predtest.c

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/*-------------------------------------------------------------------------
 *
 * predtest.c
 *    Routines to attempt to prove logical implications between predicate
 *    expressions.
 *
 * Portions Copyright (c) 1996-2013, PostgreSQL Global Development Group
 * Portions Copyright (c) 1994, Regents of the University of California
 *
 *
 * IDENTIFICATION
 *    src/backend/optimizer/util/predtest.c
 *
 *-------------------------------------------------------------------------
 */
#include "postgres.h"

#include "catalog/pg_proc.h"
#include "catalog/pg_type.h"
#include "executor/executor.h"
#include "miscadmin.h"
#include "optimizer/clauses.h"
#include "optimizer/planmain.h"
#include "optimizer/predtest.h"
#include "utils/array.h"
#include "utils/inval.h"
#include "utils/lsyscache.h"
#include "utils/syscache.h"


/*
 * Proof attempts involving large arrays in ScalarArrayOpExpr nodes are
 * likely to require O(N^2) time, and more often than not fail anyway.
 * So we set an arbitrary limit on the number of array elements that
 * we will allow to be treated as an AND or OR clause.
 * XXX is it worth exposing this as a GUC knob?
 */
#define MAX_SAOP_ARRAY_SIZE     100

/*
 * 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 Node *extract_not_arg(Node *clause);
static Node *extract_strong_not_arg(Node *clause);
static bool list_member_strip(List *list, Expr *datum);
static bool btree_predicate_proof(Expr *predicate, Node *clause,
                      bool refute_it);
static Oid  get_btree_test_op(Oid pred_op, Oid clause_op, bool refute_it);
static void InvalidateOprProofCacheCallBack(Datum arg, int cacheid, uint32 hashvalue);


/*
 * 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)
{
    Node       *p,
               *r;

    if (predicate_list == NIL)
        return true;            /* no predicate: implication is vacuous */
    if (restrictinfo_list == NIL)
        return false;           /* no restriction: implication must fail */

    /*
     * If either input is a single-element list, replace it with its lone
     * member; this avoids one useless level of AND-recursion.  We only need
     * to worry about this at top level, since eval_const_expressions should
     * have gotten rid of any trivial ANDs or ORs below that.
     */
    if (list_length(predicate_list) == 1)
        p = (Node *) linitial(predicate_list);
    else
        p = (Node *) predicate_list;
    if (list_length(restrictinfo_list) == 1)
        r = (Node *) linitial(restrictinfo_list);
    else
        r = (Node *) restrictinfo_list;

    /* And away we go ... */
    return predicate_implied_by_recurse(r, p);
}

/*
 * 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.
 *
 * An important fine point is that truth of the clauses must imply that
 * the predicate returns FALSE, not that it does not return TRUE.  This
 * is normally used to try to refute CHECK constraints, and the only
 * thing we can assume about a CHECK constraint is that it didn't return
 * FALSE --- a NULL result isn't a violation per the SQL spec.  (Someday
 * perhaps this code should be extended to support both "strong" and
 * "weak" refutation, but for now we only need "strong".)
 *
 * 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)
{
    Node       *p,
               *r;

    if (predicate_list == NIL)
        return false;           /* no predicate: no refutation is possible */
    if (restrictinfo_list == NIL)
        return false;           /* no restriction: refutation must fail */

    /*
     * If either input is a single-element list, replace it with its lone
     * member; this avoids one useless level of AND-recursion.  We only need
     * to worry about this at top level, since eval_const_expressions should
     * have gotten rid of any trivial ANDs or ORs below that.
     */
    if (list_length(predicate_list) == 1)
        p = (Node *) linitial(predicate_list);
    else
        p = (Node *) predicate_list;
    if (list_length(restrictinfo_list) == 1)
        r = (Node *) linitial(restrictinfo_list);
    else
        r = (Node *) restrictinfo_list;

    /* And away we go ... */
    return predicate_refuted_by_recurse(r, p);
}

/*----------
 * 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
 *
 * In addition, if the predicate is a NOT-clause then we can use
 *  A R=> NOT B if:                 A => B
 * This works for several different SQL constructs that assert the non-truth
 * of their argument, ie NOT, IS FALSE, IS NOT TRUE, IS UNKNOWN.
 * Unfortunately we *cannot* use
 *  NOT A R=> B if:                 B => A
 * because this type of reasoning fails to prove that B doesn't yield NULL.
 * We can however make the more limited deduction that
 *  NOT A R=> A
 *
 * 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;
    Node       *not_arg;
    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:

                    /*
                     * If B is a NOT-clause, A R=> B if A => B's arg
                     */
                    not_arg = extract_not_arg(predicate);
                    if (not_arg &&
                        predicate_implied_by_recurse(clause, not_arg))
                        return true;

                    /*
                     * 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:

                    /*
                     * If B is a NOT-clause, A R=> B if A => B's arg
                     */
                    not_arg = extract_not_arg(predicate);
                    if (not_arg &&
                        predicate_implied_by_recurse(clause, not_arg))
                        return true;

                    /*
                     * 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:

            /*
             * If A is a strong NOT-clause, A R=> B if B equals A's arg
             *
             * We cannot make the stronger conclusion that B is refuted if B
             * implies A's arg; that would only prove that B is not-TRUE, not
             * that it's not NULL either.  Hence use equal() rather than
             * predicate_implied_by_recurse().  We could do the latter if we
             * ever had a need for the weak form of refutation.
             */
            not_arg = extract_strong_not_arg(clause);
            if (not_arg && equal(predicate, not_arg))
                return true;

            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:

                    /*
                     * If B is a NOT-clause, A R=> B if A => B's arg
                     */
                    not_arg = extract_not_arg(predicate);
                    if (not_arg &&
                        predicate_implied_by_recurse(clause, not_arg))
                        return true;

                    /*
                     * 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.
 *
 * This function also implements enforcement of MAX_SAOP_ARRAY_SIZE: if a
 * ScalarArrayOpExpr's array has too many elements, we just classify it as an
 * atom.  (This will result in its being passed as-is to the simple_clause
 * functions, which will fail to prove anything about it.)  Note that we
 * cannot just stop after considering MAX_SAOP_ARRAY_SIZE elements; in general
 * that would result in wrong proofs, rather than failing to prove anything.
 */
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)
        {
            ArrayType  *arrayval;
            int         nelems;

            arrayval = DatumGetArrayTypeP(((Const *) arraynode)->constvalue);
            nelems = ArrayGetNItems(ARR_NDIM(arrayval), ARR_DIMS(arrayval));
            if (nelems <= MAX_SAOP_ARRAY_SIZE)
            {
                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;
            }
        }
        else if (arraynode && IsA(arraynode, ArrayExpr) &&
                 !((ArrayExpr *) arraynode)->multidims &&
                 list_length(((ArrayExpr *) arraynode)->elements) <= MAX_SAOP_ARRAY_SIZE)
        {
            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.opcollid = InvalidOid;
    state->opexpr.inputcollid = saop->inputcollid;
    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.consttypmod = -1;
    state->constexpr.constcollid = arrayconst->constcollid;
    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.opcollid = InvalidOid;
    state->opexpr.inputcollid = saop->inputcollid;
    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 families; this is encapsulated in btree_predicate_proof().
 *----------
 */
static bool
predicate_implied_by_simple_clause(Expr *predicate, Node *clause)
{
    /* Allow interrupting long proof attempts */
    CHECK_FOR_INTERRUPTS();

    /* 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;

        /* row IS NOT NULL does not act in the simple way we have in mind */
        if (!((NullTest *) predicate)->argisrow)
        {
            if (is_opclause(clause) &&
                list_member_strip(((OpExpr *) clause)->args, nonnullarg) &&
                op_strict(((OpExpr *) clause)->opno))
                return true;
            if (is_funcclause(clause) &&
                list_member_strip(((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.
 *
 * 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), or if the
 * clause is "foo IS NOT NULL".  A clause "foo IS NULL" refutes a predicate
 * "foo IS NOT NULL", but unfortunately does not refute strict predicates,
 * because we are looking for strong refutation.  (The motivation for covering
 * these cases is to support using IS NULL/IS NOT NULL as partition-defining
 * constraints.)
 *
 * Finally, we may be able to deduce something using knowledge about btree
 * operator families; this is encapsulated in btree_predicate_proof().
 *----------
 */
static bool
predicate_refuted_by_simple_clause(Expr *predicate, Node *clause)
{
    /* Allow interrupting long proof attempts */
    CHECK_FOR_INTERRUPTS();

    /* A simple clause can't refute itself */
    /* Worth checking because of relation_excluded_by_constraints() */
    if ((Node *) predicate == clause)
        return false;

    /* Try the predicate-IS-NULL case */
    if (predicate && IsA(predicate, NullTest) &&
        ((NullTest *) predicate)->nulltesttype == IS_NULL)
    {
        Expr       *isnullarg = ((NullTest *) predicate)->arg;

        /* row IS NULL does not act in the simple way we have in mind */
        if (((NullTest *) predicate)->argisrow)
            return false;

        /* Any strict op/func on foo refutes foo IS NULL */
        if (is_opclause(clause) &&
            list_member_strip(((OpExpr *) clause)->args, isnullarg) &&
            op_strict(((OpExpr *) clause)->opno))
            return true;
        if (is_funcclause(clause) &&
            list_member_strip(((FuncExpr *) clause)->args, isnullarg) &&
            func_strict(((FuncExpr *) clause)->funcid))
            return true;

        /* foo IS NOT NULL refutes foo IS NULL */
        if (clause && IsA(clause, NullTest) &&
            ((NullTest *) clause)->nulltesttype == IS_NOT_NULL &&
            !((NullTest *) clause)->argisrow &&
            equal(((NullTest *) clause)->arg, isnullarg))
            return true;

        return false;           /* we can't succeed below... */
    }

    /* Try the clause-IS-NULL case */
    if (clause && IsA(clause, NullTest) &&
        ((NullTest *) clause)->nulltesttype == IS_NULL)
    {
        Expr       *isnullarg = ((NullTest *) clause)->arg;

        /* row IS NULL does not act in the simple way we have in mind */
        if (((NullTest *) clause)->argisrow)
            return false;

        /* foo IS NULL refutes foo IS NOT NULL */
        if (predicate && IsA(predicate, NullTest) &&
            ((NullTest *) predicate)->nulltesttype == IS_NOT_NULL &&
            !((NullTest *) predicate)->argisrow &&
            equal(((NullTest *) predicate)->arg, isnullarg))
            return true;

        return false;           /* we can't succeed below... */
    }

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


/*
 * If clause asserts the non-truth of a subclause, return that subclause;
 * otherwise return NULL.
 */
static Node *
extract_not_arg(Node *clause)
{
    if (clause == NULL)
        return NULL;
    if (IsA(clause, BoolExpr))
    {
        BoolExpr   *bexpr = (BoolExpr *) clause;

        if (bexpr->boolop == NOT_EXPR)
            return (Node *) linitial(bexpr->args);
    }
    else if (IsA(clause, BooleanTest))
    {
        BooleanTest *btest = (BooleanTest *) clause;

        if (btest->booltesttype == IS_NOT_TRUE ||
            btest->booltesttype == IS_FALSE ||
            btest->booltesttype == IS_UNKNOWN)
            return (Node *) btest->arg;
    }
    return NULL;
}

/*
 * If clause asserts the falsity of a subclause, return that subclause;
 * otherwise return NULL.
 */
static Node *
extract_strong_not_arg(Node *clause)
{
    if (clause == NULL)
        return NULL;
    if (IsA(clause, BoolExpr))
    {
        BoolExpr   *bexpr = (BoolExpr *) clause;

        if (bexpr->boolop == NOT_EXPR)
            return (Node *) linitial(bexpr->args);
    }
    else if (IsA(clause, BooleanTest))
    {
        BooleanTest *btest = (BooleanTest *) clause;

        if (btest->booltesttype == IS_FALSE)
            return (Node *) btest->arg;
    }
    return NULL;
}


/*
 * Check whether an Expr is equal() to any member of a list, ignoring
 * any top-level RelabelType nodes.  This is legitimate for the purposes
 * we use it for (matching IS [NOT] NULL arguments to arguments of strict
 * functions) because RelabelType doesn't change null-ness.  It's helpful
 * for cases such as a varchar argument of a strict function on text.
 */
static bool
list_member_strip(List *list, Expr *datum)
{
    ListCell   *cell;

    if (datum && IsA(datum, RelabelType))
        datum = ((RelabelType *) datum)->arg;

    foreach(cell, list)
    {
        Expr       *elem = (Expr *) lfirst(cell);

        if (elem && IsA(elem, RelabelType))
            elem = ((RelabelType *) elem)->arg;

        if (equal(elem, datum))
            return true;
    }

    return false;
}


/*
 * 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 an equality operator of a btree
 * opfamily has a negator operator, the negator behaves as <> for the opfamily.
 * (This convention is also known to get_op_btree_interpretation().)
 *
 * 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 ROWCOMPARE_NE

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 family.  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 opfamilies 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;
    Oid         pred_collation,
                clause_collation;
    Oid         pred_op,
                clause_op,
                test_op;
    Expr       *test_expr;
    ExprState  *test_exprstate;
    Datum       test_result;
    bool        isNull;
    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;

    /*
     * They'd better have the same collation, too.
     */
    pred_collation = ((OpExpr *) predicate)->inputcollid;
    clause_collation = ((OpExpr *) clause)->inputcollid;
    if (pred_collation != clause_collation)
        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;
    }

    /*
     * Lookup the comparison operator using the system catalogs and the
     * operator implication tables.
     */
    test_op = get_btree_test_op(pred_op, clause_op, refute_it);

    if (!OidIsValid(test_op))
    {
        /* couldn't find a suitable comparison operator */
        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,
                              InvalidOid,
                              pred_collation);

    /* Fill in opfuncids */
    fix_opfuncids((Node *) test_expr);

    /* Prepare it for execution */
    test_exprstate = ExecInitExpr(test_expr, NULL);

    /* 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);
}


/*
 * We use a lookaside table to cache the result of btree proof operator
 * lookups, since the actual lookup is pretty expensive and doesn't change
 * for any given pair of operators (at least as long as pg_amop doesn't
 * change).  A single hash entry stores both positive and negative results
 * for a given pair of operators.
 */
typedef struct OprProofCacheKey
{
    Oid         pred_op;        /* predicate operator */
    Oid         clause_op;      /* clause operator */
} OprProofCacheKey;

typedef struct OprProofCacheEntry
{
    /* the hash lookup key MUST BE FIRST */
    OprProofCacheKey key;

    bool        have_implic;    /* do we know the implication result? */
    bool        have_refute;    /* do we know the refutation result? */
    Oid         implic_test_op; /* OID of the operator, or 0 if none */
    Oid         refute_test_op; /* OID of the operator, or 0 if none */
} OprProofCacheEntry;

static HTAB *OprProofCacheHash = NULL;


/*
 * get_btree_test_op
 *    Identify the comparison operator needed for a btree-operator
 *    proof or refutation.
 *
 * Given the truth of a predicate "var pred_op const1", we are attempting to
 * prove or refute a clause "var clause_op const2".  The identities of the two
 * operators are sufficient to determine the operator (if any) to compare
 * const2 to const1 with.
 *
 * Returns the OID of the operator to use, or InvalidOid if no proof is
 * possible.
 */
static Oid
get_btree_test_op(Oid pred_op, Oid clause_op, bool refute_it)
{
    OprProofCacheKey key;
    OprProofCacheEntry *cache_entry;
    bool        cfound;
    Oid         test_op = InvalidOid;
    bool        found = false;
    List       *pred_op_infos,
               *clause_op_infos;
    ListCell   *lcp,
               *lcc;

    /*
     * Find or make a cache entry for this pair of operators.
     */
    if (OprProofCacheHash == NULL)
    {
        /* First time through: initialize the hash table */
        HASHCTL     ctl;

        MemSet(&ctl, 0, sizeof(ctl));
        ctl.keysize = sizeof(OprProofCacheKey);
        ctl.entrysize = sizeof(OprProofCacheEntry);
        ctl.hash = tag_hash;
        OprProofCacheHash = hash_create("Btree proof lookup cache", 256,
                                        &ctl, HASH_ELEM | HASH_FUNCTION);

        /* Arrange to flush cache on pg_amop changes */
        CacheRegisterSyscacheCallback(AMOPOPID,
                                      InvalidateOprProofCacheCallBack,
                                      (Datum) 0);
    }

    key.pred_op = pred_op;
    key.clause_op = clause_op;
    cache_entry = (OprProofCacheEntry *) hash_search(OprProofCacheHash,
                                                     (void *) &key,
                                                     HASH_ENTER, &cfound);
    if (!cfound)
    {
        /* new cache entry, set it invalid */
        cache_entry->have_implic = false;
        cache_entry->have_refute = false;
    }
    else
    {
        /* pre-existing cache entry, see if we know the answer */
        if (refute_it)
        {
            if (cache_entry->have_refute)
                return cache_entry->refute_test_op;
        }
        else
        {
            if (cache_entry->have_implic)
                return cache_entry->implic_test_op;
        }
    }

    /*
     * Try to find a btree opfamily containing the given operators.
     *
     * We must find a btree opfamily that contains both operators, else the
     * implication can't be determined.  Also, the opfamily must contain a
     * suitable test operator taking the operators' righthand datatypes.
     *
     * If there are multiple matching opfamilies, 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 opfamilies.
     */
    clause_op_infos = get_op_btree_interpretation(clause_op);
    if (clause_op_infos)
        pred_op_infos = get_op_btree_interpretation(pred_op);
    else    /* no point in looking */
        pred_op_infos = NIL;

    foreach(lcp, pred_op_infos)
    {
        OpBtreeInterpretation *pred_op_info = lfirst(lcp);
        Oid         opfamily_id = pred_op_info->opfamily_id;

        foreach(lcc, clause_op_infos)
        {
            OpBtreeInterpretation *clause_op_info = lfirst(lcc);
            StrategyNumber pred_strategy,
                        clause_strategy,
                        test_strategy;

            /* Must find them in same opfamily */
            if (opfamily_id != clause_op_info->opfamily_id)
                continue;
            /* Lefttypes should match */
            Assert(clause_op_info->oplefttype == pred_op_info->oplefttype);

            pred_strategy = pred_op_info->strategy;
            clause_strategy = clause_op_info->strategy;

            /*
             * 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 opfamily has an operator for the test strategy and the
             * datatypes.
             */
            if (test_strategy == BTNE)
            {
                test_op = get_opfamily_member(opfamily_id,
                                              pred_op_info->oprighttype,
                                              clause_op_info->oprighttype,
                                              BTEqualStrategyNumber);
                if (OidIsValid(test_op))
                    test_op = get_negator(test_op);
            }
            else
            {
                test_op = get_opfamily_member(opfamily_id,
                                              pred_op_info->oprighttype,
                                              clause_op_info->oprighttype,
                                              test_strategy);
            }

            if (!OidIsValid(test_op))
                continue;

            /*
             * 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
             * opfamily is consistent even if it contains operators that are
             * merely stable.
             */
            if (op_volatile(test_op) == PROVOLATILE_IMMUTABLE)
            {
                found = true;
                break;
            }
        }

        if (found)
            break;
    }

    list_free_deep(pred_op_infos);
    list_free_deep(clause_op_infos);

    if (!found)
    {
        /* couldn't find a suitable comparison operator */
        test_op = InvalidOid;
    }

    /* Cache the result, whether positive or negative */
    if (refute_it)
    {
        cache_entry->refute_test_op = test_op;
        cache_entry->have_refute = true;
    }
    else
    {
        cache_entry->implic_test_op = test_op;
        cache_entry->have_implic = true;
    }

    return test_op;
}


/*
 * Callback for pg_amop inval events
 */
static void
InvalidateOprProofCacheCallBack(Datum arg, int cacheid, uint32 hashvalue)
{
    HASH_SEQ_STATUS status;
    OprProofCacheEntry *hentry;

    Assert(OprProofCacheHash != NULL);

    /* Currently we just reset all entries; hard to be smarter ... */
    hash_seq_init(&status, OprProofCacheHash);

    while ((hentry = (OprProofCacheEntry *) hash_seq_search(&status)) != NULL)
    {
        hentry->have_implic = false;
        hentry->have_refute = false;
    }
}