm
- |Abs (x,T,p) => Abs(x,T,(update_bounds p (i+1)))
- |t1 $ t2 => (update_bounds t1 i) $ (update_bounds t2 i)
- |_ => fm ;
-
-(*psimpl : Simplification of propositions (general purpose)*)
-fun psimpl1 fm = case fm of
- Const("Not",_) $ Const ("False",_) => HOLogic.true_const
- | Const("Not",_) $ Const ("True",_) => HOLogic.false_const
- | Const("op &",_) $ Const ("False",_) $ q => HOLogic.false_const
- | Const("op &",_) $ p $ Const ("False",_) => HOLogic.false_const
- | Const("op &",_) $ Const ("True",_) $ q => q
- | Const("op &",_) $ p $ Const ("True",_) => p
- | Const("op |",_) $ Const ("False",_) $ q => q
- | Const("op |",_) $ p $ Const ("False",_) => p
- | Const("op |",_) $ Const ("True",_) $ q => HOLogic.true_const
- | Const("op |",_) $ p $ Const ("True",_) => HOLogic.true_const
- | Const("op -->",_) $ Const ("False",_) $ q => HOLogic.true_const
- | Const("op -->",_) $ Const ("True",_) $ q => q
- | Const("op -->",_) $ p $ Const ("True",_) => HOLogic.true_const
- | Const("op -->",_) $ p $ Const ("False",_) => HOLogic.Not $ p
- | Const("op =", Type ("fun",[Type ("bool", []),_])) $ Const ("True",_) $ q => q
- | Const("op =", Type ("fun",[Type ("bool", []),_])) $ p $ Const ("True",_) => p
- | Const("op =", Type ("fun",[Type ("bool", []),_])) $ Const ("False",_) $ q => HOLogic.Not $ q
- | Const("op =", Type ("fun",[Type ("bool", []),_])) $ p $ Const ("False",_) => HOLogic.Not $ p
- | _ => fm;
-
-fun psimpl fm = case fm of
- Const ("Not",_) $ p => psimpl1 (HOLogic.Not $ (psimpl p))
- | Const("op &",_) $ p $ q => psimpl1 (HOLogic.mk_conj (psimpl p,psimpl q))
- | Const("op |",_) $ p $ q => psimpl1 (HOLogic.mk_disj (psimpl p,psimpl q))
- | Const("op -->",_) $ p $ q => psimpl1 (HOLogic.mk_imp(psimpl p,psimpl q))
- | Const("op =", Type ("fun",[Type ("bool", []),_])) $ p $ q => psimpl1 (HOLogic.mk_eq(psimpl p,psimpl q))
- | _ => fm;
-
-
-(*simpl : Simplification of Terms involving quantifiers too.
- This function is able to drop out some quantified expressions where there are no
- bound varaibles.*)
-
-fun simpl1 fm =
- case fm of
- Const("All",_) $Abs(x,_,p) => if (has_bound fm ) then fm
- else (update_bounds p 0)
- | Const("Ex",_) $ Abs (x,_,p) => if has_bound fm then fm
- else (update_bounds p 0)
- | _ => psimpl fm;
-
-fun simpl fm = case fm of
- Const ("Not",_) $ p => simpl1 (HOLogic.Not $(simpl p))
- | Const ("op &",_) $ p $ q => simpl1 (HOLogic.mk_conj (simpl p ,simpl q))
- | Const ("op |",_) $ p $ q => simpl1 (HOLogic.mk_disj (simpl p ,simpl q ))
- | Const ("op -->",_) $ p $ q => simpl1 (HOLogic.mk_imp(simpl p ,simpl q ))
- | Const("op =", Type ("fun",[Type ("bool", []),_]))$ p $ q => simpl1
- (HOLogic.mk_eq(simpl p ,simpl q ))
-(* | Const ("All",Ta) $ Abs(Vn,VT,p) => simpl1(Const("All",Ta) $
- Abs(Vn,VT,simpl p ))
- | Const ("Ex",Ta) $ Abs(Vn,VT,p) => simpl1(Const("Ex",Ta) $
- Abs(Vn,VT,simpl p ))
-*)
- | _ => fm;
-
-(* ------------------------------------------------------------------------- *)
-
-(* Puts fm into NNF*)
-
-fun nnf fm = if (is_arith_rel fm) then fm
-else (case fm of
- ( Const ("op &",_) $ p $ q) => HOLogic.conj $ (nnf p) $(nnf q)
- | (Const("op |",_) $ p $q) => HOLogic.disj $ (nnf p)$(nnf q)
- | (Const ("op -->",_) $ p $ q) => HOLogic.disj $ (nnf (HOLogic.Not $ p)) $ (nnf q)
- | ((Const ("op =", Type ("fun",[Type ("bool", []),_]))) $ p $ q) =>(HOLogic.disj $ (HOLogic.conj $ (nnf p) $ (nnf q)) $ (HOLogic.conj $ (nnf (HOLogic.Not $ p) ) $ (nnf(HOLogic.Not $ q))))
- | (Const ("Not",_)) $ ((Const ("Not",_)) $ p) => (nnf p)
- | (Const ("Not",_)) $ (( Const ("op &",_)) $ p $ q) =>HOLogic.disj $ (nnf(HOLogic.Not $ p)) $ (nnf(HOLogic.Not $q))
- | (Const ("Not",_)) $ (( Const ("op |",_)) $ p $ q) =>HOLogic.conj $ (nnf(HOLogic.Not $ p)) $ (nnf(HOLogic.Not $ q))
- | (Const ("Not",_)) $ (( Const ("op -->",_)) $ p $ q ) =>HOLogic.conj $ (nnf p) $(nnf(HOLogic.Not $ q))
- | (Const ("Not",_)) $ ((Const ("op =", Type ("fun",[Type ("bool", []),_]))) $ p $ q ) =>(HOLogic.disj $ (HOLogic.conj $(nnf p) $ (nnf(HOLogic.Not $ q))) $ (HOLogic.conj $(nnf(HOLogic.Not $ p)) $ (nnf q)))
- | _ => fm);
-
-
-(* Function remred to remove redundancy in a list while keeping the order of appearance of the
-elements. but VERY INEFFICIENT!! *)
-
-fun remred1 el [] = []
- |remred1 el (h::t) = if el=h then (remred1 el t) else h::(remred1 el t);
-
-fun remred [] = []
- |remred (x::l) = x::(remred1 x (remred l));
-
-(*Makes sure that all free Variables are of the type integer but this function is only
-used temporarily, this job must be done by the parser later on.*)
-
-fun mk_uni_vars T (node $ rest) = (case node of
- Free (name,_) => Free (name,T) $ (mk_uni_vars T rest)
- |_=> (mk_uni_vars T node) $ (mk_uni_vars T rest ) )
- |mk_uni_vars T (Free (v,_)) = Free (v,T)
- |mk_uni_vars T tm = tm;
-
-fun mk_uni_int T (Const (@{const_name HOL.zero},T2)) = if T = T2 then (mk_number 0) else (Const (@{const_name HOL.zero},T2))
- |mk_uni_int T (Const (@{const_name HOL.one},T2)) = if T = T2 then (mk_number 1) else (Const (@{const_name HOL.one},T2))
- |mk_uni_int T (node $ rest) = (mk_uni_int T node) $ (mk_uni_int T rest )
- |mk_uni_int T (Abs(AV,AT,p)) = Abs(AV,AT,mk_uni_int T p)
- |mk_uni_int T tm = tm;
-
-
-(* Minusinfinity Version*)
-fun myupto (m:IntInf.int) n = if m > n then [] else m::(myupto (m+1) n)
-
-fun coopermi vars1 fm =
- case fm of
- Const ("Ex",_) $ Abs(x0,T,p0) =>
- let
- val (xn,p1) = Syntax.variant_abs (x0,T,p0)
- val x = Free (xn,T)
- val vars = (xn::vars1)
- val p = unitycoeff x (posineq (simpl p1))
- val p_inf = simpl (minusinf x p)
- val bset = bset x p
- val js = myupto 1 (divlcm x p)
- fun p_element j b = linrep vars x (linear_add vars b (mk_number j)) p
- fun stage j = list_disj (linrep vars x (mk_number j) p_inf :: map (p_element j) bset)
- in (list_disj (map stage js))
- end
- | _ => error "cooper: not an existential formula";
-
-
-
-(* The plusinfinity version of cooper*)
-fun cooperpi vars1 fm =
- case fm of
- Const ("Ex",_) $ Abs(x0,T,p0) => let
- val (xn,p1) = Syntax.variant_abs (x0,T,p0)
- val x = Free (xn,T)
- val vars = (xn::vars1)
- val p = unitycoeff x (posineq (simpl p1))
- val p_inf = simpl (plusinf x p)
- val aset = aset x p
- val js = myupto 1 (divlcm x p)
- fun p_element j a = linrep vars x (linear_sub vars a (mk_number j)) p
- fun stage j = list_disj (linrep vars x (mk_number j) p_inf :: map (p_element j) aset)
- in (list_disj (map stage js))
- end
- | _ => error "cooper: not an existential formula";
-
-
-(* Try to find a withness for the formula *)
-
-fun inf_w mi d vars x p =
- let val f = if mi then minusinf else plusinf in
- case (simpl (minusinf x p)) of
- Const("True",_) => (SOME (mk_number 1), HOLogic.true_const)
- |Const("False",_) => (NONE,HOLogic.false_const)
- |F =>
- let
- fun h n =
- case ((simpl o evalc) (linrep vars x (mk_number n) F)) of
- Const("True",_) => (SOME (mk_number n),HOLogic.true_const)
- |F' => if n=1 then (NONE,F')
- else let val (rw,rf) = h (n-1) in
- (rw,HOLogic.mk_disj(F',rf))
- end
-
- in (h d)
- end
- end;
-
-fun set_w d b st vars x p = let
- fun h ns = case ns of
- [] => (NONE,HOLogic.false_const)
- |n::nl => ( case ((simpl o evalc) (linrep vars x n p)) of
- Const("True",_) => (SOME n,HOLogic.true_const)
- |F' => let val (rw,rf) = h nl
- in (rw,HOLogic.mk_disj(F',rf))
- end)
- val f = if b then linear_add else linear_sub
- val p_elements = fold_rev (fn i => fn l => l union (map (fn e => f [] e (mk_number i)) st)) (myupto 1 d) []
- in h p_elements
- end;
-
-fun withness d b st vars x p = case (inf_w b d vars x p) of
- (SOME n,_) => (SOME n,HOLogic.true_const)
- |(NONE,Pinf) => (case (set_w d b st vars x p) of
- (SOME n,_) => (SOME n,HOLogic.true_const)
- |(_,Pst) => (NONE,HOLogic.mk_disj(Pinf,Pst)));
-
-
-
-
-(*Cooper main procedure*)
-
-exception STAGE_TRUE;
-
-
-fun cooper vars1 fm =
- case fm of
- Const ("Ex",_) $ Abs(x0,T,p0) => let
- val (xn,p1) = Syntax.variant_abs (x0,T,p0)
- val x = Free (xn,T)
- val vars = (xn::vars1)
-(* val p = unitycoeff x (posineq (simpl p1)) *)
- val p = unitycoeff x p1
- val ast = aset x p
- val bst = bset x p
- val js = myupto 1 (divlcm x p)
- val (p_inf,f,S ) =
- if (length bst) <= (length ast)
- then (simpl (minusinf x p),linear_add,bst)
- else (simpl (plusinf x p), linear_sub,ast)
- fun p_element j a = linrep vars x (f vars a (mk_number j)) p
- fun stage j = list_disj (linrep vars x (mk_number j) p_inf :: map (p_element j) S)
- fun stageh n = ((if n = 0 then []
- else
- let
- val nth_stage = simpl (evalc (stage n))
- in
- if (nth_stage = HOLogic.true_const)
- then raise STAGE_TRUE
- else if (nth_stage = HOLogic.false_const) then stageh (n-1)
- else nth_stage::(stageh (n-1))
- end )
- handle STAGE_TRUE => [HOLogic.true_const])
- val slist = stageh (divlcm x p)
- in (list_disj slist)
- end
- | _ => error "cooper: not an existential formula";
-
-
-(* A Version of cooper that returns a withness *)
-fun cooper_w vars1 fm =
- case fm of
- Const ("Ex",_) $ Abs(x0,T,p0) => let
- val (xn,p1) = Syntax.variant_abs (x0,T,p0)
- val x = Free (xn,T)
- val vars = (xn::vars1)
-(* val p = unitycoeff x (posineq (simpl p1)) *)
- val p = unitycoeff x p1
- val ast = aset x p
- val bst = bset x p
- val d = divlcm x p
- val (p_inf,S ) =
- if (length bst) <= (length ast)
- then (true,bst)
- else (false,ast)
- in withness d p_inf S vars x p
-(* fun p_element j a = linrep vars x (f vars a (mk_number j)) p
- fun stage j = list_disj (linrep vars x (mk_number j) p_inf :: map (p_element j) S)
- in (list_disj (map stage js))
-*)
- end
- | _ => error "cooper: not an existential formula";
-
-
-(* ------------------------------------------------------------------------- *)
-(* Free variables in terms and formulas. *)
-(* ------------------------------------------------------------------------- *)
-
-fun fvt tml = case tml of
- [] => []
- | Free(x,_)::r => x::(fvt r)
-
-fun fv fm = fvt (term_frees fm);
-
-
-(* ========================================================================= *)
-(* Quantifier elimination. *)
-(* ========================================================================= *)
-(*conj[/disj]uncts lists iterated conj[disj]unctions*)
-
-fun disjuncts fm = case fm of
- Const ("op |",_) $ p $ q => (disjuncts p) @ (disjuncts q)
- | _ => [fm];
-
-fun conjuncts fm = case fm of
- Const ("op &",_) $p $ q => (conjuncts p) @ (conjuncts q)
- | _ => [fm];
-
-
-
-(* ------------------------------------------------------------------------- *)
-(* Lift procedure given literal modifier, formula normalizer & basic quelim. *)
-(* ------------------------------------------------------------------------- *)
-
-fun lift_qelim afn nfn qfn isat =
-let
-fun qelift vars fm = if (isat fm) then afn vars fm
-else
-case fm of
- Const ("Not",_) $ p => HOLogic.Not $ (qelift vars p)
- | Const ("op &",_) $ p $q => HOLogic.conj $ (qelift vars p) $ (qelift vars q)
- | Const ("op |",_) $ p $ q => HOLogic.disj $ (qelift vars p) $ (qelift vars q)
- | Const ("op -->",_) $ p $ q => HOLogic.imp $ (qelift vars p) $ (qelift vars q)
- | Const ("op =",Type ("fun",[Type ("bool", []),_])) $ p $ q => HOLogic.mk_eq ((qelift vars p),(qelift vars q))
- | Const ("All",QT) $ Abs(x,T,p) => HOLogic.Not $(qelift vars (Const ("Ex",QT) $ Abs(x,T,(HOLogic.Not $ p))))
- | (e as Const ("Ex",_)) $ Abs (x,T,p) => qfn vars (e$Abs (x,T,(nfn(qelift (x::vars) p))))
- | _ => fm
-
-in (fn fm => qelift (fv fm) fm)
-end;
-
-
-(*
-fun lift_qelim afn nfn qfn isat =
- let fun qelim x vars p =
- let val cjs = conjuncts p
- val (ycjs,ncjs) = List.partition (has_bound) cjs in
- (if ycjs = [] then p else
- let val q = (qfn vars ((HOLogic.exists_const HOLogic.intT
- ) $ Abs(x,HOLogic.intT,(list_conj ycjs)))) in
- (fold_rev conj_help ncjs q)
- end)
- end
-
- fun qelift vars fm = if (isat fm) then afn vars fm
- else
- case fm of
- Const ("Not",_) $ p => HOLogic.Not $ (qelift vars p)
- | Const ("op &",_) $ p $q => HOLogic.conj $ (qelift vars p) $ (qelift vars q)
- | Const ("op |",_) $ p $ q => HOLogic.disj $ (qelift vars p) $ (qelift vars q)
- | Const ("op -->",_) $ p $ q => HOLogic.imp $ (qelift vars p) $ (qelift vars q)
- | Const ("op =",Type ("fun",[Type ("bool", []),_])) $ p $ q => HOLogic.mk_eq ((qelift vars p),(qelift vars q))
- | Const ("All",QT) $ Abs(x,T,p) => HOLogic.Not $(qelift vars (Const ("Ex",QT) $ Abs(x,T,(HOLogic.Not $ p))))
- | Const ("Ex",_) $ Abs (x,T,p) => let val djs = disjuncts(nfn(qelift (x::vars) p)) in
- list_disj(map (qelim x vars) djs) end
- | _ => fm
-
- in (fn fm => simpl(qelift (fv fm) fm))
- end;
-*)
-
-(* ------------------------------------------------------------------------- *)
-(* Cleverer (proposisional) NNF with conditional and literal modification. *)
-(* ------------------------------------------------------------------------- *)
-
-(*Function Negate used by cnnf, negates a formula p*)
-
-fun negate (Const ("Not",_) $ p) = p
- |negate p = (HOLogic.Not $ p);
-
-fun cnnf lfn =
- let fun cnnfh fm = case fm of
- (Const ("op &",_) $ p $ q) => HOLogic.mk_conj(cnnfh p,cnnfh q)
- | (Const ("op |",_) $ p $ q) => HOLogic.mk_disj(cnnfh p,cnnfh q)
- | (Const ("op -->",_) $ p $q) => HOLogic.mk_disj(cnnfh(HOLogic.Not $ p),cnnfh q)
- | (Const ("op =",Type ("fun",[Type ("bool", []),_])) $ p $ q) => HOLogic.mk_disj(
- HOLogic.mk_conj(cnnfh p,cnnfh q),
- HOLogic.mk_conj(cnnfh(HOLogic.Not $ p),cnnfh(HOLogic.Not $q)))
-
- | (Const ("Not",_) $ (Const("Not",_) $ p)) => cnnfh p
- | (Const ("Not",_) $ (Const ("op &",_) $ p $ q)) => HOLogic.mk_disj(cnnfh(HOLogic.Not $ p),cnnfh(HOLogic.Not $ q))
- | (Const ("Not",_) $(Const ("op |",_) $ (Const ("op &",_) $ p $ q) $
- (Const ("op &",_) $ p1 $ r))) => if p1 = negate p then
- HOLogic.mk_disj(
- cnnfh (HOLogic.mk_conj(p,cnnfh(HOLogic.Not $ q))),
- cnnfh (HOLogic.mk_conj(p1,cnnfh(HOLogic.Not $ r))))
- else HOLogic.mk_conj(
- cnnfh (HOLogic.mk_disj(cnnfh (HOLogic.Not $ p),cnnfh(HOLogic.Not $ q))),
- cnnfh (HOLogic.mk_disj(cnnfh (HOLogic.Not $ p1),cnnfh(HOLogic.Not $ r)))
- )
- | (Const ("Not",_) $ (Const ("op |",_) $ p $ q)) => HOLogic.mk_conj(cnnfh(HOLogic.Not $ p),cnnfh(HOLogic.Not $ q))
- | (Const ("Not",_) $ (Const ("op -->",_) $ p $q)) => HOLogic.mk_conj(cnnfh p,cnnfh(HOLogic.Not $ q))
- | (Const ("Not",_) $ (Const ("op =",Type ("fun",[Type ("bool", []),_])) $ p $ q)) => HOLogic.mk_disj(HOLogic.mk_conj(cnnfh p,cnnfh(HOLogic.Not $ q)),HOLogic.mk_conj(cnnfh(HOLogic.Not $ p),cnnfh q))
- | _ => lfn fm
-in cnnfh
- end;
-
-(*End- function the quantifierelimination an decion procedure of presburger formulas.*)
-
-(*
-val integer_qelim = simpl o evalc o (lift_qelim linform (simpl o (cnnf posineq o evalc)) cooper is_arith_rel) ;
-*)
-
-
-val integer_qelim = simpl o evalc o (lift_qelim linform (cnnf posineq o evalc) cooper is_arith_rel) ;
-
-end;