mariadb/mysql-test/t/func_group.test

#
# simple test of all group functions
#

--disable_warnings
drop table if exists t1,t2;
--enable_warnings

set @sav_dpi= @@div_precision_increment;
set div_precision_increment= 5;
show variables like 'div_precision_increment';
create table t1 (grp int, a bigint unsigned, c char(10) not null);
insert into t1 values (1,1,"a");
insert into t1 values (2,2,"b");
insert into t1 values (2,3,"c");
insert into t1 values (3,4,"E");
insert into t1 values (3,5,"C");
insert into t1 values (3,6,"D");

# Test of MySQL field extension with and without matching records.
select a,c,sum(a) from t1 group by a;
select a,c,sum(a) from t1 where a > 10 group by a;
select sum(a) from t1 where a > 10;
select a from t1 order by rand(10);
select distinct a from t1 order by rand(10);
select count(distinct a),count(distinct grp) from t1;
insert into t1 values (null,null,'');
select count(distinct a),count(distinct grp) from t1;

select sum(all a),count(all a),avg(all a),std(all a),variance(all a),bit_or(all a),bit_and(all a),min(all a),max(all a),min(all c),max(all c) from t1;
select grp, sum(a),count(a),avg(a),std(a),variance(a),bit_or(a),bit_and(a),min(a),max(a),min(c),max(c) from t1 group by grp;
select grp, sum(a)+count(a)+avg(a)+std(a)+variance(a)+bit_or(a)+bit_and(a)+min(a)+max(a)+min(c)+max(c) as sum from t1 group by grp;

create table t2 (grp int, a bigint unsigned, c char(10));
insert into t2 select grp,max(a)+max(grp),max(c) from t1 group by grp;

# REPLACE ... SELECT doesn't yet work with PS
replace into t2 select grp, a, c from t1 limit 2,1;
select * from t2;

drop table t1,t2;

#
# Problem with std()
#

CREATE TABLE t1 (id int(11),value1 float(10,2));
INSERT INTO t1 VALUES (1,0.00),(1,1.00), (1,2.00), (2,10.00), (2,11.00), (2,12.00); 
CREATE TABLE t2 (id int(11),name char(20)); 
INSERT INTO t2 VALUES (1,'Set One'),(2,'Set Two'); 
select id, avg(value1), std(value1), variance(value1) from t1 group by id;
select name, avg(value1), std(value1), variance(value1) from t1, t2 where t1.id = t2.id group by t1.id;
drop table t1,t2;

#
# Test of bug in left join & avg
#

create table t1 (id int not null);
create table t2 (id int not null,rating int null);
insert into t1 values(1),(2),(3);
insert into t2 values(1, 3),(2, NULL),(2, NULL),(3, 2),(3, NULL);
select t1.id, avg(rating) from t1 left join t2 on ( t1.id = t2.id ) group by t1.id;
# Test different types with avg()
select sql_small_result t2.id, avg(rating) from t2 group by t2.id;
select sql_big_result t2.id, avg(rating) from t2 group by t2.id;
select sql_small_result t2.id, avg(rating+0.0e0) from t2 group by t2.id;
select sql_big_result t2.id, avg(rating+0.0e0) from t2 group by t2.id;
drop table t1,t2;

#
# test of count
#
create table t1 (a smallint(6) primary key, c char(10), b text);
INSERT INTO t1 VALUES (1,'1','1');
INSERT INTO t1 VALUES (2,'2','2');
INSERT INTO t1 VALUES (4,'4','4');

select count(*) from t1;
select count(*) from t1 where a = 1;
select count(*) from t1 where a = 100;
select count(*) from t1 where a >= 10;
select count(a) from t1 where a = 1;
select count(a) from t1 where a = 100;
select count(a) from t1 where a >= 10;
select count(b) from t1 where b >= 2;
select count(b) from t1 where b >= 10;
select count(c) from t1 where c = 10;
drop table t1;

#
# Test of bug in COUNT(i)*(i+0)
#

CREATE TABLE t1 (d DATETIME, i INT);
INSERT INTO t1 VALUES (NOW(), 1);
SELECT COUNT(i), i, COUNT(i)*i FROM t1 GROUP BY i;
SELECT COUNT(i), (i+0), COUNT(i)*(i+0) FROM t1 GROUP BY i; 
DROP TABLE t1;

#
# Another SUM() problem with 3.23.2
#

create table t1 (
        num float(5,2),
        user char(20)
);
insert into t1 values (10.3,'nem'),(20.53,'monty'),(30.23,'sinisa');
insert into t1 values (30.13,'nem'),(20.98,'monty'),(10.45,'sinisa');
insert into t1 values (5.2,'nem'),(8.64,'monty'),(11.12,'sinisa');
select sum(num) from t1;
select sum(num) from t1 group by user;
drop table t1;

#
# Test problem with MIN() optimization in case of null values
#

create table t1 (a1 int, a2 char(3), key k1(a1), key k2(a2));
insert into t1 values(10,'aaa'), (10,null), (10,'bbb'), (20,'zzz');
create table t2(a1 char(3), a2 int, a3 real, key k1(a1), key k2(a2, a1));
select * from t1;
# The following returned NULL in 4.0.10
select min(a2) from t1;
select max(t1.a1), max(t2.a2) from t1, t2;
select max(t1.a1) from t1, t2;
select max(t2.a2), max(t1.a1) from t1, t2;

explain select min(a2) from t1;
explain select max(t1.a1), max(t2.a2) from t1, t2;

insert into t2 values('AAA', 10, 0.5);
insert into t2 values('BBB', 20, 1.0);
select t1.a1, t1.a2, t2.a1, t2.a2 from t1,t2;

select max(t1.a1), max(t2.a1) from t1, t2 where t2.a2=9;
select max(t2.a1), max(t1.a1) from t1, t2 where t2.a2=9;
select t1.a1, t1.a2, t2.a1, t2.a2 from t1 left outer join t2 on t1.a1=10;
select max(t1.a2) from t1 left outer join t2 on t1.a1=10;
select max(t2.a1) from t2 left outer join t1 on t2.a2=10 where t2.a2=20;
select max(t2.a1) from t2 left outer join t1 on t2.a2=10 where t2.a2=10;
select max(t2.a1) from t1 left outer join t2 on t1.a2=t2.a1 and 1=0 where t2.a1='AAA';
select max(t1.a2),max(t2.a1) from t1 left outer join t2 on t1.a1=10;
drop table t1,t2;

#
# Test of group function and NULL values
#

CREATE TABLE t1 (a int, b int);
select count(b), sum(b), avg(b), std(b), min(b), max(b), bit_and(b), bit_or(b) from t1;
select a,count(b), sum(b), avg(b), std(b), min(b), max(b), bit_and(b), bit_or(b) from t1 group by a;
insert into t1 values (1,null);
select a,count(b), sum(b), avg(b), std(b), min(b), max(b), bit_and(b), bit_or(b) from t1 group by a;
insert into t1 values (1,null);
insert into t1 values (2,null);
select a,count(b), sum(b), avg(b), std(b), min(b), max(b), bit_and(b), bit_or(b) from t1 group by a;
select SQL_BIG_RESULT a,count(b), sum(b), avg(b), std(b), min(b), max(b), bit_and(b), bit_or(b) from t1 group by a;
insert into t1 values (2,1);
select a,count(b), sum(b), avg(b), std(b), min(b), max(b), bit_and(b), bit_or(b) from t1 group by a;
select SQL_BIG_RESULT a,count(b), sum(b), avg(b), std(b), min(b), max(b), bit_and(b), bit_or(b) from t1 group by a;
insert into t1 values (3,1);
select a,count(b), sum(b), avg(b), std(b), min(b), max(b), bit_and(b), bit_or(b) from t1 group by a;
select SQL_BIG_RESULT a,count(b), sum(b), avg(b), std(b), min(b), max(b), bit_and(b), bit_or(b), bit_xor(b) from t1 group by a;
explain extended select SQL_BIG_RESULT a,count(b), sum(b), avg(b), std(b), min(b), max(b), bit_and(b), bit_or(b), bit_xor(b) from t1 group by a;
drop table t1;

#
# Bug #1972: test for bit_and(), bit_or() and negative values
# 
create table t1 (col int);
insert into t1 values (-1), (-2), (-3);
select bit_and(col), bit_or(col) from t1;
select SQL_BIG_RESULT bit_and(col), bit_or(col) from t1 group by col;
drop table t1;

#
# Bug #3376: avg() and an empty table
#

create table t1 (a int);
select avg(2) from t1;
drop table t1;

#
# Tests to check MIN/MAX query optimization
#

# Create database schema
create table t1(
  a1 char(3) primary key,
  a2 smallint,
  a3 char(3),
  a4 real,
  a5 date,
  key k1(a2,a3),
  key k2(a4 desc,a1),
  key k3(a5,a1)
);
create table t2(
  a1 char(3) primary key,
  a2 char(17),
  a3 char(2),
  a4 char(3),
  key k1(a3, a2),
  key k2(a4)
);

# Populate table t1
insert into t1 values('AME',0,'SEA',0.100,date'1942-02-19');
insert into t1 values('HBR',1,'SEA',0.085,date'1948-03-05');
insert into t1 values('BOT',2,'SEA',0.085,date'1951-11-29');
insert into t1 values('BMC',3,'SEA',0.085,date'1958-09-08');
insert into t1 values('TWU',0,'LAX',0.080,date'1969-10-05');
insert into t1 values('BDL',0,'DEN',0.080,date'1960-11-27');
insert into t1 values('DTX',1,'NYC',0.080,date'1961-05-04');
insert into t1 values('PLS',1,'WDC',0.075,date'1949-01-02');
insert into t1 values('ZAJ',2,'CHI',0.075,date'1960-06-15');
insert into t1 values('VVV',2,'MIN',0.075,date'1959-06-28');
insert into t1 values('GTM',3,'DAL',0.070,date'1977-09-23');
insert into t1 values('SSJ',null,'CHI',null,date'1974-03-19');
insert into t1 values('KKK',3,'ATL',null,null);
insert into t1 values('XXX',null,'MIN',null,null);
insert into t1 values('WWW',1,'LED',null,null);

# Populate table t2
insert into t2 values('TKF','Seattle','WA','AME');
insert into t2 values('LCC','Los Angeles','CA','TWU');
insert into t2 values('DEN','Denver','CO','BDL');
insert into t2 values('SDC','San Diego','CA','TWU');
insert into t2 values('NOL','New Orleans','LA','GTM');
insert into t2 values('LAK','Los Angeles','CA','TWU');
insert into t2 values('AAA','AAA','AA','AME');

# Show the table contents
select * from t1;
select * from t2;

# Queries with min/max functions 
# which regular min/max optimization are applied to

explain 
select min(a1) from t1;
select min(a1) from t1;
explain 
select max(a4) from t1;
select max(a4) from t1;
explain 
select min(a5), max(a5) from t1;
select min(a5), max(a5) from t1;
explain 
select min(a3) from t1 where a2 = 2;
select min(a3) from t1 where a2 = 2;
explain 
select min(a1), max(a1) from t1 where a4 = 0.080;
select min(a1), max(a1) from t1 where a4 = 0.080;

explain 
select min(t1.a5), max(t2.a3) from t1, t2;
select min(t1.a5), max(t2.a3) from t1, t2;
explain 
select min(t1.a3), max(t2.a2) from t1, t2 where t1.a2 = 0 and t2.a3 = 'CA';
select min(t1.a3), max(t2.a2) from t1, t2 where t1.a2 = 0 and t2.a3 = 'CA';

# Queries with min/max functions 
# which extended min/max optimization are applied to

explain 
select min(a1) from t1 where a1 > 'KKK';
select min(a1) from t1 where a1 > 'KKK';
explain 
select min(a1) from t1 where a1 >= 'KKK';
select min(a1) from t1 where a1 >= 'KKK';
explain 
select max(a3) from t1 where a2 = 2 and a3 < 'SEA';
select max(a3) from t1 where a2 = 2 and a3 < 'SEA';
explain 
select max(a5) from t1 where a5 < date'1970-01-01';
select max(a5) from t1 where a5 < date'1970-01-01';
explain 
select max(a3) from t1 where a2 is null;
select max(a3) from t1 where a2 is null;
explain 
select max(a3) from t1 where a2 = 0 and a3 between 'K' and 'Q';
select max(a3) from t1 where a2 = 0 and a3 between 'K' and 'Q';
explain
select min(a1), max(a1) from t1 where a1 between 'A' and 'P';
select min(a1), max(a1) from t1 where a1 between 'A' and 'P';
explain 
select max(a3) from t1 where a3 < 'SEA' and a2 = 2 and a3 <= 'MIN';
select max(a3) from t1 where a3 < 'SEA' and a2 = 2 and a3 <= 'MIN';
explain 
select max(a3) from t1 where a3 = 'MIN' and a2 = 2;
select max(a3) from t1 where a3 = 'MIN' and a2 = 2;
explain 
select max(a3) from t1 where a3 = 'DEN' and a2 = 2;
select max(a3) from t1 where a3 = 'DEN' and a2 = 2;

explain
select max(t1.a3), min(t2.a2) from t1, t2 where t1.a2 = 2 and t1.a3 < 'MIN' and t2.a3 = 'CA';
select max(t1.a3), min(t2.a2) from t1, t2 where t1.a2 = 2 and t1.a3 < 'MIN' and t2.a3 = 'CA';

explain
select max(a3) from t1 where a2 is null and a2 = 2;
select max(a3) from t1 where a2 is null and a2 = 2;

explain
select max(a2) from t1 where a2 >= 1;
select max(a2) from t1 where a2 >= 1;
explain
select min(a3) from t1 where a2 = 2 and a3 < 'SEA';
select min(a3) from t1 where a2 = 2 and a3 < 'SEA';

explain
select min(a3) from t1 where a2 = 4;
select min(a3) from t1 where a2 = 4;
explain
select min(a3) from t1 where a2 = 2 and a3 > 'SEA';
select min(a3) from t1 where a2 = 2 and a3 > 'SEA';
explain
select (min(a4)+max(a4))/2 from t1;
select (min(a4)+max(a4))/2 from t1;
explain
select min(a3) from t1 where 2 = a2;
select min(a3) from t1 where 2 = a2;
explain
select max(a3) from t1 where a2 = 2 and 'SEA' > a3;
select max(a3) from t1 where a2 = 2 and 'SEA' > a3;
explain
select max(a3) from t1 where a2 = 2 and 'SEA' < a3;
select max(a3) from t1 where a2 = 2 and 'SEA' < a3;
explain
select min(a3) from t1 where a2 = 2 and a3 >= 'CHI';
select min(a3) from t1 where a2 = 2 and a3 >= 'CHI';
explain
select min(a3) from t1 where a2 = 2 and a3 >= 'CHI' and a3 < 'SEA';
select min(a3) from t1 where a2 = 2 and a3 >= 'CHI' and a3 < 'SEA';
explain
select min(a3) from t1 where a2 = 2 and a3 >= 'CHI' and a3 = 'MIN';
select min(a3) from t1 where a2 = 2 and a3 >= 'CHI' and a3 = 'MIN';
explain
select min(a3) from t1 where a2 = 2 and a3 >= 'SEA' and a3 = 'MIN';
select min(a3) from t1 where a2 = 2 and a3 >= 'SEA' and a3 = 'MIN';

explain
select min(t1.a1), min(t2.a4) from t1,t2 where t1.a1 < 'KKK' and t2.a4 < 'KKK';
select min(t1.a1), min(t2.a4) from t1,t2 where t1.a1 < 'KKK' and t2.a4 < 'KKK';

# Queries to which max/min optimization is not applied

explain 
select min(a1) from t1 where a1 > 'KKK' or a1 < 'XXX';
explain 
select min(a1) from t1 where a1 != 'KKK';
explain
select max(a3) from t1 where a2 < 2 and a3 < 'SEA';
explain
select max(t1.a3), min(t2.a2) from t1, t2 where t1.a2 = 2 and t1.a3 < 'MIN' and t2.a3 > 'CA';

explain
select min(a4 - 0.01) from t1;
explain
select max(a4 + 0.01) from t1;
explain
select min(a3) from t1 where (a2 +1 ) is null;
explain
select min(a3) from t1 where (a2 + 1) = 2;
explain
select min(a3) from t1 where 2 = (a2 + 1);
explain
select min(a2) from t1 where a2 < 2 * a2 - 8;
explain
select min(a1) from t1  where a1 between a3 and 'KKK';
explain
select min(a4) from t1  where (a4 + 0.01) between 0.07 and 0.08;
explain
select concat(min(t1.a1),min(t2.a4)) from t1, t2 where t2.a4 <> 'AME';
drop table t1, t2;

--disable_warnings
create table t1 (USR_ID integer not null, MAX_REQ integer not null, constraint PK_SEA_USER primary key (USR_ID)) engine=InnoDB;
--enable_warnings
insert into t1 values (1, 3);
select count(*) + MAX_REQ - MAX_REQ + MAX_REQ - MAX_REQ + MAX_REQ - MAX_REQ + MAX_REQ - MAX_REQ + MAX_REQ - MAX_REQ from t1 group by MAX_REQ;
select Case When Count(*) < MAX_REQ Then 1 Else 0 End from t1 where t1.USR_ID = 1 group by MAX_REQ;
drop table t1;

create table t1 (a char(10));
insert into t1 values ('a'),('b'),('c');
select coercibility(max(a)) from t1;
drop table t1;

#
# Bug #6658 MAX(column) returns incorrect coercibility
#
create table t1 (a char character set latin2);
insert into t1 values ('a'),('b');
select charset(max(a)), coercibility(max(a)),
       charset(min(a)), coercibility(min(a)) from t1;
show create table t1;
create table t2 select max(a),min(a) from t1;
show create table t2;
drop table t2;
create table t2 select concat(a) from t1;
show create table t2;
drop table t2,t1;

#
# aggregate functions on static tables
#
create table t1 (a int);
insert into t1 values (1);
select max(a) as b from t1 having b=1;
select a from t1 having a=1;
drop table t1;

#
# Bug #3435: variance(const), stddev(const) and an empty table
#

create table t1 (a int);
select variance(2) from t1;
select stddev(2) from t1;
drop table t1;


#
# cleunup() of optimized away count(*) and max/min
#
create table t1 (a int);
insert into t1 values (1),(2);
prepare stmt1 from 'SELECT COUNT(*) FROM t1';
execute stmt1;
execute stmt1;
execute stmt1;
deallocate prepare stmt1;
drop table t1;

create table t1 (a int, primary key(a));
insert into t1 values (1),(2);
prepare stmt1 from 'SELECT max(a) FROM t1';
execute stmt1;
execute stmt1;
execute stmt1;
deallocate prepare stmt1;
drop table t1;

#
# Bug #5406 min/max optimization for empty set
#

CREATE TABLE t1 (a int primary key);
INSERT INTO t1 VALUES (1),(2),(3),(4);

SELECT MAX(a) FROM t1 WHERE a > 5;
SELECT MIN(a) FROM t1 WHERE a < 0;

DROP TABLE t1;

#
# Bug #5555 GROUP BY enum_field" returns incorrect results
#
 
CREATE TABLE t1 (
  id int(10) unsigned NOT NULL auto_increment,
  val enum('one','two','three') NOT NULL default 'one',
  PRIMARY KEY  (id)
) ENGINE=MyISAM DEFAULT CHARSET=utf8;
 
INSERT INTO t1 VALUES
(1,'one'),(2,'two'),(3,'three'),(4,'one'),(5,'two');
 
select val, count(*) from t1 group by val;
drop table t1;

CREATE TABLE t1 (
  id int(10) unsigned NOT NULL auto_increment,
  val set('one','two','three') NOT NULL default 'one',
  PRIMARY KEY  (id)
) ENGINE=MyISAM DEFAULT CHARSET=utf8;

INSERT INTO t1 VALUES
(1,'one'),(2,'two'),(3,'three'),(4,'one'),(5,'two');

select val, count(*) from t1 group by val;
drop table t1;

#
# Bug #5615: type of aggregate function column wrong when using group by
#

create table t1(a int, b datetime);
insert into t1 values (1, NOW()), (2, NOW());
create table t2 select MAX(b) from t1 group by a;
show create table t2;
drop table t1, t2;

#
# Bug 7833:  Wrong datatype of aggregate column is returned
#

create table t1(f1 datetime);
insert into t1 values (now());
create table t2 select f2 from (select max(now()) f2 from t1) a;
show columns from t2;
drop table t2;
create table t2 select f2 from (select now() f2 from t1) a;
show columns from t2;
drop table t2, t1;

#
# Bug 8893: wrong result for min/max optimization with 2 indexes
#

CREATE TABLE t1(
  id int PRIMARY KEY,
  a  int,
  b  int,
  INDEX i_b_id(a,b,id),
  INDEX i_id(a,id)
);
INSERT INTO t1 VALUES 
  (1,1,4), (2,2,1), (3,1,3), (4,2,1), (5,1,1);
SELECT MAX(id) FROM t1 WHERE id < 3 AND a=2 AND b=6;
DROP TABLE t1;

# change the order of the last two index definitions

CREATE TABLE t1(
  id int PRIMARY KEY,
  a  int,
  b  int,
  INDEX i_id(a,id),
  INDEX i_b_id(a,b,id)
);
INSERT INTO t1 VALUES 
  (1,1,4), (2,2,1), (3,1,3), (4,2,1), (5,1,1);
SELECT MAX(id) FROM t1 WHERE id < 3 AND a=2 AND b=6;
DROP TABLE t1;

# End of 4.1 tests

#
# decimal-related tests
#
create table t2 (ff double);
insert into t2 values (2.2);
select cast(sum(distinct ff) as decimal(5,2)) from t2;
select cast(sum(distinct ff) as signed) from t2;
select cast(variance(ff) as decimal(10,3)) from t2;
select cast(min(ff) as decimal(5,2)) from t2;

create table t1 (df decimal(5,1));
insert into t1 values(1.1);
insert into t1 values(2.2);
select cast(sum(distinct df) as signed) from t1;
select cast(min(df) as signed) from t1;
select 1e8 * sum(distinct df) from t1;
select 1e8 * min(df) from t1;

create table t3 (ifl int);
insert into t3 values(1), (2);
select cast(min(ifl) as decimal(5,2)) from t3;

drop table t1, t2, t3;


#
# BUG#3190, WL#1639: Standard Deviation STDDEV - 2 different calculations
#

CREATE TABLE t1 (id int(11),value1 float(10,2));
INSERT INTO t1 VALUES (1,0.00),(1,1.00), (1,2.00), (2,10.00), (2,11.00), (2,12.00), (2,13.00);
select id, stddev_pop(value1), var_pop(value1), stddev_samp(value1), var_samp(value1) from t1 group by id;
DROP TABLE t1;

#
# BUG#8464 decimal AVG returns incorrect result
#

CREATE TABLE t1 (col1 decimal(16,12));
INSERT INTO t1 VALUES (-5.00000000001),(-5.00000000002),(-5.00000000003),(-5.00000000000),(-5.00000000001),(-5.00000000002);
insert into t1 select * from t1;
select col1,count(col1),sum(col1),avg(col1) from t1 group by col1;
DROP TABLE t1;

#
# BUG#8465 decimal MIN and MAX return incorrect result
#

create table t1 (col1 decimal(16,12));
insert into t1 values (-5.00000000001);
insert into t1 values (-5.00000000001);
select col1,sum(col1),max(col1),min(col1) from t1 group by col1;
delete from t1;
insert into t1 values (5.00000000001);
insert into t1 values (5.00000000001);
select col1,sum(col1),max(col1),min(col1) from t1 group by col1;
DROP TABLE t1;

#
# Test that new VARCHAR correctly works with COUNT(DISTINCT)
#

CREATE TABLE t1 (a VARCHAR(400));
INSERT INTO t1 (a) VALUES ("A"), ("a"), ("a "), ("a   "),
                          ("B"), ("b"), ("b "), ("b   ");
SELECT COUNT(DISTINCT a) FROM t1;
DROP TABLE t1;

#
# Test for buf #9210: GROUP BY with expression if a decimal type
#

CREATE TABLE t1 (a int, b int, c int);
INSERT INTO t1 (a, b, c) VALUES
  (1,1,1), (1,1,2), (1,1,3),
  (1,2,1), (1,2,2), (1,2,3),
  (1,3,1), (1,3,2), (1,3,3),
  (2,1,1), (2,1,2), (2,1,3),
  (2,2,1), (2,2,2), (2,2,3),
  (2,3,1), (2,3,2), (2,3,3),
  (3,1,1), (3,1,2), (3,1,3),
  (3,2,1), (3,2,2), (3,2,3),
  (3,3,1), (3,3,2), (3,3,3);

SELECT b/c as v, a FROM t1 ORDER BY v;
SELECT b/c as v, SUM(a) FROM t1 GROUP BY v;
SELECT SUM(a) FROM t1 GROUP BY b/c;

DROP TABLE t1;
set div_precision_increment= @sav_dpi;