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(******************************************************************************)
(* FILE : clausal_form.ml *)
(* DESCRIPTION : Functions for putting a formula into clausal form. *)
(* *)
(* READS FILES : <none> *)
(* WRITES FILES : <none> *)
(* *)
(* AUTHOR : R.J.Boulton *)
(* DATE : 13th May 1991 *)
(* *)
(* LAST MODIFIED : R.J.Boulton *)
(* DATE : 12th October 1992 *)
(* *)
(* LAST MODIFIED : P. Papapanagiotou (University of Edinburgh) *)
(* DATE : 2008 *)
(******************************************************************************)
let IMP_DISJ_THM = TAUT `!t1 t2. t1 ==> t2 <=> ~t1 \/ t2`;;
let RIGHT_OR_OVER_AND = TAUT `!t1 t2 t3. t2 /\ t3 \/ t1 <=> (t2 \/ t1) /\ (t3 \/ t1)`;;
let LEFT_OR_OVER_AND = TAUT `!t1 t2 t3. t1 \/ t2 /\ t3 <=> (t1 \/ t2) /\ (t1 \/ t3)`;;
(*============================================================================*)
(* Theorems for normalizing Boolean terms *)
(*============================================================================*)
(*----------------------------------------------------------------------------*)
(* EQ_EXPAND = |- (x = y) = ((~x \/ y) /\ (~y \/ x)) *)
(*----------------------------------------------------------------------------*)
let EQ_EXPAND =
prove
(`(x = y) = ((~x \/ y) /\ (~y \/ x))`,
BOOL_CASES_TAC `x:bool` THEN
BOOL_CASES_TAC `y:bool` THEN
REWRITE_TAC []);;
(*----------------------------------------------------------------------------*)
(* IMP_EXPAND = |- (x ==> y) = (~x \/ y) *)
(*----------------------------------------------------------------------------*)
let IMP_EXPAND = SPEC `y:bool` (SPEC `x:bool` IMP_DISJ_THM);;
(*----------------------------------------------------------------------------*)
(* COND_EXPAND = |- (x => y | z) = ((~x \/ y) /\ (x \/ z)) *)
(*----------------------------------------------------------------------------*)
let COND_EXPAND =
prove
(`(if x then y else z) = ((~x \/ y) /\ (x \/ z))`,
BOOL_CASES_TAC `x:bool` THEN
BOOL_CASES_TAC `y:bool` THEN
BOOL_CASES_TAC `z:bool` THEN
REWRITE_TAC []);;
(*----------------------------------------------------------------------------*)
(* NOT_NOT_NORM = |- ~~x = x *)
(*----------------------------------------------------------------------------*)
let NOT_NOT_NORM = SPEC `x:bool` (CONJUNCT1 NOT_CLAUSES);;
(*----------------------------------------------------------------------------*)
(* NOT_CONJ_NORM = |- ~(x /\ y) = (~x \/ ~y) *)
(*----------------------------------------------------------------------------*)
let NOT_CONJ_NORM = CONJUNCT1 (SPEC `y:bool` (SPEC `x:bool` DE_MORGAN_THM));;
(*----------------------------------------------------------------------------*)
(* NOT_DISJ_NORM = |- ~(x \/ y) = (~x /\ ~y) *)
(*----------------------------------------------------------------------------*)
let NOT_DISJ_NORM = CONJUNCT2 (SPEC `y:bool` (SPEC `x:bool` DE_MORGAN_THM));;
(*----------------------------------------------------------------------------*)
(* LEFT_DIST_NORM = |- x \/ (y /\ z) = (x \/ y) /\ (x \/ z) *)
(*----------------------------------------------------------------------------*)
let LEFT_DIST_NORM =
SPEC `z:bool` (SPEC `y:bool` (SPEC `x:bool` LEFT_OR_OVER_AND));;
(*----------------------------------------------------------------------------*)
(* RIGHT_DIST_NORM = |- (x /\ y) \/ z = (x \/ z) /\ (y \/ z) *)
(*----------------------------------------------------------------------------*)
let RIGHT_DIST_NORM =
SPEC `y:bool` (SPEC `x:bool` (SPEC `z:bool` RIGHT_OR_OVER_AND));;
(*----------------------------------------------------------------------------*)
(* CONJ_ASSOC_NORM = |- (x /\ y) /\ z = x /\ (y /\ z) *)
(*----------------------------------------------------------------------------*)
let CONJ_ASSOC_NORM =
SYM (SPEC `z:bool` (SPEC `y:bool` (SPEC `x:bool` CONJ_ASSOC)));;
(*----------------------------------------------------------------------------*)
(* DISJ_ASSOC_NORM = |- (x \/ y) \/ z = x \/ (y \/ z) *)
(*----------------------------------------------------------------------------*)
let DISJ_ASSOC_NORM =
SYM (SPEC `z:bool` (SPEC `y:bool` (SPEC `x:bool` DISJ_ASSOC)));;
(*============================================================================*)
(* Conversions for rewriting Boolean terms *)
(*============================================================================*)
let COND_EXPAND_CONV = REWR_CONV COND_EXPAND;;
let CONJ_ASSOC_NORM_CONV = REWR_CONV CONJ_ASSOC_NORM;;
let DISJ_ASSOC_NORM_CONV = REWR_CONV DISJ_ASSOC_NORM;;
let EQ_EXPAND_CONV = REWR_CONV EQ_EXPAND;;
let IMP_EXPAND_CONV = REWR_CONV IMP_EXPAND;;
let LEFT_DIST_NORM_CONV = REWR_CONV LEFT_DIST_NORM;;
let NOT_CONJ_NORM_CONV = REWR_CONV NOT_CONJ_NORM;;
let NOT_DISJ_NORM_CONV = REWR_CONV NOT_DISJ_NORM;;
let NOT_NOT_NORM_CONV = REWR_CONV NOT_NOT_NORM;;
let RIGHT_DIST_NORM_CONV = REWR_CONV RIGHT_DIST_NORM;;
(*----------------------------------------------------------------------------*)
(* NOT_CONV : conv *)
(* *)
(* |- !t. ~~t = t *)
(* |- ~T = F *)
(* |- ~F = T *)
(*----------------------------------------------------------------------------*)
let NOT_CONV =
try (
let [th1;th2;th3] = CONJUNCTS NOT_CLAUSES
in fun tm ->
(let arg = dest_neg tm
in if (is_T arg) then th2
else if (is_F arg) then th3
else SPEC (dest_neg arg) th1
)
) with Failure _ -> failwith "NOT_CONV";;
(*----------------------------------------------------------------------------*)
(* AND_CONV : conv *)
(* *)
(* |- T /\ t = t *)
(* |- t /\ T = t *)
(* |- F /\ t = F *)
(* |- t /\ F = F *)
(* |- t /\ t = t *)
(*----------------------------------------------------------------------------*)
let AND_CONV =
try (
let [th1;th2;th3;th4;th5] = map GEN_ALL (CONJUNCTS (SPEC_ALL AND_CLAUSES))
in fun tm ->
(let (arg1,arg2) = dest_conj tm
in if (is_T arg1) then SPEC arg2 th1
else if (is_T arg2) then SPEC arg1 th2
else if (is_F arg1) then SPEC arg2 th3
else if (is_F arg2) then SPEC arg1 th4
else if (arg1 = arg2) then SPEC arg1 th5
else failwith ""
)
) with Failure _ -> failwith "AND_CONV" ;;
(*----------------------------------------------------------------------------*)
(* OR_CONV : conv *)
(* *)
(* |- T \/ t = T *)
(* |- t \/ T = T *)
(* |- F \/ t = t *)
(* |- t \/ F = t *)
(* |- t \/ t = t *)
(*----------------------------------------------------------------------------*)
let OR_CONV =
try (
let [th1;th2;th3;th4;th5] = map GEN_ALL (CONJUNCTS (SPEC_ALL OR_CLAUSES))
in fun tm ->
(let (arg1,arg2) = dest_disj tm
in if (is_T arg1) then SPEC arg2 th1
else if (is_T arg2) then SPEC arg1 th2
else if (is_F arg1) then SPEC arg2 th3
else if (is_F arg2) then SPEC arg1 th4
else if (arg1 = arg2) then SPEC arg1 th5
else failwith ""
)
) with Failure _ -> failwith "OR_CONV";;
(*============================================================================*)
(* Conversions for obtaining `clausal' form *)
(*============================================================================*)
(*----------------------------------------------------------------------------*)
(* EQ_IMP_COND_ELIM_CONV : (term -> bool) -> conv *)
(* *)
(* Eliminates Boolean equalities, Boolean conditionals, and implications from *)
(* terms consisting of =,==>,COND,/\,\/,~ and atoms. The atoms are specified *)
(* by the predicate that the conversion takes as its first argument. *)
(*----------------------------------------------------------------------------*)
let rec EQ_IMP_COND_ELIM_CONV is_atom tm =
try
(if (is_atom tm) then ALL_CONV tm
else if (is_neg tm) then (RAND_CONV (EQ_IMP_COND_ELIM_CONV is_atom)) tm
else if (is_eq tm) then
((RATOR_CONV (RAND_CONV (EQ_IMP_COND_ELIM_CONV is_atom))) THENC
(RAND_CONV (EQ_IMP_COND_ELIM_CONV is_atom)) THENC
EQ_EXPAND_CONV) tm
else if (is_imp tm) then
((RATOR_CONV (RAND_CONV (EQ_IMP_COND_ELIM_CONV is_atom))) THENC
(RAND_CONV (EQ_IMP_COND_ELIM_CONV is_atom)) THENC
IMP_EXPAND_CONV) tm
else if (is_cond tm) then
((RATOR_CONV
(RATOR_CONV (RAND_CONV (EQ_IMP_COND_ELIM_CONV is_atom)))) THENC
(RATOR_CONV (RAND_CONV (EQ_IMP_COND_ELIM_CONV is_atom))) THENC
(RAND_CONV (EQ_IMP_COND_ELIM_CONV is_atom)) THENC
COND_EXPAND_CONV) tm
else ((RATOR_CONV (RAND_CONV (EQ_IMP_COND_ELIM_CONV is_atom))) THENC
(RAND_CONV (EQ_IMP_COND_ELIM_CONV is_atom))) tm
) with Failure _ -> failwith "EQ_IMP_COND_ELIM_CONV";;
(*----------------------------------------------------------------------------*)
(* MOVE_NOT_DOWN_CONV : (term -> bool) -> conv -> conv *)
(* *)
(* Moves negations down through a term consisting of /\,\/,~ and atoms. The *)
(* atoms are specified by a predicate (first argument). When a negation has *)
(* reached an atom, the conversion `conv' (second argument) is applied to the *)
(* negation of the atom. `conv' is also applied to any non-negated atoms *)
(* encountered. T and F are eliminated. *)
(*----------------------------------------------------------------------------*)
let rec MOVE_NOT_DOWN_CONV is_atom conv tm =
try
(if (is_atom tm) then (conv tm)
else if (is_neg tm)
then ((let tm' = rand tm
in if (is_atom tm') then ((conv THENC (TRY_CONV NOT_CONV)) tm)
else if (is_neg tm') then (NOT_NOT_NORM_CONV THENC
(MOVE_NOT_DOWN_CONV is_atom conv)) tm
else if (is_conj tm') then
(NOT_CONJ_NORM_CONV THENC
(RATOR_CONV (RAND_CONV (MOVE_NOT_DOWN_CONV is_atom conv)))
THENC
(RAND_CONV (MOVE_NOT_DOWN_CONV is_atom conv)) THENC
(TRY_CONV AND_CONV)) tm
else if (is_disj tm') then
(NOT_DISJ_NORM_CONV THENC
(RATOR_CONV (RAND_CONV (MOVE_NOT_DOWN_CONV is_atom conv)))
THENC
(RAND_CONV (MOVE_NOT_DOWN_CONV is_atom conv)) THENC
(TRY_CONV OR_CONV)) tm
else failwith ""))
else if (is_conj tm) then
((RATOR_CONV (RAND_CONV (MOVE_NOT_DOWN_CONV is_atom conv))) THENC
(RAND_CONV (MOVE_NOT_DOWN_CONV is_atom conv)) THENC
(TRY_CONV AND_CONV)) tm
else if (is_disj tm) then
((RATOR_CONV (RAND_CONV (MOVE_NOT_DOWN_CONV is_atom conv))) THENC
(RAND_CONV (MOVE_NOT_DOWN_CONV is_atom conv)) THENC
(TRY_CONV OR_CONV)) tm
else failwith ""
) with Failure _ -> failwith "MOVE_NOT_DOWN_CONV";;
(*----------------------------------------------------------------------------*)
(* CONJ_LINEAR_CONV : conv *)
(* *)
(* Linearizes conjuncts using the following conversion applied recursively: *)
(* *)
(* "(x /\ y) /\ z" *)
(* ================================ *)
(* |- (x /\ y) /\ z = x /\ (y /\ z) *)
(*----------------------------------------------------------------------------*)
let rec CONJ_LINEAR_CONV tm =
try
(if ((is_conj tm) && (is_conj (rand (rator tm))))
then (CONJ_ASSOC_NORM_CONV THENC
(RAND_CONV (TRY_CONV CONJ_LINEAR_CONV)) THENC
(TRY_CONV CONJ_LINEAR_CONV)) tm
else failwith ""
) with Failure _ -> failwith "CONJ_LINEAR_CONV";;
(*----------------------------------------------------------------------------*)
(* CONJ_NORM_FORM_CONV : conv *)
(* *)
(* Puts a term involving /\ and \/ into conjunctive normal form. Anything *)
(* other than /\ and \/ is taken to be an atom and is not processed. *)
(* *)
(* The conjunction returned is linear, i.e. the conjunctions are associated *)
(* to the right. Each conjunct is a linear disjunction. *)
(*----------------------------------------------------------------------------*)
let rec CONJ_NORM_FORM_CONV tm =
try
(if (is_disj tm) then
(if (is_conj (rand (rator tm))) then
((RATOR_CONV
(RAND_CONV ((RATOR_CONV (RAND_CONV CONJ_NORM_FORM_CONV)) THENC
(RAND_CONV CONJ_NORM_FORM_CONV)))) THENC
(RAND_CONV CONJ_NORM_FORM_CONV) THENC
RIGHT_DIST_NORM_CONV THENC
(RATOR_CONV (RAND_CONV CONJ_NORM_FORM_CONV)) THENC
(RAND_CONV CONJ_NORM_FORM_CONV) THENC
(TRY_CONV CONJ_LINEAR_CONV)) tm
else if (is_conj (rand tm)) then
((RATOR_CONV (RAND_CONV CONJ_NORM_FORM_CONV)) THENC
(RAND_CONV ((RATOR_CONV (RAND_CONV CONJ_NORM_FORM_CONV)) THENC
(RAND_CONV CONJ_NORM_FORM_CONV))) THENC
LEFT_DIST_NORM_CONV THENC
(RATOR_CONV (RAND_CONV CONJ_NORM_FORM_CONV)) THENC
(RAND_CONV CONJ_NORM_FORM_CONV) THENC
(TRY_CONV CONJ_LINEAR_CONV)) tm
else if (is_disj (rand (rator tm))) then
(DISJ_ASSOC_NORM_CONV THENC CONJ_NORM_FORM_CONV) tm
else (let th = RAND_CONV CONJ_NORM_FORM_CONV tm
in let tm' = rhs (concl th)
in if (is_conj (rand tm'))
then (TRANS th (CONJ_NORM_FORM_CONV tm'))
else th))
else if (is_conj tm) then
((RATOR_CONV (RAND_CONV CONJ_NORM_FORM_CONV)) THENC
(RAND_CONV CONJ_NORM_FORM_CONV) THENC
(TRY_CONV CONJ_LINEAR_CONV)) tm
else ALL_CONV tm
) with Failure _ -> failwith "CONJ_NORM_FORM_CONV";;
(*----------------------------------------------------------------------------*)
(* has_boolean_args_and_result : term -> bool *)
(* *)
(* Yields true if and only if the term is of type ":bool", and if it is a *)
(* function application, all the arguments are of type ":bool". *)
(*----------------------------------------------------------------------------*)
let has_boolean_args_and_result tm =
try
(let args = snd (strip_comb tm)
in let types = (type_of tm)::(map type_of args)
in (subtract (setify types) [`:bool`]) = [] )
with Failure _ -> (type_of tm = `:bool`);;
(*----------------------------------------------------------------------------*)
(* CLAUSAL_FORM_CONV : conv *)
(* *)
(* Puts into clausal form terms consisting of =,==>,COND,/\,\/,~ and atoms. *)
(*----------------------------------------------------------------------------*)
let CLAUSAL_FORM_CONV tm =
try (
let is_atom tm =
(not (has_boolean_args_and_result tm)) || (is_var tm) || (is_const tm)
in
((EQ_IMP_COND_ELIM_CONV is_atom) THENC
(MOVE_NOT_DOWN_CONV is_atom ALL_CONV) THENC
CONJ_NORM_FORM_CONV) tm
) with Failure _ -> failwith "CLAUSAL_FORM_CONV";;
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