/***
  @groupdef stdlib.logic Logical operations

  Logical operations are the standard operators of Boolean logic.
*/

/** @group stdlib.logic Return truth value of \a x ∧ \a y */
function     bool: '/\'(    bool: x,     bool: y);
/** @group stdlib.logic Return truth value of \a x &and; \a y with identity <> = True*/
function     bool: '/\'(opt bool: x, opt bool: y) = (absent(x) \/ deopt(x)) /\ (absent(y) \/ deopt(y));
/** @group stdlib.logic Return truth value of \a x &and; \a y */
function var bool: '/\'(var bool: x, var bool: y);
/** @group stdlib.logic Return truth value of \a x &and; \a y with identity <> = True*/
function var bool: '/\'(var opt bool: x, var opt bool: y) = (absent(x) \/ deopt(x)) /\ (absent(y) \/ deopt(y));

/** @group stdlib.logic Return truth value of \a x &or; \a y */
function     bool: '\/'(    bool: x,     bool: y);
/** @group stdlib.logic Return truth value of \a x &or; \a y with identity <> = False*/
function     bool: '\/'(opt bool: x, opt bool: y) = (occurs(x) /\ deopt(x)) \/ (occurs(y) /\ deopt(y));
/** @group stdlib.logic Return truth value of \a x &or; \a y */
function var bool: '\/'(var bool: x, var bool: y);
/** @group stdlib.logic Return truth value of \a x &or; \a y with identity <> = False*/
function var bool: '\/'(var opt bool: x, var opt bool: y) = (occurs(x) /\ deopt(x)) \/ (occurs(y) /\ deopt(y));

/** @group stdlib.logic Return truth value of \a x implies \a y */
function     bool: '->'(    bool: x,     bool: y);
/** @group stdlib.logic Return truth value of \a x implies \a y */
function var bool: '->'(var bool: x, var bool: y);
/** @group stdlib.logic Return truth value of \a y implies \a x */
function     bool: '<-'(    bool: x,     bool: y);
/** @group stdlib.logic Return truth value of \a y implies \a x */
function var bool: '<-'(var bool: x, var bool: y);
/** @group stdlib.logic Return truth value of \a x if-and-only-if \a y */
function     bool: '<->'(    bool: x,     bool: y);
/** @group stdlib.logic Return truth value of \a x if-and-only-if \a y */
function var bool: '<->'(var bool: x, var bool: y);
/** @group stdlib.logic Return truth value of \a x xor \a y */
function     bool: 'xor'(    bool: x,     bool: y);
/** @group stdlib.logic Return truth value of \a x xor \a y */
function var bool: 'xor'(var bool: x, var bool: y);
/** @group stdlib.logic Return truth value of the negation of \a x */
function     bool: 'not'(    bool: x);
/** @group stdlib.logic Negation of \a x if it occurs, otherwise absent */
function opt bool: 'not'(opt bool: x) = if absent(x) then <> else not deopt(x) endif;
/** @group stdlib.logic Return truth value of the negation of \a x */
function var bool: 'not'(var bool: x);
/** @group stdlib.logic Negation of \a x if it occurs, otherwise absent */
function var opt bool: 'not'(var opt bool: x) = if absent(x) then <> else not deopt(x) endif;

/** @group stdlib.logic Return truth value of \(\bigwedge_i \a x[i]\) */
function     bool: forall(array[$T] of     bool: x);
/** @group stdlib.logic Return truth value of \(\bigwedge_i \a x[i]\) */
function var bool: forall(array[$T] of var bool: x);
/** @group stdlib.logic True iff for any \p i, \a x[i] is absent or true */
predicate forall (array[int] of var opt bool: x) =
  forall ([absent(x[i]) \/ deopt(x[i]) | i in index_set(x)]);

/** @group stdlib.logic Return truth value of \(\bigvee_i \a x[i]\) */
function     bool: exists(array[$T] of     bool: x);
/** @group stdlib.logic Return truth value of \(\bigvee_i \a x[i]\) */
function var bool: exists(array[$T] of var bool: x);
/** @group stdlib.logic True iff for at least one \p i, \a x[i] occurs and is true */
predicate exists (array[int] of var opt bool: x) =
  exists ([occurs(x[i]) /\ deopt(x[i]) | i in index_set(x)]);

/** @group stdlib.logic Return truth value of \(\oplus_i \a x[i]\) */
function     bool: xorall(array[$T] of     bool: x);
/** @group stdlib.logic Return truth value of \(\oplus_i \a x[i]\) */
function var bool: xorall(array[$T] of var bool: x) =
  array_bool_xor(array1d(x));
/** @group stdlib.logic Return truth value of \(\text{true}\oplus (\oplus_i \a x[i])\) */
function     bool: iffall(array[$T] of     bool: x);
/** @group stdlib.logic Return truth value of \(\text{true}\oplus (\oplus_i \a x[i])\) */
function var bool: iffall(array[$T] of var bool: x) =
  array_bool_xor(array1d(x)++[true]);
/** @group stdlib.logic Return truth value of \((\bigvee_i \a x[i]) \lor (\bigvee_j \lnot \a y[j])\) */
function var bool: clause(array[$T] of var bool: x, array[$T] of var bool: y);
/** @group stdlib.logic Return truth value of \((\bigvee_i \a x[i]) \lor (\bigvee_j \lnot \a y[j])\) */
function var bool: clause(array[$T] of     bool: x, array[$T] of     bool: y);

/** @group stdlib.logic Return negation of \a b */
function var bool: bool_not(var bool: b);