author | haftmann |
Fri, 10 Mar 2006 15:33:48 +0100 | |
changeset 19233 | 77ca20b0ed77 |
parent 17521 | 0f1c48de39f5 |
child 19277 | f7602e74d948 |
permissions | -rw-r--r-- |
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(* Title: HOL/Integ/cooper_dec.ML |
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ID: $Id$ |
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Author: Amine Chaieb and Tobias Nipkow, TU Muenchen |
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File containing the implementation of Cooper Algorithm |
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decision procedure (intensively inspired from J.Harrison) |
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*) |
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An oracle is built in. The tactic will not generate any proofs any more, if the quick_and_dirty flag is set on.
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signature COOPER_DEC = |
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sig |
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exception COOPER |
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val is_arith_rel : term -> bool |
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val mk_numeral : IntInf.int -> term |
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val dest_numeral : term -> IntInf.int |
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val is_numeral : term -> bool |
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val zero : term |
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val one : term |
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val linear_cmul : IntInf.int -> term -> term |
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val linear_add : string list -> term -> term -> term |
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val linear_sub : string list -> term -> term -> term |
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val linear_neg : term -> term |
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val lint : string list -> term -> term |
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val linform : string list -> term -> term |
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val formlcm : term -> term -> IntInf.int |
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val adjustcoeff : term -> IntInf.int -> term -> term |
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val unitycoeff : term -> term -> term |
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val divlcm : term -> term -> IntInf.int |
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val bset : term -> term -> term list |
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val aset : term -> term -> term list |
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val linrep : string list -> term -> term -> term -> term |
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val list_disj : term list -> term |
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val list_conj : term list -> term |
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val simpl : term -> term |
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val fv : term -> string list |
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val negate : term -> term |
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val operations : (string * (IntInf.int * IntInf.int -> bool)) list |
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A new implementation for presburger arithmetic following the one suggested in technical report Chaieb Amine and Tobias Nipkow. It is generic an smaller.
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val conjuncts : term -> term list |
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A new implementation for presburger arithmetic following the one suggested in technical report Chaieb Amine and Tobias Nipkow. It is generic an smaller.
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val disjuncts : term -> term list |
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A new implementation for presburger arithmetic following the one suggested in technical report Chaieb Amine and Tobias Nipkow. It is generic an smaller.
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val has_bound : term -> bool |
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A new implementation for presburger arithmetic following the one suggested in technical report Chaieb Amine and Tobias Nipkow. It is generic an smaller.
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val minusinf : term -> term -> term |
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A new implementation for presburger arithmetic following the one suggested in technical report Chaieb Amine and Tobias Nipkow. It is generic an smaller.
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val plusinf : term -> term -> term |
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val onatoms : (term -> term) -> term -> term |
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val evalc : term -> term |
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val cooper_w : string list -> term -> (term option * term) |
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val integer_qelim : Term.term -> Term.term |
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end; |
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structure CooperDec : COOPER_DEC = |
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struct |
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(* ========================================================================= *) |
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(* Cooper's algorithm for Presburger arithmetic. *) |
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(* ========================================================================= *) |
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exception COOPER; |
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(* ------------------------------------------------------------------------- *) |
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(* Lift operations up to numerals. *) |
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(* ------------------------------------------------------------------------- *) |
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(*Assumption : The construction of atomar formulas in linearl arithmetic is based on |
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relation operations of Type : [IntInf.int,IntInf.int]---> bool *) |
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(* ------------------------------------------------------------------------- *) |
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(*Function is_arith_rel returns true if and only if the term is an atomar presburger |
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formula *) |
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fun is_arith_rel tm = case tm of |
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Const(p,Type ("fun",[Type ("Numeral.bin", []),Type ("fun",[Type ("Numeral.bin", |
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[]),Type ("bool",[])] )])) $ _ $_ => true |
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|Const(p,Type ("fun",[Type ("IntDef.int", []),Type ("fun",[Type ("IntDef.int", |
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[]),Type ("bool",[])] )])) $ _ $_ => true |
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|_ => false; |
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(*Function is_arith_rel returns true if and only if the term is an operation of the |
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form [int,int]---> int*) |
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(*Transform a natural number to a term*) |
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fun mk_numeral 0 = Const("0",HOLogic.intT) |
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|mk_numeral 1 = Const("1",HOLogic.intT) |
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|mk_numeral n = (HOLogic.number_of_const HOLogic.intT) $ (HOLogic.mk_bin n); |
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(*Transform an Term to an natural number*) |
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fun dest_numeral (Const("0",Type ("IntDef.int", []))) = 0 |
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|dest_numeral (Const("1",Type ("IntDef.int", []))) = 1 |
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|dest_numeral (Const ("Numeral.number_of",_) $ n) = |
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HOLogic.dest_binum n; |
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(*Some terms often used for pattern matching*) |
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val zero = mk_numeral 0; |
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val one = mk_numeral 1; |
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(*Tests if a Term is representing a number*) |
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fun is_numeral t = (t = zero) orelse (t = one) orelse (can dest_numeral t); |
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(*maps a unary natural function on a term containing an natural number*) |
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fun numeral1 f n = mk_numeral (f(dest_numeral n)); |
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(*maps a binary natural function on 2 term containing natural numbers*) |
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fun numeral2 f m n = mk_numeral(f(dest_numeral m) (dest_numeral n)); |
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(* ------------------------------------------------------------------------- *) |
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(* Operations on canonical linear terms c1 * x1 + ... + cn * xn + k *) |
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(* *) |
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(* Note that we're quite strict: the ci must be present even if 1 *) |
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(* (but if 0 we expect the monomial to be omitted) and k must be there *) |
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(* even if it's zero. Thus, it's a constant iff not an addition term. *) |
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(* ------------------------------------------------------------------------- *) |
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fun linear_cmul n tm = if n = 0 then zero else let fun times n k = n*k in |
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( case tm of |
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(Const("HOL.plus",T) $ (Const ("HOL.times",T1 ) $c1 $ x1) $ rest) => |
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Const("HOL.plus",T) $ ((Const("HOL.times",T1) $ (numeral1 (times n) c1) $ x1)) $ (linear_cmul n rest) |
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|_ => numeral1 (times n) tm) |
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end ; |
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(* Whether the first of two items comes earlier in the list *) |
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fun earlier [] x y = false |
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|earlier (h::t) x y =if h = y then false |
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else if h = x then true |
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else earlier t x y ; |
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fun earlierv vars (Bound i) (Bound j) = i < j |
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|earlierv vars (Bound _) _ = true |
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|earlierv vars _ (Bound _) = false |
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|earlierv vars (Free (x,_)) (Free (y,_)) = earlier vars x y; |
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fun linear_add vars tm1 tm2 = |
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let fun addwith x y = x + y in |
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(case (tm1,tm2) of |
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((Const ("HOL.plus",T1) $ ( Const("HOL.times",T2) $ c1 $ x1) $ rest1),(Const |
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("HOL.plus",T3)$( Const("HOL.times",T4) $ c2 $ x2) $ rest2)) => |
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if x1 = x2 then |
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let val c = (numeral2 (addwith) c1 c2) |
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in |
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if c = zero then (linear_add vars rest1 rest2) |
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else (Const("HOL.plus",T1) $ (Const("HOL.times",T2) $ c $ x1) $ (linear_add vars rest1 rest2)) |
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end |
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else |
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if earlierv vars x1 x2 then (Const("HOL.plus",T1) $ |
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(Const("HOL.times",T2)$ c1 $ x1) $ (linear_add vars rest1 tm2)) |
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else (Const("HOL.plus",T1) $ (Const("HOL.times",T2) $ c2 $ x2) $ (linear_add vars tm1 rest2)) |
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|((Const("HOL.plus",T1) $ (Const("HOL.times",T2) $ c1 $ x1) $ rest1) ,_) => |
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(Const("HOL.plus",T1)$ (Const("HOL.times",T2) $ c1 $ x1) $ (linear_add vars |
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rest1 tm2)) |
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|(_, (Const("HOL.plus",T1) $(Const("HOL.times",T2) $ c2 $ x2) $ rest2)) => |
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(Const("HOL.plus",T1) $ (Const("HOL.times",T2) $ c2 $ x2) $ (linear_add vars tm1 |
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rest2)) |
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| (_,_) => numeral2 (addwith) tm1 tm2) |
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end; |
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(*To obtain the unary - applyed on a formula*) |
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fun linear_neg tm = linear_cmul (0 - 1) tm; |
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(*Substraction of two terms *) |
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fun linear_sub vars tm1 tm2 = linear_add vars tm1 (linear_neg tm2); |
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(* ------------------------------------------------------------------------- *) |
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(* Linearize a term. *) |
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(* ------------------------------------------------------------------------- *) |
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(* linearises a term from the point of view of Variable Free (x,T). |
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After this fuction the all expressions containig ths variable will have the form |
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c*Free(x,T) + t where c is a constant ant t is a Term which is not containing |
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Free(x,T)*) |
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fun lint vars tm = if is_numeral tm then tm else case tm of |
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(Free (x,T)) => (HOLogic.mk_binop "HOL.plus" ((HOLogic.mk_binop "HOL.times" ((mk_numeral 1),Free (x,T))), zero)) |
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|(Bound i) => (Const("HOL.plus",HOLogic.intT -->HOLogic.intT -->HOLogic.intT) $ |
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(Const("HOL.times",HOLogic.intT -->HOLogic.intT -->HOLogic.intT) $ (mk_numeral 1) $ (Bound i)) $ zero) |
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|(Const("HOL.uminus",_) $ t ) => (linear_neg (lint vars t)) |
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|(Const("HOL.plus",_) $ s $ t) => (linear_add vars (lint vars s) (lint vars t)) |
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|(Const("HOL.minus",_) $ s $ t) => (linear_sub vars (lint vars s) (lint vars t)) |
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|(Const ("HOL.times",_) $ s $ t) => |
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let val s' = lint vars s |
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val t' = lint vars t |
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in |
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if is_numeral s' then (linear_cmul (dest_numeral s') t') |
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else if is_numeral t' then (linear_cmul (dest_numeral t') s') |
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else raise COOPER |
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end |
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|_ => raise COOPER; |
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(* ------------------------------------------------------------------------- *) |
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(* Linearize the atoms in a formula, and eliminate non-strict inequalities. *) |
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(* ------------------------------------------------------------------------- *) |
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fun mkatom vars p t = Const(p,HOLogic.intT --> HOLogic.intT --> HOLogic.boolT) $ zero $ (lint vars t); |
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fun linform vars (Const ("Divides.op dvd",_) $ c $ t) = |
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if is_numeral c then |
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let val c' = (mk_numeral(abs(dest_numeral c))) |
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in (HOLogic.mk_binrel "Divides.op dvd" (c,lint vars t)) |
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end |
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else (warning "Nonlinear term --- Non numeral leftside at dvd" |
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;raise COOPER) |
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|linform vars (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ s $ t ) = (mkatom vars "op =" (Const ("HOL.minus",HOLogic.intT --> HOLogic.intT --> HOLogic.intT) $ t $ s) ) |
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|linform vars (Const("op <",_)$ s $t ) = (mkatom vars "op <" (Const ("HOL.minus",HOLogic.intT --> HOLogic.intT --> HOLogic.intT) $ t $ s)) |
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|linform vars (Const("op >",_) $ s $ t ) = (mkatom vars "op <" (Const ("HOL.minus",HOLogic.intT --> HOLogic.intT --> HOLogic.intT) $ s $ t)) |
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|linform vars (Const("op <=",_)$ s $ t ) = |
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(mkatom vars "op <" (Const ("HOL.minus",HOLogic.intT --> HOLogic.intT --> HOLogic.intT) $ (Const("HOL.plus",HOLogic.intT --> HOLogic.intT --> HOLogic.intT) $t $(mk_numeral 1)) $ s)) |
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|linform vars (Const("op >=",_)$ s $ t ) = |
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(mkatom vars "op <" (Const ("HOL.minus",HOLogic.intT --> HOLogic.intT --> |
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HOLogic.intT) $ (Const("HOL.plus",HOLogic.intT --> HOLogic.intT --> |
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HOLogic.intT) $s $(mk_numeral 1)) $ t)) |
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|linform vars fm = fm; |
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(* ------------------------------------------------------------------------- *) |
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(* Post-NNF transformation eliminating negated inequalities. *) |
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229 |
(* ------------------------------------------------------------------------- *) |
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fun posineq fm = case fm of |
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(Const ("Not",_)$(Const("op <",_)$ c $ t)) => |
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(HOLogic.mk_binrel "op <" (zero , (linear_sub [] (mk_numeral 1) (linear_add [] c t ) ))) |
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| ( Const ("op &",_) $ p $ q) => HOLogic.mk_conj (posineq p,posineq q) |
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| ( Const ("op |",_) $ p $ q ) => HOLogic.mk_disj (posineq p,posineq q) |
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| _ => fm; |
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(* ------------------------------------------------------------------------- *) |
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(* Find the LCM of the coefficients of x. *) |
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(* ------------------------------------------------------------------------- *) |
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(*gcd calculates gcd (a,b) and helps lcm_num calculating lcm (a,b)*) |
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243 |
||
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(*BEWARE: replaces Library.gcd!! There is also Library.lcm!*) |
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fun gcd (a:IntInf.int) b = if a=0 then b else gcd (b mod a) a ; |
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fun lcm_num a b = (abs a*b) div (gcd (abs a) (abs b)); |
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248 |
fun formlcm x fm = case fm of |
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(Const (p,_)$ _ $(Const ("HOL.plus", _)$(Const ("HOL.times",_)$ c $ y ) $z ) ) => if |
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250 |
(is_arith_rel fm) andalso (x = y) then (abs(dest_numeral c)) else 1 |
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| ( Const ("Not", _) $p) => formlcm x p |
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| ( Const ("op &",_) $ p $ q) => lcm_num (formlcm x p) (formlcm x q) |
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| ( Const ("op |",_) $ p $ q )=> lcm_num (formlcm x p) (formlcm x q) |
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| _ => 1; |
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256 |
(* ------------------------------------------------------------------------- *) |
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257 |
(* Adjust all coefficients of x in formula; fold in reduction to +/- 1. *) |
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258 |
(* ------------------------------------------------------------------------- *) |
|
259 |
||
260 |
fun adjustcoeff x l fm = |
|
261 |
case fm of |
|
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|
262 |
(Const(p,_) $d $( Const ("HOL.plus", _)$(Const ("HOL.times",_) $ |
13876 | 263 |
c $ y ) $z )) => if (is_arith_rel fm) andalso (x = y) then |
264 |
let val m = l div (dest_numeral c) |
|
265 |
val n = (if p = "op <" then abs(m) else m) |
|
19233
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parents:
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changeset
|
266 |
val xtm = HOLogic.mk_binop "HOL.times" ((mk_numeral (m div n)), x) |
13876 | 267 |
in |
19233
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|
268 |
(HOLogic.mk_binrel p ((linear_cmul n d),(HOLogic.mk_binop "HOL.plus" ( xtm ,( linear_cmul n z) )))) |
13876 | 269 |
end |
270 |
else fm |
|
271 |
|( Const ("Not", _) $ p) => HOLogic.Not $ (adjustcoeff x l p) |
|
272 |
|( Const ("op &",_) $ p $ q) => HOLogic.conj$(adjustcoeff x l p) $(adjustcoeff x l q) |
|
273 |
|( Const ("op |",_) $ p $ q) => HOLogic.disj $(adjustcoeff x l p)$ (adjustcoeff x l q) |
|
274 |
|_ => fm; |
|
275 |
||
276 |
(* ------------------------------------------------------------------------- *) |
|
277 |
(* Hence make coefficient of x one in existential formula. *) |
|
278 |
(* ------------------------------------------------------------------------- *) |
|
279 |
||
280 |
fun unitycoeff x fm = |
|
15965
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changeset
|
281 |
let val l = formlcm x fm |
13876 | 282 |
val fm' = adjustcoeff x l fm in |
15267 | 283 |
if l = 1 then fm' |
284 |
else |
|
19233
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renamed HOL + - * etc. to HOL.plus HOL.minus HOL.times etc.
haftmann
parents:
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diff
changeset
|
285 |
let val xp = (HOLogic.mk_binop "HOL.plus" |
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|
286 |
((HOLogic.mk_binop "HOL.times" ((mk_numeral 1), x )), zero)) |
15267 | 287 |
in |
13876 | 288 |
HOLogic.conj $(HOLogic.mk_binrel "Divides.op dvd" ((mk_numeral l) , xp )) $ (adjustcoeff x l fm) |
289 |
end |
|
290 |
end; |
|
291 |
||
292 |
(* adjustcoeffeq l fm adjusts the coeffitients c_i of x overall in fm to l*) |
|
293 |
(* Here l must be a multiple of all c_i otherwise the obtained formula is not equivalent*) |
|
294 |
(* |
|
295 |
fun adjustcoeffeq x l fm = |
|
296 |
case fm of |
|
19233
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parents:
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changeset
|
297 |
(Const(p,_) $d $( Const ("HOL.plus", _)$(Const ("HOL.times",_) $ |
13876 | 298 |
c $ y ) $z )) => if (is_arith_rel fm) andalso (x = y) then |
299 |
let val m = l div (dest_numeral c) |
|
300 |
val n = (if p = "op <" then abs(m) else m) |
|
19233
77ca20b0ed77
renamed HOL + - * etc. to HOL.plus HOL.minus HOL.times etc.
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parents:
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changeset
|
301 |
val xtm = (HOLogic.mk_binop "HOL.times" ((mk_numeral ((m div n)*l) ), x)) |
77ca20b0ed77
renamed HOL + - * etc. to HOL.plus HOL.minus HOL.times etc.
haftmann
parents:
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diff
changeset
|
302 |
in (HOLogic.mk_binrel p ((linear_cmul n d),(HOLogic.mk_binop "HOL.plus" ( xtm ,( linear_cmul n z) )))) |
13876 | 303 |
end |
304 |
else fm |
|
305 |
|( Const ("Not", _) $ p) => HOLogic.Not $ (adjustcoeffeq x l p) |
|
306 |
|( Const ("op &",_) $ p $ q) => HOLogic.conj$(adjustcoeffeq x l p) $(adjustcoeffeq x l q) |
|
307 |
|( Const ("op |",_) $ p $ q) => HOLogic.disj $(adjustcoeffeq x l p)$ (adjustcoeffeq x l q) |
|
308 |
|_ => fm; |
|
309 |
||
310 |
||
311 |
*) |
|
312 |
||
313 |
(* ------------------------------------------------------------------------- *) |
|
314 |
(* The "minus infinity" version. *) |
|
315 |
(* ------------------------------------------------------------------------- *) |
|
316 |
||
317 |
fun minusinf x fm = case fm of |
|
19233
77ca20b0ed77
renamed HOL + - * etc. to HOL.plus HOL.minus HOL.times etc.
haftmann
parents:
17521
diff
changeset
|
318 |
(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ (c1 ) $(Const ("HOL.plus", _) $(Const ("HOL.times",_) $ c2 $ y) $z)) => |
13876 | 319 |
if (is_arith_rel fm) andalso (x=y) andalso (c2 = one) andalso (c1 =zero) then HOLogic.false_const |
320 |
else fm |
|
321 |
||
19233
77ca20b0ed77
renamed HOL + - * etc. to HOL.plus HOL.minus HOL.times etc.
haftmann
parents:
17521
diff
changeset
|
322 |
|(Const("op <",_) $ c $(Const ("HOL.plus", _) $(Const ("HOL.times",_) $ pm1 $ y ) $ z |
15859 | 323 |
)) => if (x = y) |
324 |
then if (pm1 = one) andalso (c = zero) then HOLogic.false_const |
|
325 |
else if (dest_numeral pm1 = ~1) andalso (c = zero) then HOLogic.true_const |
|
326 |
else error "minusinf : term not in normal form!!!" |
|
327 |
else fm |
|
13876 | 328 |
|
329 |
|(Const ("Not", _) $ p) => HOLogic.Not $ (minusinf x p) |
|
330 |
|(Const ("op &",_) $ p $ q) => HOLogic.conj $ (minusinf x p) $ (minusinf x q) |
|
331 |
|(Const ("op |",_) $ p $ q) => HOLogic.disj $ (minusinf x p) $ (minusinf x q) |
|
332 |
|_ => fm; |
|
333 |
||
334 |
(* ------------------------------------------------------------------------- *) |
|
335 |
(* The "Plus infinity" version. *) |
|
336 |
(* ------------------------------------------------------------------------- *) |
|
337 |
||
338 |
fun plusinf x fm = case fm of |
|
19233
77ca20b0ed77
renamed HOL + - * etc. to HOL.plus HOL.minus HOL.times etc.
haftmann
parents:
17521
diff
changeset
|
339 |
(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ (c1 ) $(Const ("HOL.plus", _) $(Const ("HOL.times",_) $ c2 $ y) $z)) => |
13876 | 340 |
if (is_arith_rel fm) andalso (x=y) andalso (c2 = one) andalso (c1 =zero) then HOLogic.false_const |
341 |
else fm |
|
342 |
||
19233
77ca20b0ed77
renamed HOL + - * etc. to HOL.plus HOL.minus HOL.times etc.
haftmann
parents:
17521
diff
changeset
|
343 |
|(Const("op <",_) $ c $(Const ("HOL.plus", _) $(Const ("HOL.times",_) $ pm1 $ y ) $ z |
15859 | 344 |
)) => if (x = y) |
345 |
then if (pm1 = one) andalso (c = zero) then HOLogic.true_const |
|
346 |
else if (dest_numeral pm1 = ~1) andalso (c = zero) then HOLogic.false_const |
|
347 |
else error "plusinf : term not in normal form!!!" |
|
348 |
else fm |
|
13876 | 349 |
|
350 |
|(Const ("Not", _) $ p) => HOLogic.Not $ (plusinf x p) |
|
351 |
|(Const ("op &",_) $ p $ q) => HOLogic.conj $ (plusinf x p) $ (plusinf x q) |
|
352 |
|(Const ("op |",_) $ p $ q) => HOLogic.disj $ (plusinf x p) $ (plusinf x q) |
|
353 |
|_ => fm; |
|
354 |
||
355 |
(* ------------------------------------------------------------------------- *) |
|
356 |
(* The LCM of all the divisors that involve x. *) |
|
357 |
(* ------------------------------------------------------------------------- *) |
|
358 |
||
19233
77ca20b0ed77
renamed HOL + - * etc. to HOL.plus HOL.minus HOL.times etc.
haftmann
parents:
17521
diff
changeset
|
359 |
fun divlcm x (Const("Divides.op dvd",_)$ d $ (Const ("HOL.plus",_) $ (Const ("HOL.times",_) $ c $ y ) $ z ) ) = |
13876 | 360 |
if x = y then abs(dest_numeral d) else 1 |
361 |
|divlcm x ( Const ("Not", _) $ p) = divlcm x p |
|
362 |
|divlcm x ( Const ("op &",_) $ p $ q) = lcm_num (divlcm x p) (divlcm x q) |
|
363 |
|divlcm x ( Const ("op |",_) $ p $ q ) = lcm_num (divlcm x p) (divlcm x q) |
|
364 |
|divlcm x _ = 1; |
|
365 |
||
366 |
(* ------------------------------------------------------------------------- *) |
|
367 |
(* Construct the B-set. *) |
|
368 |
(* ------------------------------------------------------------------------- *) |
|
369 |
||
370 |
fun bset x fm = case fm of |
|
371 |
(Const ("Not", _) $ p) => if (is_arith_rel p) then |
|
372 |
(case p of |
|
19233
77ca20b0ed77
renamed HOL + - * etc. to HOL.plus HOL.minus HOL.times etc.
haftmann
parents:
17521
diff
changeset
|
373 |
(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("HOL.plus", _) $(Const ("HOL.times",_) $c2 $y) $a ) ) |
13876 | 374 |
=> if (is_arith_rel p) andalso (x= y) andalso (c2 = one) andalso (c1 = zero) |
375 |
then [linear_neg a] |
|
376 |
else bset x p |
|
377 |
|_ =>[]) |
|
378 |
||
379 |
else bset x p |
|
19233
77ca20b0ed77
renamed HOL + - * etc. to HOL.plus HOL.minus HOL.times etc.
haftmann
parents:
17521
diff
changeset
|
380 |
|(Const ("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("HOL.plus",_) $ (Const ("HOL.times",_) $c2 $ x) $ a)) => if (c1 =zero) andalso (c2 = one) then [linear_neg(linear_add [] a (mk_numeral 1))] else [] |
77ca20b0ed77
renamed HOL + - * etc. to HOL.plus HOL.minus HOL.times etc.
haftmann
parents:
17521
diff
changeset
|
381 |
|(Const ("op <",_) $ c1$ (Const ("HOL.plus",_) $(Const ("HOL.times",_)$ c2 $ x) $ a)) => if (c1 =zero) andalso (c2 = one) then [linear_neg a] else [] |
13876 | 382 |
|(Const ("op &",_) $ p $ q) => (bset x p) union (bset x q) |
383 |
|(Const ("op |",_) $ p $ q) => (bset x p) union (bset x q) |
|
384 |
|_ => []; |
|
385 |
||
386 |
(* ------------------------------------------------------------------------- *) |
|
387 |
(* Construct the A-set. *) |
|
388 |
(* ------------------------------------------------------------------------- *) |
|
389 |
||
390 |
fun aset x fm = case fm of |
|
391 |
(Const ("Not", _) $ p) => if (is_arith_rel p) then |
|
392 |
(case p of |
|
19233
77ca20b0ed77
renamed HOL + - * etc. to HOL.plus HOL.minus HOL.times etc.
haftmann
parents:
17521
diff
changeset
|
393 |
(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("HOL.plus", _) $(Const ("HOL.times",_) $c2 $y) $a ) ) |
13876 | 394 |
=> if (x= y) andalso (c2 = one) andalso (c1 = zero) |
395 |
then [linear_neg a] |
|
396 |
else [] |
|
397 |
|_ =>[]) |
|
398 |
||
399 |
else aset x p |
|
19233
77ca20b0ed77
renamed HOL + - * etc. to HOL.plus HOL.minus HOL.times etc.
haftmann
parents:
17521
diff
changeset
|
400 |
|(Const ("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("HOL.plus",_) $ (Const ("HOL.times",_) $c2 $ x) $ a)) => if (c1 =zero) andalso (c2 = one) then [linear_sub [] (mk_numeral 1) a] else [] |
77ca20b0ed77
renamed HOL + - * etc. to HOL.plus HOL.minus HOL.times etc.
haftmann
parents:
17521
diff
changeset
|
401 |
|(Const ("op <",_) $ c1$ (Const ("HOL.plus",_) $(Const ("HOL.times",_)$ c2 $ x) $ a)) => if (c1 =zero) andalso (c2 = (mk_numeral (~1))) then [a] else [] |
13876 | 402 |
|(Const ("op &",_) $ p $ q) => (aset x p) union (aset x q) |
403 |
|(Const ("op |",_) $ p $ q) => (aset x p) union (aset x q) |
|
404 |
|_ => []; |
|
405 |
||
406 |
||
407 |
(* ------------------------------------------------------------------------- *) |
|
408 |
(* Replace top variable with another linear form, retaining canonicality. *) |
|
409 |
(* ------------------------------------------------------------------------- *) |
|
410 |
||
411 |
fun linrep vars x t fm = case fm of |
|
19233
77ca20b0ed77
renamed HOL + - * etc. to HOL.plus HOL.minus HOL.times etc.
haftmann
parents:
17521
diff
changeset
|
412 |
((Const(p,_)$ d $ (Const("HOL.plus",_)$(Const("HOL.times",_)$ c $ y) $ z))) => |
13876 | 413 |
if (x = y) andalso (is_arith_rel fm) |
414 |
then |
|
415 |
let val ct = linear_cmul (dest_numeral c) t |
|
416 |
in (HOLogic.mk_binrel p (d, linear_add vars ct z)) |
|
417 |
end |
|
418 |
else fm |
|
419 |
|(Const ("Not", _) $ p) => HOLogic.Not $ (linrep vars x t p) |
|
420 |
|(Const ("op &",_) $ p $ q) => HOLogic.conj $ (linrep vars x t p) $ (linrep vars x t q) |
|
421 |
|(Const ("op |",_) $ p $ q) => HOLogic.disj $ (linrep vars x t p) $ (linrep vars x t q) |
|
15267 | 422 |
|_ => fm; |
13876 | 423 |
|
424 |
(* ------------------------------------------------------------------------- *) |
|
425 |
(* Evaluation of constant expressions. *) |
|
426 |
(* ------------------------------------------------------------------------- *) |
|
15107 | 427 |
|
428 |
(* An other implementation of divides, that covers more cases*) |
|
429 |
||
430 |
exception DVD_UNKNOWN |
|
431 |
||
432 |
fun dvd_op (d, t) = |
|
433 |
if not(is_numeral d) then raise DVD_UNKNOWN |
|
434 |
else let |
|
435 |
val dn = dest_numeral d |
|
436 |
fun coeffs_of x = case x of |
|
437 |
Const(p,_) $ tl $ tr => |
|
19233
77ca20b0ed77
renamed HOL + - * etc. to HOL.plus HOL.minus HOL.times etc.
haftmann
parents:
17521
diff
changeset
|
438 |
if p = "HOL.plus" then (coeffs_of tl) union (coeffs_of tr) |
77ca20b0ed77
renamed HOL + - * etc. to HOL.plus HOL.minus HOL.times etc.
haftmann
parents:
17521
diff
changeset
|
439 |
else if p = "HOL.times" |
15107 | 440 |
then if (is_numeral tr) |
441 |
then [(dest_numeral tr) * (dest_numeral tl)] |
|
442 |
else [dest_numeral tl] |
|
443 |
else [] |
|
444 |
|_ => if (is_numeral t) then [dest_numeral t] else [] |
|
445 |
val ts = coeffs_of t |
|
446 |
in case ts of |
|
447 |
[] => raise DVD_UNKNOWN |
|
15574
b1d1b5bfc464
Removed practically all references to Library.foldr.
skalberg
parents:
15570
diff
changeset
|
448 |
|_ => foldr (fn(k,r) => r andalso (k mod dn = 0)) true ts |
15107 | 449 |
end; |
450 |
||
451 |
||
13876 | 452 |
val operations = |
16398 | 453 |
[("op =",op=), ("op <",IntInf.<), ("op >",IntInf.>), ("op <=",IntInf.<=) , |
454 |
("op >=",IntInf.>=), |
|
455 |
("Divides.op dvd",fn (x,y) =>((IntInf.mod(y, x)) = 0))]; |
|
13876 | 456 |
|
15531 | 457 |
fun applyoperation (SOME f) (a,b) = f (a, b) |
13876 | 458 |
|applyoperation _ (_, _) = false; |
459 |
||
460 |
(*Evaluation of constant atomic formulas*) |
|
15107 | 461 |
(*FIXME : This is an optimation but still incorrect !! *) |
462 |
(* |
|
13876 | 463 |
fun evalc_atom at = case at of |
15107 | 464 |
(Const (p,_) $ s $ t) => |
465 |
(if p="Divides.op dvd" then |
|
466 |
((if dvd_op(s,t) then HOLogic.true_const |
|
467 |
else HOLogic.false_const) |
|
468 |
handle _ => at) |
|
469 |
else |
|
17521 | 470 |
case AList.lookup (op =) operations p of |
15531 | 471 |
SOME f => ((if (f ((dest_numeral s),(dest_numeral t))) then HOLogic.true_const else HOLogic.false_const) |
15107 | 472 |
handle _ => at) |
473 |
| _ => at) |
|
474 |
|Const("Not",_)$(Const (p,_) $ s $ t) =>( |
|
17521 | 475 |
case AList.lookup (op =) operations p of |
15531 | 476 |
SOME f => ((if (f ((dest_numeral s),(dest_numeral t))) then |
15107 | 477 |
HOLogic.false_const else HOLogic.true_const) |
478 |
handle _ => at) |
|
479 |
| _ => at) |
|
480 |
| _ => at; |
|
481 |
||
482 |
*) |
|
483 |
||
484 |
fun evalc_atom at = case at of |
|
485 |
(Const (p,_) $ s $ t) => |
|
17485 | 486 |
( case AList.lookup (op =) operations p of |
15965
f422f8283491
Use of IntInf.int instead of int in most numeric simprocs; avoids
paulson
parents:
15859
diff
changeset
|
487 |
SOME f => ((if (f ((dest_numeral s),(dest_numeral t))) then HOLogic.true_const |
f422f8283491
Use of IntInf.int instead of int in most numeric simprocs; avoids
paulson
parents:
15859
diff
changeset
|
488 |
else HOLogic.false_const) |
15107 | 489 |
handle _ => at) |
490 |
| _ => at) |
|
491 |
|Const("Not",_)$(Const (p,_) $ s $ t) =>( |
|
17485 | 492 |
case AList.lookup (op =) operations p of |
15965
f422f8283491
Use of IntInf.int instead of int in most numeric simprocs; avoids
paulson
parents:
15859
diff
changeset
|
493 |
SOME f => ((if (f ((dest_numeral s),(dest_numeral t))) |
f422f8283491
Use of IntInf.int instead of int in most numeric simprocs; avoids
paulson
parents:
15859
diff
changeset
|
494 |
then HOLogic.false_const else HOLogic.true_const) |
15107 | 495 |
handle _ => at) |
496 |
| _ => at) |
|
497 |
| _ => at; |
|
498 |
||
499 |
(*Function onatoms apllys function f on the atomic formulas involved in a.*) |
|
13876 | 500 |
|
501 |
fun onatoms f a = if (is_arith_rel a) then f a else case a of |
|
502 |
||
503 |
(Const ("Not",_) $ p) => if is_arith_rel p then HOLogic.Not $ (f p) |
|
504 |
||
505 |
else HOLogic.Not $ (onatoms f p) |
|
506 |
|(Const ("op &",_) $ p $ q) => HOLogic.conj $ (onatoms f p) $ (onatoms f q) |
|
507 |
|(Const ("op |",_) $ p $ q) => HOLogic.disj $ (onatoms f p) $ (onatoms f q) |
|
508 |
|(Const ("op -->",_) $ p $ q) => HOLogic.imp $ (onatoms f p) $ (onatoms f q) |
|
509 |
|((Const ("op =", Type ("fun",[Type ("bool", []),_]))) $ p $ q) => (Const ("op =", [HOLogic.boolT, HOLogic.boolT] ---> HOLogic.boolT)) $ (onatoms f p) $ (onatoms f q) |
|
510 |
|(Const("All",_) $ Abs(x,T,p)) => Const("All", [HOLogic.intT --> |
|
511 |
HOLogic.boolT] ---> HOLogic.boolT)$ Abs (x ,T, (onatoms f p)) |
|
512 |
|(Const("Ex",_) $ Abs(x,T,p)) => Const("Ex", [HOLogic.intT --> HOLogic.boolT]---> HOLogic.boolT) $ Abs( x ,T, (onatoms f p)) |
|
513 |
|_ => a; |
|
514 |
||
515 |
val evalc = onatoms evalc_atom; |
|
516 |
||
517 |
(* ------------------------------------------------------------------------- *) |
|
518 |
(* Hence overall quantifier elimination. *) |
|
519 |
(* ------------------------------------------------------------------------- *) |
|
520 |
||
521 |
||
522 |
(*list_disj[conj] makes a disj[conj] of a given list. used with conjucts or disjuncts |
|
523 |
it liearises iterated conj[disj]unctions. *) |
|
524 |
||
525 |
fun disj_help p q = HOLogic.disj $ p $ q ; |
|
526 |
||
527 |
fun list_disj l = |
|
16837 | 528 |
if l = [] then HOLogic.false_const else Utils.end_itlist disj_help l; |
13876 | 529 |
|
530 |
fun conj_help p q = HOLogic.conj $ p $ q ; |
|
531 |
||
532 |
fun list_conj l = |
|
16837 | 533 |
if l = [] then HOLogic.true_const else Utils.end_itlist conj_help l; |
13876 | 534 |
|
535 |
(*Simplification of Formulas *) |
|
536 |
||
537 |
(*Function q_bnd_chk checks if a quantified Formula makes sens : Means if in |
|
538 |
the body of the existential quantifier there are bound variables to the |
|
539 |
existential quantifier.*) |
|
540 |
||
541 |
fun has_bound fm =let fun has_boundh fm i = case fm of |
|
542 |
Bound n => (i = n) |
|
543 |
|Abs (_,_,p) => has_boundh p (i+1) |
|
544 |
|t1 $ t2 => (has_boundh t1 i) orelse (has_boundh t2 i) |
|
545 |
|_ =>false |
|
546 |
||
547 |
in case fm of |
|
548 |
Bound _ => true |
|
549 |
|Abs (_,_,p) => has_boundh p 0 |
|
550 |
|t1 $ t2 => (has_bound t1 ) orelse (has_bound t2 ) |
|
551 |
|_ =>false |
|
552 |
end; |
|
553 |
||
554 |
(*has_sub_abs checks if in a given Formula there are subformulas which are quantifed |
|
555 |
too. Is no used no more.*) |
|
556 |
||
557 |
fun has_sub_abs fm = case fm of |
|
558 |
Abs (_,_,_) => true |
|
559 |
|t1 $ t2 => (has_bound t1 ) orelse (has_bound t2 ) |
|
560 |
|_ =>false ; |
|
561 |
||
562 |
(*update_bounds called with i=0 udates the numeration of bounded variables because the |
|
563 |
formula will not be quantified any more.*) |
|
564 |
||
565 |
fun update_bounds fm i = case fm of |
|
566 |
Bound n => if n >= i then Bound (n-1) else fm |
|
567 |
|Abs (x,T,p) => Abs(x,T,(update_bounds p (i+1))) |
|
568 |
|t1 $ t2 => (update_bounds t1 i) $ (update_bounds t2 i) |
|
569 |
|_ => fm ; |
|
570 |
||
571 |
(*psimpl : Simplification of propositions (general purpose)*) |
|
572 |
fun psimpl1 fm = case fm of |
|
573 |
Const("Not",_) $ Const ("False",_) => HOLogic.true_const |
|
574 |
| Const("Not",_) $ Const ("True",_) => HOLogic.false_const |
|
575 |
| Const("op &",_) $ Const ("False",_) $ q => HOLogic.false_const |
|
576 |
| Const("op &",_) $ p $ Const ("False",_) => HOLogic.false_const |
|
577 |
| Const("op &",_) $ Const ("True",_) $ q => q |
|
578 |
| Const("op &",_) $ p $ Const ("True",_) => p |
|
579 |
| Const("op |",_) $ Const ("False",_) $ q => q |
|
580 |
| Const("op |",_) $ p $ Const ("False",_) => p |
|
581 |
| Const("op |",_) $ Const ("True",_) $ q => HOLogic.true_const |
|
582 |
| Const("op |",_) $ p $ Const ("True",_) => HOLogic.true_const |
|
583 |
| Const("op -->",_) $ Const ("False",_) $ q => HOLogic.true_const |
|
584 |
| Const("op -->",_) $ Const ("True",_) $ q => q |
|
585 |
| Const("op -->",_) $ p $ Const ("True",_) => HOLogic.true_const |
|
586 |
| Const("op -->",_) $ p $ Const ("False",_) => HOLogic.Not $ p |
|
587 |
| Const("op =", Type ("fun",[Type ("bool", []),_])) $ Const ("True",_) $ q => q |
|
588 |
| Const("op =", Type ("fun",[Type ("bool", []),_])) $ p $ Const ("True",_) => p |
|
589 |
| Const("op =", Type ("fun",[Type ("bool", []),_])) $ Const ("False",_) $ q => HOLogic.Not $ q |
|
590 |
| Const("op =", Type ("fun",[Type ("bool", []),_])) $ p $ Const ("False",_) => HOLogic.Not $ p |
|
591 |
| _ => fm; |
|
592 |
||
593 |
fun psimpl fm = case fm of |
|
594 |
Const ("Not",_) $ p => psimpl1 (HOLogic.Not $ (psimpl p)) |
|
595 |
| Const("op &",_) $ p $ q => psimpl1 (HOLogic.mk_conj (psimpl p,psimpl q)) |
|
596 |
| Const("op |",_) $ p $ q => psimpl1 (HOLogic.mk_disj (psimpl p,psimpl q)) |
|
597 |
| Const("op -->",_) $ p $ q => psimpl1 (HOLogic.mk_imp(psimpl p,psimpl q)) |
|
15267 | 598 |
| Const("op =", Type ("fun",[Type ("bool", []),_])) $ p $ q => psimpl1 (HOLogic.mk_eq(psimpl p,psimpl q)) |
13876 | 599 |
| _ => fm; |
600 |
||
601 |
||
602 |
(*simpl : Simplification of Terms involving quantifiers too. |
|
603 |
This function is able to drop out some quantified expressions where there are no |
|
604 |
bound varaibles.*) |
|
605 |
||
606 |
fun simpl1 fm = |
|
607 |
case fm of |
|
608 |
Const("All",_) $Abs(x,_,p) => if (has_bound fm ) then fm |
|
609 |
else (update_bounds p 0) |
|
610 |
| Const("Ex",_) $ Abs (x,_,p) => if has_bound fm then fm |
|
611 |
else (update_bounds p 0) |
|
15267 | 612 |
| _ => psimpl fm; |
13876 | 613 |
|
614 |
fun simpl fm = case fm of |
|
615 |
Const ("Not",_) $ p => simpl1 (HOLogic.Not $(simpl p)) |
|
616 |
| Const ("op &",_) $ p $ q => simpl1 (HOLogic.mk_conj (simpl p ,simpl q)) |
|
617 |
| Const ("op |",_) $ p $ q => simpl1 (HOLogic.mk_disj (simpl p ,simpl q )) |
|
618 |
| Const ("op -->",_) $ p $ q => simpl1 (HOLogic.mk_imp(simpl p ,simpl q )) |
|
619 |
| Const("op =", Type ("fun",[Type ("bool", []),_]))$ p $ q => simpl1 |
|
620 |
(HOLogic.mk_eq(simpl p ,simpl q )) |
|
14920
a7525235e20f
An oracle is built in. The tactic will not generate any proofs any more, if the quick_and_dirty flag is set on.
chaieb
parents:
14877
diff
changeset
|
621 |
(* | Const ("All",Ta) $ Abs(Vn,VT,p) => simpl1(Const("All",Ta) $ |
13876 | 622 |
Abs(Vn,VT,simpl p )) |
623 |
| Const ("Ex",Ta) $ Abs(Vn,VT,p) => simpl1(Const("Ex",Ta) $ |
|
624 |
Abs(Vn,VT,simpl p )) |
|
14920
a7525235e20f
An oracle is built in. The tactic will not generate any proofs any more, if the quick_and_dirty flag is set on.
chaieb
parents:
14877
diff
changeset
|
625 |
*) |
13876 | 626 |
| _ => fm; |
627 |
||
628 |
(* ------------------------------------------------------------------------- *) |
|
629 |
||
630 |
(* Puts fm into NNF*) |
|
631 |
||
632 |
fun nnf fm = if (is_arith_rel fm) then fm |
|
633 |
else (case fm of |
|
634 |
( Const ("op &",_) $ p $ q) => HOLogic.conj $ (nnf p) $(nnf q) |
|
635 |
| (Const("op |",_) $ p $q) => HOLogic.disj $ (nnf p)$(nnf q) |
|
636 |
| (Const ("op -->",_) $ p $ q) => HOLogic.disj $ (nnf (HOLogic.Not $ p)) $ (nnf q) |
|
637 |
| ((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)))) |
|
638 |
| (Const ("Not",_)) $ ((Const ("Not",_)) $ p) => (nnf p) |
|
639 |
| (Const ("Not",_)) $ (( Const ("op &",_)) $ p $ q) =>HOLogic.disj $ (nnf(HOLogic.Not $ p)) $ (nnf(HOLogic.Not $q)) |
|
640 |
| (Const ("Not",_)) $ (( Const ("op |",_)) $ p $ q) =>HOLogic.conj $ (nnf(HOLogic.Not $ p)) $ (nnf(HOLogic.Not $ q)) |
|
641 |
| (Const ("Not",_)) $ (( Const ("op -->",_)) $ p $ q ) =>HOLogic.conj $ (nnf p) $(nnf(HOLogic.Not $ q)) |
|
642 |
| (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))) |
|
643 |
| _ => fm); |
|
644 |
||
645 |
||
646 |
(* Function remred to remove redundancy in a list while keeping the order of appearance of the |
|
647 |
elements. but VERY INEFFICIENT!! *) |
|
648 |
||
649 |
fun remred1 el [] = [] |
|
650 |
|remred1 el (h::t) = if el=h then (remred1 el t) else h::(remred1 el t); |
|
651 |
||
652 |
fun remred [] = [] |
|
653 |
|remred (x::l) = x::(remred1 x (remred l)); |
|
654 |
||
655 |
(*Makes sure that all free Variables are of the type integer but this function is only |
|
656 |
used temporarily, this job must be done by the parser later on.*) |
|
657 |
||
658 |
fun mk_uni_vars T (node $ rest) = (case node of |
|
659 |
Free (name,_) => Free (name,T) $ (mk_uni_vars T rest) |
|
660 |
|_=> (mk_uni_vars T node) $ (mk_uni_vars T rest ) ) |
|
661 |
|mk_uni_vars T (Free (v,_)) = Free (v,T) |
|
662 |
|mk_uni_vars T tm = tm; |
|
663 |
||
664 |
fun mk_uni_int T (Const ("0",T2)) = if T = T2 then (mk_numeral 0) else (Const ("0",T2)) |
|
665 |
|mk_uni_int T (Const ("1",T2)) = if T = T2 then (mk_numeral 1) else (Const ("1",T2)) |
|
666 |
|mk_uni_int T (node $ rest) = (mk_uni_int T node) $ (mk_uni_int T rest ) |
|
667 |
|mk_uni_int T (Abs(AV,AT,p)) = Abs(AV,AT,mk_uni_int T p) |
|
668 |
|mk_uni_int T tm = tm; |
|
669 |
||
670 |
||
16398 | 671 |
(* Minusinfinity Version*) |
672 |
fun myupto (m:IntInf.int) n = if m > n then [] else m::(myupto (m+1) n) |
|
673 |
||
13876 | 674 |
fun coopermi vars1 fm = |
675 |
case fm of |
|
15965
f422f8283491
Use of IntInf.int instead of int in most numeric simprocs; avoids
paulson
parents:
15859
diff
changeset
|
676 |
Const ("Ex",_) $ Abs(x0,T,p0) => |
f422f8283491
Use of IntInf.int instead of int in most numeric simprocs; avoids
paulson
parents:
15859
diff
changeset
|
677 |
let |
13876 | 678 |
val (xn,p1) = variant_abs (x0,T,p0) |
679 |
val x = Free (xn,T) |
|
680 |
val vars = (xn::vars1) |
|
681 |
val p = unitycoeff x (posineq (simpl p1)) |
|
682 |
val p_inf = simpl (minusinf x p) |
|
683 |
val bset = bset x p |
|
16398 | 684 |
val js = myupto 1 (divlcm x p) |
13876 | 685 |
fun p_element j b = linrep vars x (linear_add vars b (mk_numeral j)) p |
686 |
fun stage j = list_disj (linrep vars x (mk_numeral j) p_inf :: map (p_element j) bset) |
|
687 |
in (list_disj (map stage js)) |
|
688 |
end |
|
689 |
| _ => error "cooper: not an existential formula"; |
|
690 |
||
691 |
||
692 |
||
693 |
(* The plusinfinity version of cooper*) |
|
694 |
fun cooperpi vars1 fm = |
|
695 |
case fm of |
|
696 |
Const ("Ex",_) $ Abs(x0,T,p0) => let |
|
697 |
val (xn,p1) = variant_abs (x0,T,p0) |
|
698 |
val x = Free (xn,T) |
|
699 |
val vars = (xn::vars1) |
|
700 |
val p = unitycoeff x (posineq (simpl p1)) |
|
701 |
val p_inf = simpl (plusinf x p) |
|
702 |
val aset = aset x p |
|
16398 | 703 |
val js = myupto 1 (divlcm x p) |
13876 | 704 |
fun p_element j a = linrep vars x (linear_sub vars a (mk_numeral j)) p |
705 |
fun stage j = list_disj (linrep vars x (mk_numeral j) p_inf :: map (p_element j) aset) |
|
706 |
in (list_disj (map stage js)) |
|
707 |
end |
|
708 |
| _ => error "cooper: not an existential formula"; |
|
709 |
||
710 |
||
15107 | 711 |
(* Try to find a withness for the formula *) |
712 |
||
713 |
fun inf_w mi d vars x p = |
|
714 |
let val f = if mi then minusinf else plusinf in |
|
715 |
case (simpl (minusinf x p)) of |
|
15531 | 716 |
Const("True",_) => (SOME (mk_numeral 1), HOLogic.true_const) |
717 |
|Const("False",_) => (NONE,HOLogic.false_const) |
|
15107 | 718 |
|F => |
719 |
let |
|
720 |
fun h n = |
|
721 |
case ((simpl o evalc) (linrep vars x (mk_numeral n) F)) of |
|
15531 | 722 |
Const("True",_) => (SOME (mk_numeral n),HOLogic.true_const) |
723 |
|F' => if n=1 then (NONE,F') |
|
15107 | 724 |
else let val (rw,rf) = h (n-1) in |
725 |
(rw,HOLogic.mk_disj(F',rf)) |
|
726 |
end |
|
727 |
||
728 |
in (h d) |
|
729 |
end |
|
730 |
end; |
|
731 |
||
732 |
fun set_w d b st vars x p = let |
|
733 |
fun h ns = case ns of |
|
15531 | 734 |
[] => (NONE,HOLogic.false_const) |
15107 | 735 |
|n::nl => ( case ((simpl o evalc) (linrep vars x n p)) of |
15531 | 736 |
Const("True",_) => (SOME n,HOLogic.true_const) |
15107 | 737 |
|F' => let val (rw,rf) = h nl |
738 |
in (rw,HOLogic.mk_disj(F',rf)) |
|
739 |
end) |
|
740 |
val f = if b then linear_add else linear_sub |
|
16398 | 741 |
val p_elements = foldr (fn (i,l) => l union (map (fn e => f [] e (mk_numeral i)) st)) [] (myupto 1 d) |
15107 | 742 |
in h p_elements |
743 |
end; |
|
744 |
||
745 |
fun withness d b st vars x p = case (inf_w b d vars x p) of |
|
15531 | 746 |
(SOME n,_) => (SOME n,HOLogic.true_const) |
747 |
|(NONE,Pinf) => (case (set_w d b st vars x p) of |
|
748 |
(SOME n,_) => (SOME n,HOLogic.true_const) |
|
749 |
|(_,Pst) => (NONE,HOLogic.mk_disj(Pinf,Pst))); |
|
15107 | 750 |
|
751 |
||
752 |
||
13876 | 753 |
|
754 |
(*Cooper main procedure*) |
|
15267 | 755 |
|
756 |
exception STAGE_TRUE; |
|
757 |
||
13876 | 758 |
|
759 |
fun cooper vars1 fm = |
|
760 |
case fm of |
|
761 |
Const ("Ex",_) $ Abs(x0,T,p0) => let |
|
762 |
val (xn,p1) = variant_abs (x0,T,p0) |
|
763 |
val x = Free (xn,T) |
|
764 |
val vars = (xn::vars1) |
|
14920
a7525235e20f
An oracle is built in. The tactic will not generate any proofs any more, if the quick_and_dirty flag is set on.
chaieb
parents:
14877
diff
changeset
|
765 |
(* val p = unitycoeff x (posineq (simpl p1)) *) |
a7525235e20f
An oracle is built in. The tactic will not generate any proofs any more, if the quick_and_dirty flag is set on.
chaieb
parents:
14877
diff
changeset
|
766 |
val p = unitycoeff x p1 |
13876 | 767 |
val ast = aset x p |
768 |
val bst = bset x p |
|
16398 | 769 |
val js = myupto 1 (divlcm x p) |
13876 | 770 |
val (p_inf,f,S ) = |
15267 | 771 |
if (length bst) <= (length ast) |
772 |
then (simpl (minusinf x p),linear_add,bst) |
|
773 |
else (simpl (plusinf x p), linear_sub,ast) |
|
13876 | 774 |
fun p_element j a = linrep vars x (f vars a (mk_numeral j)) p |
775 |
fun stage j = list_disj (linrep vars x (mk_numeral j) p_inf :: map (p_element j) S) |
|
15267 | 776 |
fun stageh n = ((if n = 0 then [] |
777 |
else |
|
778 |
let |
|
779 |
val nth_stage = simpl (evalc (stage n)) |
|
780 |
in |
|
781 |
if (nth_stage = HOLogic.true_const) |
|
782 |
then raise STAGE_TRUE |
|
783 |
else if (nth_stage = HOLogic.false_const) then stageh (n-1) |
|
784 |
else nth_stage::(stageh (n-1)) |
|
785 |
end ) |
|
786 |
handle STAGE_TRUE => [HOLogic.true_const]) |
|
787 |
val slist = stageh (divlcm x p) |
|
788 |
in (list_disj slist) |
|
13876 | 789 |
end |
790 |
| _ => error "cooper: not an existential formula"; |
|
791 |
||
792 |
||
15107 | 793 |
(* A Version of cooper that returns a withness *) |
794 |
fun cooper_w vars1 fm = |
|
795 |
case fm of |
|
796 |
Const ("Ex",_) $ Abs(x0,T,p0) => let |
|
797 |
val (xn,p1) = variant_abs (x0,T,p0) |
|
798 |
val x = Free (xn,T) |
|
799 |
val vars = (xn::vars1) |
|
800 |
(* val p = unitycoeff x (posineq (simpl p1)) *) |
|
801 |
val p = unitycoeff x p1 |
|
802 |
val ast = aset x p |
|
803 |
val bst = bset x p |
|
804 |
val d = divlcm x p |
|
805 |
val (p_inf,S ) = |
|
806 |
if (length bst) <= (length ast) |
|
807 |
then (true,bst) |
|
808 |
else (false,ast) |
|
809 |
in withness d p_inf S vars x p |
|
810 |
(* fun p_element j a = linrep vars x (f vars a (mk_numeral j)) p |
|
811 |
fun stage j = list_disj (linrep vars x (mk_numeral j) p_inf :: map (p_element j) S) |
|
812 |
in (list_disj (map stage js)) |
|
813 |
*) |
|
814 |
end |
|
815 |
| _ => error "cooper: not an existential formula"; |
|
13876 | 816 |
|
817 |
||
818 |
(* ------------------------------------------------------------------------- *) |
|
819 |
(* Free variables in terms and formulas. *) |
|
820 |
(* ------------------------------------------------------------------------- *) |
|
821 |
||
822 |
fun fvt tml = case tml of |
|
823 |
[] => [] |
|
824 |
| Free(x,_)::r => x::(fvt r) |
|
825 |
||
826 |
fun fv fm = fvt (term_frees fm); |
|
827 |
||
828 |
||
829 |
(* ========================================================================= *) |
|
830 |
(* Quantifier elimination. *) |
|
831 |
(* ========================================================================= *) |
|
832 |
(*conj[/disj]uncts lists iterated conj[disj]unctions*) |
|
833 |
||
834 |
fun disjuncts fm = case fm of |
|
835 |
Const ("op |",_) $ p $ q => (disjuncts p) @ (disjuncts q) |
|
836 |
| _ => [fm]; |
|
837 |
||
838 |
fun conjuncts fm = case fm of |
|
839 |
Const ("op &",_) $p $ q => (conjuncts p) @ (conjuncts q) |
|
840 |
| _ => [fm]; |
|
841 |
||
842 |
||
843 |
||
844 |
(* ------------------------------------------------------------------------- *) |
|
845 |
(* Lift procedure given literal modifier, formula normalizer & basic quelim. *) |
|
14920
a7525235e20f
An oracle is built in. The tactic will not generate any proofs any more, if the quick_and_dirty flag is set on.
chaieb
parents:
14877
diff
changeset
|
846 |
(* ------------------------------------------------------------------------- *) |
15267 | 847 |
|
14920
a7525235e20f
An oracle is built in. The tactic will not generate any proofs any more, if the quick_and_dirty flag is set on.
chaieb
parents:
14877
diff
changeset
|
848 |
fun lift_qelim afn nfn qfn isat = |
a7525235e20f
An oracle is built in. The tactic will not generate any proofs any more, if the quick_and_dirty flag is set on.
chaieb
parents:
14877
diff
changeset
|
849 |
let |
a7525235e20f
An oracle is built in. The tactic will not generate any proofs any more, if the quick_and_dirty flag is set on.
chaieb
parents:
14877
diff
changeset
|
850 |
fun qelift vars fm = if (isat fm) then afn vars fm |
a7525235e20f
An oracle is built in. The tactic will not generate any proofs any more, if the quick_and_dirty flag is set on.
chaieb
parents:
14877
diff
changeset
|
851 |
else |
a7525235e20f
An oracle is built in. The tactic will not generate any proofs any more, if the quick_and_dirty flag is set on.
chaieb
parents:
14877
diff
changeset
|
852 |
case fm of |
a7525235e20f
An oracle is built in. The tactic will not generate any proofs any more, if the quick_and_dirty flag is set on.
chaieb
parents:
14877
diff
changeset
|
853 |
Const ("Not",_) $ p => HOLogic.Not $ (qelift vars p) |
a7525235e20f
An oracle is built in. The tactic will not generate any proofs any more, if the quick_and_dirty flag is set on.
chaieb
parents:
14877
diff
changeset
|
854 |
| Const ("op &",_) $ p $q => HOLogic.conj $ (qelift vars p) $ (qelift vars q) |
a7525235e20f
An oracle is built in. The tactic will not generate any proofs any more, if the quick_and_dirty flag is set on.
chaieb
parents:
14877
diff
changeset
|
855 |
| Const ("op |",_) $ p $ q => HOLogic.disj $ (qelift vars p) $ (qelift vars q) |
a7525235e20f
An oracle is built in. The tactic will not generate any proofs any more, if the quick_and_dirty flag is set on.
chaieb
parents:
14877
diff
changeset
|
856 |
| Const ("op -->",_) $ p $ q => HOLogic.imp $ (qelift vars p) $ (qelift vars q) |
a7525235e20f
An oracle is built in. The tactic will not generate any proofs any more, if the quick_and_dirty flag is set on.
chaieb
parents:
14877
diff
changeset
|
857 |
| Const ("op =",Type ("fun",[Type ("bool", []),_])) $ p $ q => HOLogic.mk_eq ((qelift vars p),(qelift vars q)) |
a7525235e20f
An oracle is built in. The tactic will not generate any proofs any more, if the quick_and_dirty flag is set on.
chaieb
parents:
14877
diff
changeset
|
858 |
| Const ("All",QT) $ Abs(x,T,p) => HOLogic.Not $(qelift vars (Const ("Ex",QT) $ Abs(x,T,(HOLogic.Not $ p)))) |
a7525235e20f
An oracle is built in. The tactic will not generate any proofs any more, if the quick_and_dirty flag is set on.
chaieb
parents:
14877
diff
changeset
|
859 |
| (e as Const ("Ex",_)) $ Abs (x,T,p) => qfn vars (e$Abs (x,T,(nfn(qelift (x::vars) p)))) |
a7525235e20f
An oracle is built in. The tactic will not generate any proofs any more, if the quick_and_dirty flag is set on.
chaieb
parents:
14877
diff
changeset
|
860 |
| _ => fm |
a7525235e20f
An oracle is built in. The tactic will not generate any proofs any more, if the quick_and_dirty flag is set on.
chaieb
parents:
14877
diff
changeset
|
861 |
|
a7525235e20f
An oracle is built in. The tactic will not generate any proofs any more, if the quick_and_dirty flag is set on.
chaieb
parents:
14877
diff
changeset
|
862 |
in (fn fm => qelift (fv fm) fm) |
a7525235e20f
An oracle is built in. The tactic will not generate any proofs any more, if the quick_and_dirty flag is set on.
chaieb
parents:
14877
diff
changeset
|
863 |
end; |
15267 | 864 |
|
14920
a7525235e20f
An oracle is built in. The tactic will not generate any proofs any more, if the quick_and_dirty flag is set on.
chaieb
parents:
14877
diff
changeset
|
865 |
|
15267 | 866 |
(* |
13876 | 867 |
fun lift_qelim afn nfn qfn isat = |
868 |
let fun qelim x vars p = |
|
869 |
let val cjs = conjuncts p |
|
15570 | 870 |
val (ycjs,ncjs) = List.partition (has_bound) cjs in |
13876 | 871 |
(if ycjs = [] then p else |
872 |
let val q = (qfn vars ((HOLogic.exists_const HOLogic.intT |
|
873 |
) $ Abs(x,HOLogic.intT,(list_conj ycjs)))) in |
|
16837 | 874 |
(fold_rev conj_help ncjs q) |
13876 | 875 |
end) |
876 |
end |
|
877 |
||
878 |
fun qelift vars fm = if (isat fm) then afn vars fm |
|
879 |
else |
|
880 |
case fm of |
|
881 |
Const ("Not",_) $ p => HOLogic.Not $ (qelift vars p) |
|
882 |
| Const ("op &",_) $ p $q => HOLogic.conj $ (qelift vars p) $ (qelift vars q) |
|
883 |
| Const ("op |",_) $ p $ q => HOLogic.disj $ (qelift vars p) $ (qelift vars q) |
|
884 |
| Const ("op -->",_) $ p $ q => HOLogic.imp $ (qelift vars p) $ (qelift vars q) |
|
885 |
| Const ("op =",Type ("fun",[Type ("bool", []),_])) $ p $ q => HOLogic.mk_eq ((qelift vars p),(qelift vars q)) |
|
886 |
| Const ("All",QT) $ Abs(x,T,p) => HOLogic.Not $(qelift vars (Const ("Ex",QT) $ Abs(x,T,(HOLogic.Not $ p)))) |
|
887 |
| Const ("Ex",_) $ Abs (x,T,p) => let val djs = disjuncts(nfn(qelift (x::vars) p)) in |
|
888 |
list_disj(map (qelim x vars) djs) end |
|
889 |
| _ => fm |
|
890 |
||
891 |
in (fn fm => simpl(qelift (fv fm) fm)) |
|
892 |
end; |
|
15267 | 893 |
*) |
13876 | 894 |
|
895 |
(* ------------------------------------------------------------------------- *) |
|
896 |
(* Cleverer (proposisional) NNF with conditional and literal modification. *) |
|
897 |
(* ------------------------------------------------------------------------- *) |
|
898 |
||
899 |
(*Function Negate used by cnnf, negates a formula p*) |
|
900 |
||
901 |
fun negate (Const ("Not",_) $ p) = p |
|
902 |
|negate p = (HOLogic.Not $ p); |
|
903 |
||
904 |
fun cnnf lfn = |
|
905 |
let fun cnnfh fm = case fm of |
|
906 |
(Const ("op &",_) $ p $ q) => HOLogic.mk_conj(cnnfh p,cnnfh q) |
|
907 |
| (Const ("op |",_) $ p $ q) => HOLogic.mk_disj(cnnfh p,cnnfh q) |
|
908 |
| (Const ("op -->",_) $ p $q) => HOLogic.mk_disj(cnnfh(HOLogic.Not $ p),cnnfh q) |
|
909 |
| (Const ("op =",Type ("fun",[Type ("bool", []),_])) $ p $ q) => HOLogic.mk_disj( |
|
910 |
HOLogic.mk_conj(cnnfh p,cnnfh q), |
|
911 |
HOLogic.mk_conj(cnnfh(HOLogic.Not $ p),cnnfh(HOLogic.Not $q))) |
|
912 |
||
913 |
| (Const ("Not",_) $ (Const("Not",_) $ p)) => cnnfh p |
|
914 |
| (Const ("Not",_) $ (Const ("op &",_) $ p $ q)) => HOLogic.mk_disj(cnnfh(HOLogic.Not $ p),cnnfh(HOLogic.Not $ q)) |
|
915 |
| (Const ("Not",_) $(Const ("op |",_) $ (Const ("op &",_) $ p $ q) $ |
|
916 |
(Const ("op &",_) $ p1 $ r))) => if p1 = negate p then |
|
917 |
HOLogic.mk_disj( |
|
918 |
cnnfh (HOLogic.mk_conj(p,cnnfh(HOLogic.Not $ q))), |
|
919 |
cnnfh (HOLogic.mk_conj(p1,cnnfh(HOLogic.Not $ r)))) |
|
920 |
else HOLogic.mk_conj( |
|
921 |
cnnfh (HOLogic.mk_disj(cnnfh (HOLogic.Not $ p),cnnfh(HOLogic.Not $ q))), |
|
922 |
cnnfh (HOLogic.mk_disj(cnnfh (HOLogic.Not $ p1),cnnfh(HOLogic.Not $ r))) |
|
923 |
) |
|
924 |
| (Const ("Not",_) $ (Const ("op |",_) $ p $ q)) => HOLogic.mk_conj(cnnfh(HOLogic.Not $ p),cnnfh(HOLogic.Not $ q)) |
|
925 |
| (Const ("Not",_) $ (Const ("op -->",_) $ p $q)) => HOLogic.mk_conj(cnnfh p,cnnfh(HOLogic.Not $ q)) |
|
926 |
| (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)) |
|
927 |
| _ => lfn fm |
|
14920
a7525235e20f
An oracle is built in. The tactic will not generate any proofs any more, if the quick_and_dirty flag is set on.
chaieb
parents:
14877
diff
changeset
|
928 |
in cnnfh |
a7525235e20f
An oracle is built in. The tactic will not generate any proofs any more, if the quick_and_dirty flag is set on.
chaieb
parents:
14877
diff
changeset
|
929 |
end; |
13876 | 930 |
|
931 |
(*End- function the quantifierelimination an decion procedure of presburger formulas.*) |
|
14920
a7525235e20f
An oracle is built in. The tactic will not generate any proofs any more, if the quick_and_dirty flag is set on.
chaieb
parents:
14877
diff
changeset
|
932 |
|
a7525235e20f
An oracle is built in. The tactic will not generate any proofs any more, if the quick_and_dirty flag is set on.
chaieb
parents:
14877
diff
changeset
|
933 |
(* |
13876 | 934 |
val integer_qelim = simpl o evalc o (lift_qelim linform (simpl o (cnnf posineq o evalc)) cooper is_arith_rel) ; |
14920
a7525235e20f
An oracle is built in. The tactic will not generate any proofs any more, if the quick_and_dirty flag is set on.
chaieb
parents:
14877
diff
changeset
|
935 |
*) |
15267 | 936 |
|
937 |
||
14920
a7525235e20f
An oracle is built in. The tactic will not generate any proofs any more, if the quick_and_dirty flag is set on.
chaieb
parents:
14877
diff
changeset
|
938 |
val integer_qelim = simpl o evalc o (lift_qelim linform (cnnf posineq o evalc) cooper is_arith_rel) ; |
a7525235e20f
An oracle is built in. The tactic will not generate any proofs any more, if the quick_and_dirty flag is set on.
chaieb
parents:
14877
diff
changeset
|
939 |
|
13876 | 940 |
end; |