/*-------------------------------------------------------------------------
 *
 * predtest.c
 *	  Routines to attempt to prove logical implications between predicate
 *	  expressions.
 *
 * Portions Copyright (c) 1996-2005, PostgreSQL Global Development Group
 * Portions Copyright (c) 1994, Regents of the University of California
 *
 *
 * IDENTIFICATION
 *	  $PostgreSQL: pgsql/src/backend/optimizer/util/predtest.c,v 1.5 2005/11/27 22:15:42 tgl Exp $
 *
 *-------------------------------------------------------------------------
 */
#include "postgres.h"

#include "catalog/pg_amop.h"
#include "catalog/pg_proc.h"
#include "catalog/pg_type.h"
#include "executor/executor.h"
#include "optimizer/clauses.h"
#include "optimizer/predtest.h"
#include "utils/array.h"
#include "utils/catcache.h"
#include "utils/lsyscache.h"
#include "utils/syscache.h"


/*
 * To avoid redundant coding in predicate_implied_by_recurse and
 * predicate_refuted_by_recurse, we need to abstract out the notion of
 * iterating over the components of an expression that is logically an AND
 * or OR structure.  There are multiple sorts of expression nodes that can
 * be treated as ANDs or ORs, and we don't want to code each one separately.
 * Hence, these types and support routines.
 */
typedef enum
{
	CLASS_ATOM,					/* expression that's not AND or OR */
	CLASS_AND,					/* expression with AND semantics */
	CLASS_OR					/* expression with OR semantics */
} PredClass;

typedef struct PredIterInfoData *PredIterInfo;

typedef struct PredIterInfoData
{
	/* node-type-specific iteration state */
	void	   *state;
	/* initialize to do the iteration */
	void		(*startup_fn) (Node *clause, PredIterInfo info);
	/* next-component iteration function */
	Node	   *(*next_fn) (PredIterInfo info);
	/* release resources when done with iteration */
	void		(*cleanup_fn) (PredIterInfo info);
} PredIterInfoData;

#define iterate_begin(item, clause, info)	\
	do { \
		Node   *item; \
		(info).startup_fn((clause), &(info)); \
		while ((item = (info).next_fn(&(info))) != NULL)

#define iterate_end(info)	\
		(info).cleanup_fn(&(info)); \
	} while (0)


static bool predicate_implied_by_recurse(Node *clause, Node *predicate);
static bool predicate_refuted_by_recurse(Node *clause, Node *predicate);
static PredClass predicate_classify(Node *clause, PredIterInfo info);
static void list_startup_fn(Node *clause, PredIterInfo info);
static Node *list_next_fn(PredIterInfo info);
static void list_cleanup_fn(PredIterInfo info);
static void boolexpr_startup_fn(Node *clause, PredIterInfo info);
static void arrayconst_startup_fn(Node *clause, PredIterInfo info);
static Node *arrayconst_next_fn(PredIterInfo info);
static void arrayconst_cleanup_fn(PredIterInfo info);
static void arrayexpr_startup_fn(Node *clause, PredIterInfo info);
static Node *arrayexpr_next_fn(PredIterInfo info);
static void arrayexpr_cleanup_fn(PredIterInfo info);
static bool predicate_implied_by_simple_clause(Expr *predicate, Node *clause);
static bool predicate_refuted_by_simple_clause(Expr *predicate, Node *clause);
static bool btree_predicate_proof(Expr *predicate, Node *clause,
					  bool refute_it);


/*
 * predicate_implied_by
 *	  Recursively checks whether the clauses in restrictinfo_list imply
 *	  that the given predicate is true.
 *
 * The top-level List structure of each list corresponds to an AND list.
 * We assume that eval_const_expressions() has been applied and so there
 * are no un-flattened ANDs or ORs (e.g., no AND immediately within an AND,
 * including AND just below the top-level List structure).
 * If this is not true we might fail to prove an implication that is
 * valid, but no worse consequences will ensue.
 *
 * We assume the predicate has already been checked to contain only
 * immutable functions and operators.  (In most current uses this is true
 * because the predicate is part of an index predicate that has passed
 * CheckPredicate().)  We dare not make deductions based on non-immutable
 * functions, because they might change answers between the time we make
 * the plan and the time we execute the plan.
 */
bool
predicate_implied_by(List *predicate_list, List *restrictinfo_list)
{
	if (predicate_list == NIL)
		return true;			/* no predicate: implication is vacuous */
	if (restrictinfo_list == NIL)
		return false;			/* no restriction: implication must fail */

	/* Otherwise, away we go ... */
	return predicate_implied_by_recurse((Node *) restrictinfo_list,
										(Node *) predicate_list);
}

/*
 * predicate_refuted_by
 *	  Recursively checks whether the clauses in restrictinfo_list refute
 *	  the given predicate (that is, prove it false).
 *
 * This is NOT the same as !(predicate_implied_by), though it is similar
 * in the technique and structure of the code.
 *
 * The top-level List structure of each list corresponds to an AND list.
 * We assume that eval_const_expressions() has been applied and so there
 * are no un-flattened ANDs or ORs (e.g., no AND immediately within an AND,
 * including AND just below the top-level List structure).
 * If this is not true we might fail to prove an implication that is
 * valid, but no worse consequences will ensue.
 *
 * We assume the predicate has already been checked to contain only
 * immutable functions and operators.  We dare not make deductions based on
 * non-immutable functions, because they might change answers between the
 * time we make the plan and the time we execute the plan.
 */
bool
predicate_refuted_by(List *predicate_list, List *restrictinfo_list)
{
	if (predicate_list == NIL)
		return false;			/* no predicate: no refutation is possible */
	if (restrictinfo_list == NIL)
		return false;			/* no restriction: refutation must fail */

	/* Otherwise, away we go ... */
	return predicate_refuted_by_recurse((Node *) restrictinfo_list,
										(Node *) predicate_list);
}

/*----------
 * predicate_implied_by_recurse
 *	  Does the predicate implication test for non-NULL restriction and
 *	  predicate clauses.
 *
 * The logic followed here is ("=>" means "implies"):
 *	atom A => atom B iff:			predicate_implied_by_simple_clause says so
 *	atom A => AND-expr B iff:		A => each of B's components
 *	atom A => OR-expr B iff:		A => any of B's components
 *	AND-expr A => atom B iff:		any of A's components => B
 *	AND-expr A => AND-expr B iff:	A => each of B's components
 *	AND-expr A => OR-expr B iff:	A => any of B's components,
 *									*or* any of A's components => B
 *	OR-expr A => atom B iff:		each of A's components => B
 *	OR-expr A => AND-expr B iff:	A => each of B's components
 *	OR-expr A => OR-expr B iff:		each of A's components => any of B's
 *
 * An "atom" is anything other than an AND or OR node.	Notice that we don't
 * have any special logic to handle NOT nodes; these should have been pushed
 * down or eliminated where feasible by prepqual.c.
 *
 * We can't recursively expand either side first, but have to interleave
 * the expansions per the above rules, to be sure we handle all of these
 * examples:
 *		(x OR y) => (x OR y OR z)
 *		(x AND y AND z) => (x AND y)
 *		(x AND y) => ((x AND y) OR z)
 *		((x OR y) AND z) => (x OR y)
 * This is still not an exhaustive test, but it handles most normal cases
 * under the assumption that both inputs have been AND/OR flattened.
 *
 * We have to be prepared to handle RestrictInfo nodes in the restrictinfo
 * tree, though not in the predicate tree.
 *----------
 */
static bool
predicate_implied_by_recurse(Node *clause, Node *predicate)
{
	PredIterInfoData clause_info;
	PredIterInfoData pred_info;
	PredClass	pclass;
	bool		result;

	/* skip through RestrictInfo */
	Assert(clause != NULL);
	if (IsA(clause, RestrictInfo))
		clause = (Node *) ((RestrictInfo *) clause)->clause;

	pclass = predicate_classify(predicate, &pred_info);

	switch (predicate_classify(clause, &clause_info))
	{
		case CLASS_AND:
			switch (pclass)
			{
				case CLASS_AND:
					/*
					 * AND-clause => AND-clause if A implies each of B's items
					 */
					result = true;
					iterate_begin(pitem, predicate, pred_info)
					{
						if (!predicate_implied_by_recurse(clause, pitem))
						{
							result = false;
							break;
						}
					}
					iterate_end(pred_info);
					return result;

				case CLASS_OR:
					/*
					 * AND-clause => OR-clause if A implies any of B's items
					 *
					 * Needed to handle (x AND y) => ((x AND y) OR z)
					 */
					result = false;
					iterate_begin(pitem, predicate, pred_info)
					{
						if (predicate_implied_by_recurse(clause, pitem))
						{
							result = true;
							break;
						}
					}
					iterate_end(pred_info);
					if (result)
						return result;
					/*
					 * Also check if any of A's items implies B
					 *
					 * Needed to handle ((x OR y) AND z) => (x OR y)
					 */
					iterate_begin(citem, clause, clause_info)
					{
						if (predicate_implied_by_recurse(citem, predicate))
						{
							result = true;
							break;
						}
					}
					iterate_end(clause_info);
					return result;

				case CLASS_ATOM:
					/*
					 * AND-clause => atom if any of A's items implies B
					 */
					result = false;
					iterate_begin(citem, clause, clause_info)
					{
						if (predicate_implied_by_recurse(citem, predicate))
						{
							result = true;
							break;
						}
					}
					iterate_end(clause_info);
					return result;
			}
			break;

		case CLASS_OR:
			switch (pclass)
			{
				case CLASS_OR:
					/*
					 * OR-clause => OR-clause if each of A's items implies any
					 * of B's items.  Messy but can't do it any more simply.
					 */
					result = true;
					iterate_begin(citem, clause, clause_info)
					{
						bool	presult = false;

						iterate_begin(pitem, predicate, pred_info)
						{
							if (predicate_implied_by_recurse(citem, pitem))
							{
								presult = true;
								break;
							}
						}
						iterate_end(pred_info);
						if (!presult)
						{
							result = false;		/* doesn't imply any of B's */
							break;
						}
					}
					iterate_end(clause_info);
					return result;

				case CLASS_AND:
				case CLASS_ATOM:
					/*
					 * OR-clause => AND-clause if each of A's items implies B
					 *
					 * OR-clause => atom if each of A's items implies B
					 */
					result = true;
					iterate_begin(citem, clause, clause_info)
					{
						if (!predicate_implied_by_recurse(citem, predicate))
						{
							result = false;
							break;
						}
					}
					iterate_end(clause_info);
					return result;
			}
			break;

		case CLASS_ATOM:
			switch (pclass)
			{
				case CLASS_AND:
					/*
					 * atom => AND-clause if A implies each of B's items
					 */
					result = true;
					iterate_begin(pitem, predicate, pred_info)
					{
						if (!predicate_implied_by_recurse(clause, pitem))
						{
							result = false;
							break;
						}
					}
					iterate_end(pred_info);
					return result;

				case CLASS_OR:
					/*
					 * atom => OR-clause if A implies any of B's items
					 */
					result = false;
					iterate_begin(pitem, predicate, pred_info)
					{
						if (predicate_implied_by_recurse(clause, pitem))
						{
							result = true;
							break;
						}
					}
					iterate_end(pred_info);
					return result;

				case CLASS_ATOM:
					/*
					 * atom => atom is the base case
					 */
					return
						predicate_implied_by_simple_clause((Expr *) predicate,
														   clause);
			}
			break;
	}

	/* can't get here */
	elog(ERROR, "predicate_classify returned a bogus value");
	return false;
}

/*----------
 * predicate_refuted_by_recurse
 *	  Does the predicate refutation test for non-NULL restriction and
 *	  predicate clauses.
 *
 * The logic followed here is ("R=>" means "refutes"):
 *	atom A R=> atom B iff:			predicate_refuted_by_simple_clause says so
 *	atom A R=> AND-expr B iff:		A R=> any of B's components
 *	atom A R=> OR-expr B iff:		A R=> each of B's components
 *	AND-expr A R=> atom B iff:		any of A's components R=> B
 *	AND-expr A R=> AND-expr B iff:	A R=> any of B's components,
 *									*or* any of A's components R=> B
 *	AND-expr A R=> OR-expr B iff:	A R=> each of B's components
 *	OR-expr A R=> atom B iff:		each of A's components R=> B
 *	OR-expr A R=> AND-expr B iff:	each of A's components R=> any of B's
 *	OR-expr A R=> OR-expr B iff:	A R=> each of B's components
 *
 * Other comments are as for predicate_implied_by_recurse().
 *----------
 */
static bool
predicate_refuted_by_recurse(Node *clause, Node *predicate)
{
	PredIterInfoData clause_info;
	PredIterInfoData pred_info;
	PredClass	pclass;
	bool		result;

	/* skip through RestrictInfo */
	Assert(clause != NULL);
	if (IsA(clause, RestrictInfo))
		clause = (Node *) ((RestrictInfo *) clause)->clause;

	pclass = predicate_classify(predicate, &pred_info);

	switch (predicate_classify(clause, &clause_info))
	{
		case CLASS_AND:
			switch (pclass)
			{
				case CLASS_AND:
					/*
					 * AND-clause R=> AND-clause if A refutes any of B's items
					 *
					 * Needed to handle (x AND y) R=> ((!x OR !y) AND z)
					 */
					result = false;
					iterate_begin(pitem, predicate, pred_info)
					{
						if (predicate_refuted_by_recurse(clause, pitem))
						{
							result = true;
							break;
						}
					}
					iterate_end(pred_info);
					if (result)
						return result;
					/*
					 * Also check if any of A's items refutes B
					 *
					 * Needed to handle ((x OR y) AND z) R=> (!x AND !y)
					 */
					iterate_begin(citem, clause, clause_info)
					{
						if (predicate_refuted_by_recurse(citem, predicate))
						{
							result = true;
							break;
						}
					}
					iterate_end(clause_info);
					return result;

				case CLASS_OR:
					/*
					 * AND-clause R=> OR-clause if A refutes each of B's items
					 */
					result = true;
					iterate_begin(pitem, predicate, pred_info)
					{
						if (!predicate_refuted_by_recurse(clause, pitem))
						{
							result = false;
							break;
						}
					}
					iterate_end(pred_info);
					return result;

				case CLASS_ATOM:
					/*
					 * AND-clause R=> atom if any of A's items refutes B
					 */
					result = false;
					iterate_begin(citem, clause, clause_info)
					{
						if (predicate_refuted_by_recurse(citem, predicate))
						{
							result = true;
							break;
						}
					}
					iterate_end(clause_info);
					return result;
			}
			break;

		case CLASS_OR:
			switch (pclass)
			{
				case CLASS_OR:
					/*
					 * OR-clause R=> OR-clause if A refutes each of B's items
					 */
					result = true;
					iterate_begin(pitem, predicate, pred_info)
					{
						if (!predicate_refuted_by_recurse(clause, pitem))
						{
							result = false;
							break;
						}
					}
					iterate_end(pred_info);
					return result;

				case CLASS_AND:
					/*
					 * OR-clause R=> AND-clause if each of A's items refutes
					 * any of B's items.
					 */
					result = true;
					iterate_begin(citem, clause, clause_info)
					{
						bool	presult = false;

						iterate_begin(pitem, predicate, pred_info)
						{
							if (predicate_refuted_by_recurse(citem, pitem))
							{
								presult = true;
								break;
							}
						}
						iterate_end(pred_info);
						if (!presult)
						{
							result = false;		/* citem refutes nothing */
							break;
						}
					}
					iterate_end(clause_info);
					return result;

				case CLASS_ATOM:
					/*
					 * OR-clause R=> atom if each of A's items refutes B
					 */
					result = true;
					iterate_begin(citem, clause, clause_info)
					{
						if (!predicate_refuted_by_recurse(citem, predicate))
						{
							result = false;
							break;
						}
					}
					iterate_end(clause_info);
					return result;
			}
			break;

		case CLASS_ATOM:
			switch (pclass)
			{
				case CLASS_AND:
					/*
					 * atom R=> AND-clause if A refutes any of B's items
					 */
					result = false;
					iterate_begin(pitem, predicate, pred_info)
					{
						if (predicate_refuted_by_recurse(clause, pitem))
						{
							result = true;
							break;
						}
					}
					iterate_end(pred_info);
					return result;

				case CLASS_OR:
					/*
					 * atom R=> OR-clause if A refutes each of B's items
					 */
					result = true;
					iterate_begin(pitem, predicate, pred_info)
					{
						if (!predicate_refuted_by_recurse(clause, pitem))
						{
							result = false;
							break;
						}
					}
					iterate_end(pred_info);
					return result;

				case CLASS_ATOM:
					/*
					 * atom R=> atom is the base case
					 */
					return
						predicate_refuted_by_simple_clause((Expr *) predicate,
														   clause);
			}
			break;
	}

	/* can't get here */
	elog(ERROR, "predicate_classify returned a bogus value");
	return false;
}


/*
 * predicate_classify
 *	  Classify an expression node as AND-type, OR-type, or neither (an atom).
 *
 * If the expression is classified as AND- or OR-type, then *info is filled
 * in with the functions needed to iterate over its components.
 */
static PredClass
predicate_classify(Node *clause, PredIterInfo info)
{
	/* Caller should not pass us NULL, nor a RestrictInfo clause */
	Assert(clause != NULL);
	Assert(!IsA(clause, RestrictInfo));

	/*
	 * If we see a List, assume it's an implicit-AND list; this is the
	 * correct semantics for lists of RestrictInfo nodes.
	 */
	if (IsA(clause, List))
	{
		info->startup_fn = list_startup_fn;
		info->next_fn = list_next_fn;
		info->cleanup_fn = list_cleanup_fn;
		return CLASS_AND;
	}

	/* Handle normal AND and OR boolean clauses */
	if (and_clause(clause))
	{
		info->startup_fn = boolexpr_startup_fn;
		info->next_fn = list_next_fn;
		info->cleanup_fn = list_cleanup_fn;
		return CLASS_AND;
	}
	if (or_clause(clause))
	{
		info->startup_fn = boolexpr_startup_fn;
		info->next_fn = list_next_fn;
		info->cleanup_fn = list_cleanup_fn;
		return CLASS_OR;
	}

	/* Handle ScalarArrayOpExpr */
	if (IsA(clause, ScalarArrayOpExpr))
	{
		ScalarArrayOpExpr *saop = (ScalarArrayOpExpr *) clause;
		Node   *arraynode = (Node *) lsecond(saop->args);

		/*
		 * We can break this down into an AND or OR structure, but only if
		 * we know how to iterate through expressions for the array's
		 * elements.  We can do that if the array operand is a non-null
		 * constant or a simple ArrayExpr.
		 */
		if (arraynode && IsA(arraynode, Const) &&
			!((Const *) arraynode)->constisnull)
		{
			info->startup_fn = arrayconst_startup_fn;
			info->next_fn = arrayconst_next_fn;
			info->cleanup_fn = arrayconst_cleanup_fn;
			return saop->useOr ? CLASS_OR : CLASS_AND;
		}
		if (arraynode && IsA(arraynode, ArrayExpr) &&
			!((ArrayExpr *) arraynode)->multidims)
		{
			info->startup_fn = arrayexpr_startup_fn;
			info->next_fn = arrayexpr_next_fn;
			info->cleanup_fn = arrayexpr_cleanup_fn;
			return saop->useOr ? CLASS_OR : CLASS_AND;
		}
	}

	/* None of the above, so it's an atom */
	return CLASS_ATOM;
}

/*
 * PredIterInfo routines for iterating over regular Lists.  The iteration
 * state variable is the next ListCell to visit.
 */
static void
list_startup_fn(Node *clause, PredIterInfo info)
{
	info->state = (void *) list_head((List *) clause);
}

static Node *
list_next_fn(PredIterInfo info)
{
	ListCell   *l = (ListCell *) info->state;
	Node	   *n;

	if (l == NULL)
		return NULL;
	n = lfirst(l);
	info->state = (void *) lnext(l);
	return n;
}

static void
list_cleanup_fn(PredIterInfo info)
{
	/* Nothing to clean up */
}

/*
 * BoolExpr needs its own startup function, but can use list_next_fn and
 * list_cleanup_fn.
 */
static void
boolexpr_startup_fn(Node *clause, PredIterInfo info)
{
	info->state = (void *) list_head(((BoolExpr *) clause)->args);
}

/*
 * PredIterInfo routines for iterating over a ScalarArrayOpExpr with a
 * constant array operand.
 */
typedef struct
{
	OpExpr		opexpr;
	Const		constexpr;
	int			next_elem;
	int			num_elems;
	Datum	   *elem_values;
	bool	   *elem_nulls;
} ArrayConstIterState;

static void
arrayconst_startup_fn(Node *clause, PredIterInfo info)
{
	ScalarArrayOpExpr *saop = (ScalarArrayOpExpr *) clause;
	ArrayConstIterState *state;
	Const	   *arrayconst;
	ArrayType  *arrayval;
	int16		elmlen;
	bool		elmbyval;
	char		elmalign;

	/* Create working state struct */
	state = (ArrayConstIterState *) palloc(sizeof(ArrayConstIterState));
	info->state = (void *) state;

	/* Deconstruct the array literal */
	arrayconst = (Const *) lsecond(saop->args);
	arrayval = DatumGetArrayTypeP(arrayconst->constvalue);
	get_typlenbyvalalign(ARR_ELEMTYPE(arrayval),
						 &elmlen, &elmbyval, &elmalign);
	deconstruct_array(arrayval,
					  ARR_ELEMTYPE(arrayval),
					  elmlen, elmbyval, elmalign,
					  &state->elem_values, &state->elem_nulls,
					  &state->num_elems);

	/* Set up a dummy OpExpr to return as the per-item node */
	state->opexpr.xpr.type = T_OpExpr;
	state->opexpr.opno = saop->opno;
	state->opexpr.opfuncid = saop->opfuncid;
	state->opexpr.opresulttype = BOOLOID;
	state->opexpr.opretset = false;
	state->opexpr.args = list_copy(saop->args);

	/* Set up a dummy Const node to hold the per-element values */
	state->constexpr.xpr.type = T_Const;
	state->constexpr.consttype = ARR_ELEMTYPE(arrayval);
	state->constexpr.constlen = elmlen;
	state->constexpr.constbyval = elmbyval;
	lsecond(state->opexpr.args) = &state->constexpr;

	/* Initialize iteration state */
	state->next_elem = 0;
}

static Node *
arrayconst_next_fn(PredIterInfo info)
{
	ArrayConstIterState *state = (ArrayConstIterState *) info->state;

	if (state->next_elem >= state->num_elems)
		return NULL;
	state->constexpr.constvalue = state->elem_values[state->next_elem];
	state->constexpr.constisnull = state->elem_nulls[state->next_elem];
	state->next_elem++;
	return (Node *) &(state->opexpr);
}

static void
arrayconst_cleanup_fn(PredIterInfo info)
{
	ArrayConstIterState *state = (ArrayConstIterState *) info->state;

	pfree(state->elem_values);
	pfree(state->elem_nulls);
	list_free(state->opexpr.args);
	pfree(state);
}

/*
 * PredIterInfo routines for iterating over a ScalarArrayOpExpr with a
 * one-dimensional ArrayExpr array operand.
 */
typedef struct
{
	OpExpr		opexpr;
	ListCell   *next;
} ArrayExprIterState;

static void
arrayexpr_startup_fn(Node *clause, PredIterInfo info)
{
	ScalarArrayOpExpr *saop = (ScalarArrayOpExpr *) clause;
	ArrayExprIterState *state;
	ArrayExpr *arrayexpr;

	/* Create working state struct */
	state = (ArrayExprIterState *) palloc(sizeof(ArrayExprIterState));
	info->state = (void *) state;

	/* Set up a dummy OpExpr to return as the per-item node */
	state->opexpr.xpr.type = T_OpExpr;
	state->opexpr.opno = saop->opno;
	state->opexpr.opfuncid = saop->opfuncid;
	state->opexpr.opresulttype = BOOLOID;
	state->opexpr.opretset = false;
	state->opexpr.args = list_copy(saop->args);

	/* Initialize iteration variable to first member of ArrayExpr */
	arrayexpr = (ArrayExpr *) lsecond(saop->args);
	state->next = list_head(arrayexpr->elements);
}

static Node *
arrayexpr_next_fn(PredIterInfo info)
{
	ArrayExprIterState *state = (ArrayExprIterState *) info->state;

	if (state->next == NULL)
		return NULL;
	lsecond(state->opexpr.args) = lfirst(state->next);
	state->next = lnext(state->next);
	return (Node *) &(state->opexpr);
}

static void
arrayexpr_cleanup_fn(PredIterInfo info)
{
	ArrayExprIterState *state = (ArrayExprIterState *) info->state;

	list_free(state->opexpr.args);
	pfree(state);
}


/*----------
 * predicate_implied_by_simple_clause
 *	  Does the predicate implication test for a "simple clause" predicate
 *	  and a "simple clause" restriction.
 *
 * We return TRUE if able to prove the implication, FALSE if not.
 *
 * We have three strategies for determining whether one simple clause
 * implies another:
 *
 * A simple and general way is to see if they are equal(); this works for any
 * kind of expression.	(Actually, there is an implied assumption that the
 * functions in the expression are immutable, ie dependent only on their input
 * arguments --- but this was checked for the predicate by the caller.)
 *
 * When the predicate is of the form "foo IS NOT NULL", we can conclude that
 * the predicate is implied if the clause is a strict operator or function
 * that has "foo" as an input.	In this case the clause must yield NULL when
 * "foo" is NULL, which we can take as equivalent to FALSE because we know
 * we are within an AND/OR subtree of a WHERE clause.  (Again, "foo" is
 * already known immutable, so the clause will certainly always fail.)
 *
 * Finally, we may be able to deduce something using knowledge about btree
 * operator classes; this is encapsulated in btree_predicate_proof().
 *----------
 */
static bool
predicate_implied_by_simple_clause(Expr *predicate, Node *clause)
{
	/* First try the equal() test */
	if (equal((Node *) predicate, clause))
		return true;

	/* Next try the IS NOT NULL case */
	if (predicate && IsA(predicate, NullTest) &&
		((NullTest *) predicate)->nulltesttype == IS_NOT_NULL)
	{
		Expr	   *nonnullarg = ((NullTest *) predicate)->arg;

		if (is_opclause(clause) &&
			list_member(((OpExpr *) clause)->args, nonnullarg) &&
			op_strict(((OpExpr *) clause)->opno))
			return true;
		if (is_funcclause(clause) &&
			list_member(((FuncExpr *) clause)->args, nonnullarg) &&
			func_strict(((FuncExpr *) clause)->funcid))
			return true;
		return false;			/* we can't succeed below... */
	}

	/* Else try btree operator knowledge */
	return btree_predicate_proof(predicate, clause, false);
}

/*----------
 * predicate_refuted_by_simple_clause
 *	  Does the predicate refutation test for a "simple clause" predicate
 *	  and a "simple clause" restriction.
 *
 * We return TRUE if able to prove the refutation, FALSE if not.
 *
 * Unlike the implication case, checking for equal() clauses isn't
 * helpful.  (XXX is it worth looking at "x vs NOT x" cases?  Probably
 * not seeing that canonicalization tries to get rid of NOTs.)
 *
 * When the predicate is of the form "foo IS NULL", we can conclude that
 * the predicate is refuted if the clause is a strict operator or function
 * that has "foo" as an input.	See notes for implication case.
 *
 * Finally, we may be able to deduce something using knowledge about btree
 * operator classes; this is encapsulated in btree_predicate_proof().
 *----------
 */
static bool
predicate_refuted_by_simple_clause(Expr *predicate, Node *clause)
{
	/* First try the IS NULL case */
	if (predicate && IsA(predicate, NullTest) &&
		((NullTest *) predicate)->nulltesttype == IS_NULL)
	{
		Expr	   *isnullarg = ((NullTest *) predicate)->arg;

		if (is_opclause(clause) &&
			list_member(((OpExpr *) clause)->args, isnullarg) &&
			op_strict(((OpExpr *) clause)->opno))
			return true;
		if (is_funcclause(clause) &&
			list_member(((FuncExpr *) clause)->args, isnullarg) &&
			func_strict(((FuncExpr *) clause)->funcid))
			return true;
		return false;			/* we can't succeed below... */
	}

	/* Else try btree operator knowledge */
	return btree_predicate_proof(predicate, clause, true);
}


/*
 * Define an "operator implication table" for btree operators ("strategies"),
 * and a similar table for refutation.
 *
 * The strategy numbers defined by btree indexes (see access/skey.h) are:
 *		(1) <	(2) <=	 (3) =	 (4) >=   (5) >
 * and in addition we use (6) to represent <>.	<> is not a btree-indexable
 * operator, but we assume here that if the equality operator of a btree
 * opclass has a negator operator, the negator behaves as <> for the opclass.
 *
 * The interpretation of:
 *
 *		test_op = BT_implic_table[given_op-1][target_op-1]
 *
 * where test_op, given_op and target_op are strategy numbers (from 1 to 6)
 * of btree operators, is as follows:
 *
 *	 If you know, for some ATTR, that "ATTR given_op CONST1" is true, and you
 *	 want to determine whether "ATTR target_op CONST2" must also be true, then
 *	 you can use "CONST2 test_op CONST1" as a test.  If this test returns true,
 *	 then the target expression must be true; if the test returns false, then
 *	 the target expression may be false.
 *
 * For example, if clause is "Quantity > 10" and pred is "Quantity > 5"
 * then we test "5 <= 10" which evals to true, so clause implies pred.
 *
 * Similarly, the interpretation of a BT_refute_table entry is:
 *
 *	 If you know, for some ATTR, that "ATTR given_op CONST1" is true, and you
 *	 want to determine whether "ATTR target_op CONST2" must be false, then
 *	 you can use "CONST2 test_op CONST1" as a test.  If this test returns true,
 *	 then the target expression must be false; if the test returns false, then
 *	 the target expression may be true.
 *
 * For example, if clause is "Quantity > 10" and pred is "Quantity < 5"
 * then we test "5 <= 10" which evals to true, so clause refutes pred.
 *
 * An entry where test_op == 0 means the implication cannot be determined.
 */

#define BTLT BTLessStrategyNumber
#define BTLE BTLessEqualStrategyNumber
#define BTEQ BTEqualStrategyNumber
#define BTGE BTGreaterEqualStrategyNumber
#define BTGT BTGreaterStrategyNumber
#define BTNE 6

static const StrategyNumber BT_implic_table[6][6] = {
/*
 *			The target operator:
 *
 *	 LT    LE	 EQ    GE	 GT    NE
 */
	{BTGE, BTGE, 0, 0, 0, BTGE},	/* LT */
	{BTGT, BTGE, 0, 0, 0, BTGT},	/* LE */
	{BTGT, BTGE, BTEQ, BTLE, BTLT, BTNE},		/* EQ */
	{0, 0, 0, BTLE, BTLT, BTLT},	/* GE */
	{0, 0, 0, BTLE, BTLE, BTLE},	/* GT */
	{0, 0, 0, 0, 0, BTEQ}		/* NE */
};

static const StrategyNumber BT_refute_table[6][6] = {
/*
 *			The target operator:
 *
 *	 LT    LE	 EQ    GE	 GT    NE
 */
	{0, 0, BTGE, BTGE, BTGE, 0},	/* LT */
	{0, 0, BTGT, BTGT, BTGE, 0},	/* LE */
	{BTLE, BTLT, BTNE, BTGT, BTGE, BTEQ},		/* EQ */
	{BTLE, BTLT, BTLT, 0, 0, 0},	/* GE */
	{BTLE, BTLE, BTLE, 0, 0, 0},	/* GT */
	{0, 0, BTEQ, 0, 0, 0}		/* NE */
};


/*----------
 * btree_predicate_proof
 *	  Does the predicate implication or refutation test for a "simple clause"
 *	  predicate and a "simple clause" restriction, when both are simple
 *	  operator clauses using related btree operators.
 *
 * When refute_it == false, we want to prove the predicate true;
 * when refute_it == true, we want to prove the predicate false.
 * (There is enough common code to justify handling these two cases
 * in one routine.)  We return TRUE if able to make the proof, FALSE
 * if not able to prove it.
 *
 * What we look for here is binary boolean opclauses of the form
 * "foo op constant", where "foo" is the same in both clauses.	The operators
 * and constants can be different but the operators must be in the same btree
 * operator class.	We use the above operator implication tables to
 * derive implications between nonidentical clauses.  (Note: "foo" is known
 * immutable, and constants are surely immutable, but we have to check that
 * the operators are too.  As of 8.0 it's possible for opclasses to contain
 * operators that are merely stable, and we dare not make deductions with
 * these.)
 *----------
 */
static bool
btree_predicate_proof(Expr *predicate, Node *clause, bool refute_it)
{
	Node	   *leftop,
			   *rightop;
	Node	   *pred_var,
			   *clause_var;
	Const	   *pred_const,
			   *clause_const;
	bool		pred_var_on_left,
				clause_var_on_left,
				pred_op_negated;
	Oid			pred_op,
				clause_op,
				pred_op_negator,
				clause_op_negator,
				test_op = InvalidOid;
	Oid			opclass_id;
	bool		found = false;
	StrategyNumber pred_strategy,
				clause_strategy,
				test_strategy;
	Oid			clause_subtype;
	Expr	   *test_expr;
	ExprState  *test_exprstate;
	Datum		test_result;
	bool		isNull;
	CatCList   *catlist;
	int			i;
	EState	   *estate;
	MemoryContext oldcontext;

	/*
	 * Both expressions must be binary opclauses with a Const on one side, and
	 * identical subexpressions on the other sides. Note we don't have to
	 * think about binary relabeling of the Const node, since that would have
	 * been folded right into the Const.
	 *
	 * If either Const is null, we also fail right away; this assumes that the
	 * test operator will always be strict.
	 */
	if (!is_opclause(predicate))
		return false;
	leftop = get_leftop(predicate);
	rightop = get_rightop(predicate);
	if (rightop == NULL)
		return false;			/* not a binary opclause */
	if (IsA(rightop, Const))
	{
		pred_var = leftop;
		pred_const = (Const *) rightop;
		pred_var_on_left = true;
	}
	else if (IsA(leftop, Const))
	{
		pred_var = rightop;
		pred_const = (Const *) leftop;
		pred_var_on_left = false;
	}
	else
		return false;			/* no Const to be found */
	if (pred_const->constisnull)
		return false;

	if (!is_opclause(clause))
		return false;
	leftop = get_leftop((Expr *) clause);
	rightop = get_rightop((Expr *) clause);
	if (rightop == NULL)
		return false;			/* not a binary opclause */
	if (IsA(rightop, Const))
	{
		clause_var = leftop;
		clause_const = (Const *) rightop;
		clause_var_on_left = true;
	}
	else if (IsA(leftop, Const))
	{
		clause_var = rightop;
		clause_const = (Const *) leftop;
		clause_var_on_left = false;
	}
	else
		return false;			/* no Const to be found */
	if (clause_const->constisnull)
		return false;

	/*
	 * Check for matching subexpressions on the non-Const sides.  We used to
	 * only allow a simple Var, but it's about as easy to allow any
	 * expression.	Remember we already know that the pred expression does not
	 * contain any non-immutable functions, so identical expressions should
	 * yield identical results.
	 */
	if (!equal(pred_var, clause_var))
		return false;

	/*
	 * Okay, get the operators in the two clauses we're comparing. Commute
	 * them if needed so that we can assume the variables are on the left.
	 */
	pred_op = ((OpExpr *) predicate)->opno;
	if (!pred_var_on_left)
	{
		pred_op = get_commutator(pred_op);
		if (!OidIsValid(pred_op))
			return false;
	}

	clause_op = ((OpExpr *) clause)->opno;
	if (!clause_var_on_left)
	{
		clause_op = get_commutator(clause_op);
		if (!OidIsValid(clause_op))
			return false;
	}

	/*
	 * Try to find a btree opclass containing the needed operators.
	 *
	 * We must find a btree opclass that contains both operators, else the
	 * implication can't be determined.  Also, the pred_op has to be of
	 * default subtype (implying left and right input datatypes are the same);
	 * otherwise it's unsafe to put the pred_const on the left side of the
	 * test.  Also, the opclass must contain a suitable test operator matching
	 * the clause_const's type (which we take to mean that it has the same
	 * subtype as the original clause_operator).
	 *
	 * If there are multiple matching opclasses, assume we can use any one to
	 * determine the logical relationship of the two operators and the correct
	 * corresponding test operator.  This should work for any logically
	 * consistent opclasses.
	 */
	catlist = SearchSysCacheList(AMOPOPID, 1,
								 ObjectIdGetDatum(pred_op),
								 0, 0, 0);

	/*
	 * If we couldn't find any opclass containing the pred_op, perhaps it is a
	 * <> operator.  See if it has a negator that is in an opclass.
	 */
	pred_op_negated = false;
	if (catlist->n_members == 0)
	{
		pred_op_negator = get_negator(pred_op);
		if (OidIsValid(pred_op_negator))
		{
			pred_op_negated = true;
			ReleaseSysCacheList(catlist);
			catlist = SearchSysCacheList(AMOPOPID, 1,
										 ObjectIdGetDatum(pred_op_negator),
										 0, 0, 0);
		}
	}

	/* Also may need the clause_op's negator */
	clause_op_negator = get_negator(clause_op);

	/* Now search the opclasses */
	for (i = 0; i < catlist->n_members; i++)
	{
		HeapTuple	pred_tuple = &catlist->members[i]->tuple;
		Form_pg_amop pred_form = (Form_pg_amop) GETSTRUCT(pred_tuple);
		HeapTuple	clause_tuple;

		opclass_id = pred_form->amopclaid;

		/* must be btree */
		if (!opclass_is_btree(opclass_id))
			continue;
		/* predicate operator must be default within this opclass */
		if (pred_form->amopsubtype != InvalidOid)
			continue;

		/* Get the predicate operator's btree strategy number */
		pred_strategy = (StrategyNumber) pred_form->amopstrategy;
		Assert(pred_strategy >= 1 && pred_strategy <= 5);

		if (pred_op_negated)
		{
			/* Only consider negators that are = */
			if (pred_strategy != BTEqualStrategyNumber)
				continue;
			pred_strategy = BTNE;
		}

		/*
		 * From the same opclass, find a strategy number for the clause_op, if
		 * possible
		 */
		clause_tuple = SearchSysCache(AMOPOPID,
									  ObjectIdGetDatum(clause_op),
									  ObjectIdGetDatum(opclass_id),
									  0, 0);
		if (HeapTupleIsValid(clause_tuple))
		{
			Form_pg_amop clause_form = (Form_pg_amop) GETSTRUCT(clause_tuple);

			/* Get the restriction clause operator's strategy/subtype */
			clause_strategy = (StrategyNumber) clause_form->amopstrategy;
			Assert(clause_strategy >= 1 && clause_strategy <= 5);
			clause_subtype = clause_form->amopsubtype;
			ReleaseSysCache(clause_tuple);
		}
		else if (OidIsValid(clause_op_negator))
		{
			clause_tuple = SearchSysCache(AMOPOPID,
										  ObjectIdGetDatum(clause_op_negator),
										  ObjectIdGetDatum(opclass_id),
										  0, 0);
			if (HeapTupleIsValid(clause_tuple))
			{
				Form_pg_amop clause_form = (Form_pg_amop) GETSTRUCT(clause_tuple);

				/* Get the restriction clause operator's strategy/subtype */
				clause_strategy = (StrategyNumber) clause_form->amopstrategy;
				Assert(clause_strategy >= 1 && clause_strategy <= 5);
				clause_subtype = clause_form->amopsubtype;
				ReleaseSysCache(clause_tuple);

				/* Only consider negators that are = */
				if (clause_strategy != BTEqualStrategyNumber)
					continue;
				clause_strategy = BTNE;
			}
			else
				continue;
		}
		else
			continue;

		/*
		 * Look up the "test" strategy number in the implication table
		 */
		if (refute_it)
			test_strategy = BT_refute_table[clause_strategy - 1][pred_strategy - 1];
		else
			test_strategy = BT_implic_table[clause_strategy - 1][pred_strategy - 1];

		if (test_strategy == 0)
		{
			/* Can't determine implication using this interpretation */
			continue;
		}

		/*
		 * See if opclass has an operator for the test strategy and the clause
		 * datatype.
		 */
		if (test_strategy == BTNE)
		{
			test_op = get_opclass_member(opclass_id, clause_subtype,
										 BTEqualStrategyNumber);
			if (OidIsValid(test_op))
				test_op = get_negator(test_op);
		}
		else
		{
			test_op = get_opclass_member(opclass_id, clause_subtype,
										 test_strategy);
		}
		if (OidIsValid(test_op))
		{
			/*
			 * Last check: test_op must be immutable.
			 *
			 * Note that we require only the test_op to be immutable, not the
			 * original clause_op.	(pred_op is assumed to have been checked
			 * immutable by the caller.)  Essentially we are assuming that the
			 * opclass is consistent even if it contains operators that are
			 * merely stable.
			 */
			if (op_volatile(test_op) == PROVOLATILE_IMMUTABLE)
			{
				found = true;
				break;
			}
		}
	}

	ReleaseSysCacheList(catlist);

	if (!found)
	{
		/* couldn't find a btree opclass to interpret the operators */
		return false;
	}

	/*
	 * Evaluate the test.  For this we need an EState.
	 */
	estate = CreateExecutorState();

	/* We can use the estate's working context to avoid memory leaks. */
	oldcontext = MemoryContextSwitchTo(estate->es_query_cxt);

	/* Build expression tree */
	test_expr = make_opclause(test_op,
							  BOOLOID,
							  false,
							  (Expr *) pred_const,
							  (Expr *) clause_const);

	/* Prepare it for execution */
	test_exprstate = ExecPrepareExpr(test_expr, estate);

	/* And execute it. */
	test_result = ExecEvalExprSwitchContext(test_exprstate,
											GetPerTupleExprContext(estate),
											&isNull, NULL);

	/* Get back to outer memory context */
	MemoryContextSwitchTo(oldcontext);

	/* Release all the junk we just created */
	FreeExecutorState(estate);

	if (isNull)
	{
		/* Treat a null result as non-proof ... but it's a tad fishy ... */
		elog(DEBUG2, "null predicate test result");
		return false;
	}
	return DatumGetBool(test_result);
}
