src/librustc_mir/borrow_check/places_conflict.rs

   1 use crate::borrow_check::ArtificialField;
   2 use crate::borrow_check::Overlap;
   3 use crate::borrow_check::{Deep, Shallow, AccessDepth};
   4 use rustc::hir;
   5 use rustc::mir::{
   6     BorrowKind, Body, Place, PlaceBase, Projection, ProjectionElem, ProjectionsIter,
   7     StaticKind
   8 };
   9 use rustc::ty::{self, TyCtxt};
  10 use std::cmp::max;
  11
  12 /// When checking if a place conflicts with another place, this enum is used to influence decisions
  13 /// where a place might be equal or disjoint with another place, such as if `a[i] == a[j]`.
  14 /// `PlaceConflictBias::Overlap` would bias toward assuming that `i` might equal `j` and that these
  15 /// places overlap. `PlaceConflictBias::NoOverlap` assumes that for the purposes of the predicate
  16 /// being run in the calling context, the conservative choice is to assume the compared indices
  17 /// are disjoint (and therefore, do not overlap).
  18 #[derive(Copy, Clone, Debug, Eq, PartialEq)]
  19 crate enum PlaceConflictBias {
  20     Overlap,
  21     NoOverlap,
  22 }
  23
  24 /// Helper function for checking if places conflict with a mutable borrow and deep access depth.
  25 /// This is used to check for places conflicting outside of the borrow checking code (such as in
  26 /// dataflow).
  27 crate fn places_conflict<'gcx, 'tcx>(
  28     tcx: TyCtxt<'_, 'gcx, 'tcx>,
  29     mir: &Body<'tcx>,
  30     borrow_place: &Place<'tcx>,
  31     access_place: &Place<'tcx>,
  32     bias: PlaceConflictBias,
  33 ) -> bool {
  34     borrow_conflicts_with_place(
  35         tcx,
  36         mir,
  37         borrow_place,
  38         BorrowKind::Mut { allow_two_phase_borrow: true },
  39         access_place,
  40         AccessDepth::Deep,
  41         bias,
  42     )
  43 }
  44
  45 /// Checks whether the `borrow_place` conflicts with the `access_place` given a borrow kind and
  46 /// access depth. The `bias` parameter is used to determine how the unknowable (comparing runtime
  47 /// array indices, for example) should be interpreted - this depends on what the caller wants in
  48 /// order to make the conservative choice and preserve soundness.
  49 pub(super) fn borrow_conflicts_with_place<'gcx, 'tcx>(
  50     tcx: TyCtxt<'_, 'gcx, 'tcx>,
  51     mir: &Body<'tcx>,
  52     borrow_place: &Place<'tcx>,
  53     borrow_kind: BorrowKind,
  54     access_place: &Place<'tcx>,
  55     access: AccessDepth,
  56     bias: PlaceConflictBias,
  57 ) -> bool {
  58     debug!(
  59         "borrow_conflicts_with_place({:?}, {:?}, {:?}, {:?})",
  60         borrow_place, access_place, access, bias,
  61     );
  62
  63     // This Local/Local case is handled by the more general code below, but
  64     // it's so common that it's a speed win to check for it first.
  65     if let Place::Base(PlaceBase::Local(l1)) = borrow_place {
  66         if let Place::Base(PlaceBase::Local(l2)) = access_place {
  67             return l1 == l2;
  68         }
  69     }
  70
  71     borrow_place.iterate(|borrow_base, borrow_projections| {
  72         access_place.iterate(|access_base, access_projections| {
  73             place_components_conflict(
  74                 tcx,
  75                 mir,
  76                 (borrow_base, borrow_projections),
  77                 borrow_kind,
  78                 (access_base, access_projections),
  79                 access,
  80                 bias,
  81             )
  82         })
  83     })
  84 }
  85
  86 fn place_components_conflict<'gcx, 'tcx>(
  87     tcx: TyCtxt<'_, 'gcx, 'tcx>,
  88     mir: &Body<'tcx>,
  89     borrow_projections: (&PlaceBase<'tcx>, ProjectionsIter<'_, 'tcx>),
  90     borrow_kind: BorrowKind,
  91     access_projections: (&PlaceBase<'tcx>, ProjectionsIter<'_, 'tcx>),
  92     access: AccessDepth,
  93     bias: PlaceConflictBias,
  94 ) -> bool {
  95     // The borrowck rules for proving disjointness are applied from the "root" of the
  96     // borrow forwards, iterating over "similar" projections in lockstep until
  97     // we can prove overlap one way or another. Essentially, we treat `Overlap` as
  98     // a monoid and report a conflict if the product ends up not being `Disjoint`.
  99     //
 100     // At each step, if we didn't run out of borrow or place, we know that our elements
 101     // have the same type, and that they only overlap if they are the identical.
 102     //
 103     // For example, if we are comparing these:
 104     // BORROW:  (*x1[2].y).z.a
 105     // ACCESS:  (*x1[i].y).w.b
 106     //
 107     // Then our steps are:
 108     //       x1         |   x1          -- places are the same
 109     //       x1[2]      |   x1[i]       -- equal or disjoint (disjoint if indexes differ)
 110     //       x1[2].y    |   x1[i].y     -- equal or disjoint
 111     //      *x1[2].y    |  *x1[i].y     -- equal or disjoint
 112     //     (*x1[2].y).z | (*x1[i].y).w  -- we are disjoint and don't need to check more!
 113     //
 114     // Because `zip` does potentially bad things to the iterator inside, this loop
 115     // also handles the case where the access might be a *prefix* of the borrow, e.g.
 116     //
 117     // BORROW:  (*x1[2].y).z.a
 118     // ACCESS:  x1[i].y
 119     //
 120     // Then our steps are:
 121     //       x1         |   x1          -- places are the same
 122     //       x1[2]      |   x1[i]       -- equal or disjoint (disjoint if indexes differ)
 123     //       x1[2].y    |   x1[i].y     -- equal or disjoint
 124     //
 125     // -- here we run out of access - the borrow can access a part of it. If this
 126     // is a full deep access, then we *know* the borrow conflicts with it. However,
 127     // if the access is shallow, then we can proceed:
 128     //
 129     //       x1[2].y    | (*x1[i].y)    -- a deref! the access can't get past this, so we
 130     //                                     are disjoint
 131     //
 132     // Our invariant is, that at each step of the iteration:
 133     //  - If we didn't run out of access to match, our borrow and access are comparable
 134     //    and either equal or disjoint.
 135     //  - If we did run out of access, the borrow can access a part of it.
 136
 137     let borrow_base = borrow_projections.0;
 138     let access_base = access_projections.0;
 139
 140     match place_base_conflict(tcx, borrow_base, access_base) {
 141         Overlap::Arbitrary => {
 142             bug!("Two base can't return Arbitrary");
 143         }
 144         Overlap::EqualOrDisjoint => {
 145             // This is the recursive case - proceed to the next element.
 146         }
 147         Overlap::Disjoint => {
 148             // We have proven the borrow disjoint - further
 149             // projections will remain disjoint.
 150             debug!("borrow_conflicts_with_place: disjoint");
 151             return false;
 152         }
 153     }
 154
 155     let mut borrow_projections = borrow_projections.1;
 156     let mut access_projections = access_projections.1;
 157
 158     loop {
 159         // loop invariant: borrow_c is always either equal to access_c or disjoint from it.
 160         if let Some(borrow_c) = borrow_projections.next() {
 161             debug!("borrow_conflicts_with_place: borrow_c = {:?}", borrow_c);
 162
 163             if let Some(access_c) = access_projections.next() {
 164                 debug!("borrow_conflicts_with_place: access_c = {:?}", access_c);
 165
 166                 // Borrow and access path both have more components.
 167                 //
 168                 // Examples:
 169                 //
 170                 // - borrow of `a.(...)`, access to `a.(...)`
 171                 // - borrow of `a.(...)`, access to `b.(...)`
 172                 //
 173                 // Here we only see the components we have checked so
 174                 // far (in our examples, just the first component). We
 175                 // check whether the components being borrowed vs
 176                 // accessed are disjoint (as in the second example,
 177                 // but not the first).
 178                 match place_projection_conflict(tcx, mir, borrow_c, access_c, bias) {
 179                     Overlap::Arbitrary => {
 180                         // We have encountered different fields of potentially
 181                         // the same union - the borrow now partially overlaps.
 182                         //
 183                         // There is no *easy* way of comparing the fields
 184                         // further on, because they might have different types
 185                         // (e.g., borrows of `u.a.0` and `u.b.y` where `.0` and
 186                         // `.y` come from different structs).
 187                         //
 188                         // We could try to do some things here - e.g., count
 189                         // dereferences - but that's probably not a good
 190                         // idea, at least for now, so just give up and
 191                         // report a conflict. This is unsafe code anyway so
 192                         // the user could always use raw pointers.
 193                         debug!("borrow_conflicts_with_place: arbitrary -> conflict");
 194                         return true;
 195                     }
 196                     Overlap::EqualOrDisjoint => {
 197                         // This is the recursive case - proceed to the next element.
 198                     }
 199                     Overlap::Disjoint => {
 200                         // We have proven the borrow disjoint - further
 201                         // projections will remain disjoint.
 202                         debug!("borrow_conflicts_with_place: disjoint");
 203                         return false;
 204                     }
 205                 }
 206             } else {
 207                 // Borrow path is longer than the access path. Examples:
 208                 //
 209                 // - borrow of `a.b.c`, access to `a.b`
 210                 //
 211                 // Here, we know that the borrow can access a part of
 212                 // our place. This is a conflict if that is a part our
 213                 // access cares about.
 214
 215                 let base = &borrow_c.base;
 216                 let elem = &borrow_c.elem;
 217                 let base_ty = base.ty(mir, tcx).ty;
 218
 219                 match (elem, &base_ty.sty, access) {
 220                     (_, _, Shallow(Some(ArtificialField::ArrayLength)))
 221                     | (_, _, Shallow(Some(ArtificialField::ShallowBorrow))) => {
 222                         // The array length is like  additional fields on the
 223                         // type; it does not overlap any existing data there.
 224                         // Furthermore, if cannot actually be a prefix of any
 225                         // borrowed place (at least in MIR as it is currently.)
 226                         //
 227                         // e.g., a (mutable) borrow of `a[5]` while we read the
 228                         // array length of `a`.
 229                         debug!("borrow_conflicts_with_place: implicit field");
 230                         return false;
 231                     }
 232
 233                     (ProjectionElem::Deref, _, Shallow(None)) => {
 234                         // e.g., a borrow of `*x.y` while we shallowly access `x.y` or some
 235                         // prefix thereof - the shallow access can't touch anything behind
 236                         // the pointer.
 237                         debug!("borrow_conflicts_with_place: shallow access behind ptr");
 238                         return false;
 239                     }
 240                     (ProjectionElem::Deref, ty::Ref(_, _, hir::MutImmutable), _) => {
 241                         // Shouldn't be tracked
 242                         bug!("Tracking borrow behind shared reference.");
 243                     }
 244                     (ProjectionElem::Deref, ty::Ref(_, _, hir::MutMutable), AccessDepth::Drop) => {
 245                         // Values behind a mutable reference are not access either by dropping a
 246                         // value, or by StorageDead
 247                         debug!("borrow_conflicts_with_place: drop access behind ptr");
 248                         return false;
 249                     }
 250
 251                     (ProjectionElem::Field { .. }, ty::Adt(def, _), AccessDepth::Drop) => {
 252                         // Drop can read/write arbitrary projections, so places
 253                         // conflict regardless of further projections.
 254                         if def.has_dtor(tcx) {
 255                             return true;
 256                         }
 257                     }
 258
 259                     (ProjectionElem::Deref, _, Deep)
 260                     | (ProjectionElem::Deref, _, AccessDepth::Drop)
 261                     | (ProjectionElem::Field { .. }, _, _)
 262                     | (ProjectionElem::Index { .. }, _, _)
 263                     | (ProjectionElem::ConstantIndex { .. }, _, _)
 264                     | (ProjectionElem::Subslice { .. }, _, _)
 265                     | (ProjectionElem::Downcast { .. }, _, _) => {
 266                         // Recursive case. This can still be disjoint on a
 267                         // further iteration if this a shallow access and
 268                         // there's a deref later on, e.g., a borrow
 269                         // of `*x.y` while accessing `x`.
 270                     }
 271                 }
 272             }
 273         } else {
 274             // Borrow path ran out but access path may not
 275             // have. Examples:
 276             //
 277             // - borrow of `a.b`, access to `a.b.c`
 278             // - borrow of `a.b`, access to `a.b`
 279             //
 280             // In the first example, where we didn't run out of
 281             // access, the borrow can access all of our place, so we
 282             // have a conflict.
 283             //
 284             // If the second example, where we did, then we still know
 285             // that the borrow can access a *part* of our place that
 286             // our access cares about, so we still have a conflict.
 287             if borrow_kind == BorrowKind::Shallow && access_projections.next().is_some() {
 288                 debug!("borrow_conflicts_with_place: shallow borrow");
 289                 return false;
 290             } else {
 291                 debug!("borrow_conflicts_with_place: full borrow, CONFLICT");
 292                 return true;
 293             }
 294         }
 295     }
 296 }
 297
 298 // Given that the bases of `elem1` and `elem2` are always either equal
 299 // or disjoint (and have the same type!), return the overlap situation
 300 // between `elem1` and `elem2`.
 301 fn place_base_conflict<'a, 'gcx: 'tcx, 'tcx>(
 302     tcx: TyCtxt<'a, 'gcx, 'tcx>,
 303     elem1: &PlaceBase<'tcx>,
 304     elem2: &PlaceBase<'tcx>,
 305 ) -> Overlap {
 306     match (elem1, elem2) {
 307         (PlaceBase::Local(l1), PlaceBase::Local(l2)) => {
 308             if l1 == l2 {
 309                 // the same local - base case, equal
 310                 debug!("place_element_conflict: DISJOINT-OR-EQ-LOCAL");
 311                 Overlap::EqualOrDisjoint
 312             } else {
 313                 // different locals - base case, disjoint
 314                 debug!("place_element_conflict: DISJOINT-LOCAL");
 315                 Overlap::Disjoint
 316             }
 317         }
 318         (PlaceBase::Static(s1), PlaceBase::Static(s2)) => {
 319             match (&s1.kind, &s2.kind) {
 320                 (StaticKind::Static(def_id_1), StaticKind::Static(def_id_2)) => {
 321                     if def_id_1 != def_id_2 {
 322                         debug!("place_element_conflict: DISJOINT-STATIC");
 323                         Overlap::Disjoint
 324                     } else if tcx.is_mutable_static(*def_id_1) {
 325                         // We ignore mutable statics - they can only be unsafe code.
 326                         debug!("place_element_conflict: IGNORE-STATIC-MUT");
 327                         Overlap::Disjoint
 328                     } else {
 329                         debug!("place_element_conflict: DISJOINT-OR-EQ-STATIC");
 330                         Overlap::EqualOrDisjoint
 331                     }
 332                 },
 333                 (StaticKind::Promoted(promoted_1), StaticKind::Promoted(promoted_2)) => {
 334                     if promoted_1 == promoted_2 {
 335                         if let ty::Array(_, len) = s1.ty.sty {
 336                             if let Some(0) = len.assert_usize(tcx) {
 337                                 // Ignore conflicts with promoted [T; 0].
 338                                 debug!("place_element_conflict: IGNORE-LEN-0-PROMOTED");
 339                                 return Overlap::Disjoint;
 340                             }
 341                         }
 342                         // the same promoted - base case, equal
 343                         debug!("place_element_conflict: DISJOINT-OR-EQ-PROMOTED");
 344                         Overlap::EqualOrDisjoint
 345                     } else {
 346                         // different promoteds - base case, disjoint
 347                         debug!("place_element_conflict: DISJOINT-PROMOTED");
 348                         Overlap::Disjoint
 349                     }
 350                 },
 351                 (_, _) => {
 352                     debug!("place_element_conflict: DISJOINT-STATIC-PROMOTED");
 353                     Overlap::Disjoint
 354                 }
 355             }
 356         }
 357         (PlaceBase::Local(_), PlaceBase::Static(_)) |
 358         (PlaceBase::Static(_), PlaceBase::Local(_)) => {
 359             debug!("place_element_conflict: DISJOINT-STATIC-LOCAL-PROMOTED");
 360             Overlap::Disjoint
 361         }
 362     }
 363 }
 364
 365 // Given that the bases of `elem1` and `elem2` are always either equal
 366 // or disjoint (and have the same type!), return the overlap situation
 367 // between `elem1` and `elem2`.
 368 fn place_projection_conflict<'a, 'gcx: 'tcx, 'tcx>(
 369     tcx: TyCtxt<'a, 'gcx, 'tcx>,
 370     mir: &Body<'tcx>,
 371     pi1: &Projection<'tcx>,
 372     pi2: &Projection<'tcx>,
 373     bias: PlaceConflictBias,
 374 ) -> Overlap {
 375     match (&pi1.elem, &pi2.elem) {
 376         (ProjectionElem::Deref, ProjectionElem::Deref) => {
 377             // derefs (e.g., `*x` vs. `*x`) - recur.
 378             debug!("place_element_conflict: DISJOINT-OR-EQ-DEREF");
 379             Overlap::EqualOrDisjoint
 380         }
 381         (ProjectionElem::Field(f1, _), ProjectionElem::Field(f2, _)) => {
 382             if f1 == f2 {
 383                 // same field (e.g., `a.y` vs. `a.y`) - recur.
 384                 debug!("place_element_conflict: DISJOINT-OR-EQ-FIELD");
 385                 Overlap::EqualOrDisjoint
 386             } else {
 387                 let ty = pi1.base.ty(mir, tcx).ty;
 388                 match ty.sty {
 389                     ty::Adt(def, _) if def.is_union() => {
 390                         // Different fields of a union, we are basically stuck.
 391                         debug!("place_element_conflict: STUCK-UNION");
 392                         Overlap::Arbitrary
 393                     }
 394                     _ => {
 395                         // Different fields of a struct (`a.x` vs. `a.y`). Disjoint!
 396                         debug!("place_element_conflict: DISJOINT-FIELD");
 397                         Overlap::Disjoint
 398                     }
 399                 }
 400             }
 401         }
 402         (ProjectionElem::Downcast(_, v1), ProjectionElem::Downcast(_, v2)) => {
 403             // different variants are treated as having disjoint fields,
 404             // even if they occupy the same "space", because it's
 405             // impossible for 2 variants of the same enum to exist
 406             // (and therefore, to be borrowed) at the same time.
 407             //
 408             // Note that this is different from unions - we *do* allow
 409             // this code to compile:
 410             //
 411             // ```
 412             // fn foo(x: &mut Result<i32, i32>) {
 413             //     let mut v = None;
 414             //     if let Ok(ref mut a) = *x {
 415             //         v = Some(a);
 416             //     }
 417             //     // here, you would *think* that the
 418             //     // *entirety* of `x` would be borrowed,
 419             //     // but in fact only the `Ok` variant is,
 420             //     // so the `Err` variant is *entirely free*:
 421             //     if let Err(ref mut a) = *x {
 422             //         v = Some(a);
 423             //     }
 424             //     drop(v);
 425             // }
 426             // ```
 427             if v1 == v2 {
 428                 debug!("place_element_conflict: DISJOINT-OR-EQ-FIELD");
 429                 Overlap::EqualOrDisjoint
 430             } else {
 431                 debug!("place_element_conflict: DISJOINT-FIELD");
 432                 Overlap::Disjoint
 433             }
 434         }
 435         (ProjectionElem::Index(..), ProjectionElem::Index(..))
 436         | (ProjectionElem::Index(..), ProjectionElem::ConstantIndex { .. })
 437         | (ProjectionElem::Index(..), ProjectionElem::Subslice { .. })
 438         | (ProjectionElem::ConstantIndex { .. }, ProjectionElem::Index(..))
 439         | (ProjectionElem::Subslice { .. }, ProjectionElem::Index(..)) => {
 440             // Array indexes (`a[0]` vs. `a[i]`). These can either be disjoint
 441             // (if the indexes differ) or equal (if they are the same).
 442             match bias {
 443                 PlaceConflictBias::Overlap => {
 444                     // If we are biased towards overlapping, then this is the recursive
 445                     // case that gives "equal *or* disjoint" its meaning.
 446                     debug!("place_element_conflict: DISJOINT-OR-EQ-ARRAY-INDEX");
 447                     Overlap::EqualOrDisjoint
 448                 }
 449                 PlaceConflictBias::NoOverlap => {
 450                     // If we are biased towards no overlapping, then this is disjoint.
 451                     debug!("place_element_conflict: DISJOINT-ARRAY-INDEX");
 452                     Overlap::Disjoint
 453                 }
 454             }
 455         }
 456         (ProjectionElem::ConstantIndex { offset: o1, min_length: _, from_end: false },
 457             ProjectionElem::ConstantIndex { offset: o2, min_length: _, from_end: false })
 458         | (ProjectionElem::ConstantIndex { offset: o1, min_length: _, from_end: true },
 459             ProjectionElem::ConstantIndex {
 460                 offset: o2, min_length: _, from_end: true }) => {
 461             if o1 == o2 {
 462                 debug!("place_element_conflict: DISJOINT-OR-EQ-ARRAY-CONSTANT-INDEX");
 463                 Overlap::EqualOrDisjoint
 464             } else {
 465                 debug!("place_element_conflict: DISJOINT-ARRAY-CONSTANT-INDEX");
 466                 Overlap::Disjoint
 467             }
 468         }
 469         (ProjectionElem::ConstantIndex {
 470             offset: offset_from_begin, min_length: min_length1, from_end: false },
 471             ProjectionElem::ConstantIndex {
 472                 offset: offset_from_end, min_length: min_length2, from_end: true })
 473         | (ProjectionElem::ConstantIndex {
 474             offset: offset_from_end, min_length: min_length1, from_end: true },
 475            ProjectionElem::ConstantIndex {
 476                offset: offset_from_begin, min_length: min_length2, from_end: false }) => {
 477             // both patterns matched so it must be at least the greater of the two
 478             let min_length = max(min_length1, min_length2);
 479             // `offset_from_end` can be in range `[1..min_length]`, 1 indicates the last
 480             // element (like -1 in Python) and `min_length` the first.
 481             // Therefore, `min_length - offset_from_end` gives the minimal possible
 482             // offset from the beginning
 483             if *offset_from_begin >= min_length - offset_from_end {
 484                 debug!("place_element_conflict: DISJOINT-OR-EQ-ARRAY-CONSTANT-INDEX-FE");
 485                 Overlap::EqualOrDisjoint
 486             } else {
 487                 debug!("place_element_conflict: DISJOINT-ARRAY-CONSTANT-INDEX-FE");
 488                 Overlap::Disjoint
 489             }
 490         }
 491         (ProjectionElem::ConstantIndex { offset, min_length: _, from_end: false },
 492          ProjectionElem::Subslice {from, .. })
 493         | (ProjectionElem::Subslice {from, .. },
 494             ProjectionElem::ConstantIndex { offset, min_length: _, from_end: false }) => {
 495             if offset >= from {
 496                 debug!(
 497                     "place_element_conflict: DISJOINT-OR-EQ-ARRAY-CONSTANT-INDEX-SUBSLICE");
 498                 Overlap::EqualOrDisjoint
 499             } else {
 500                 debug!("place_element_conflict: DISJOINT-ARRAY-CONSTANT-INDEX-SUBSLICE");
 501                 Overlap::Disjoint
 502             }
 503         }
 504         (ProjectionElem::ConstantIndex { offset, min_length: _, from_end: true },
 505          ProjectionElem::Subslice {from: _, to })
 506         | (ProjectionElem::Subslice {from: _, to },
 507             ProjectionElem::ConstantIndex { offset, min_length: _, from_end: true }) => {
 508             if offset > to {
 509                 debug!("place_element_conflict: \
 510                        DISJOINT-OR-EQ-ARRAY-CONSTANT-INDEX-SUBSLICE-FE");
 511                 Overlap::EqualOrDisjoint
 512             } else {
 513                 debug!("place_element_conflict: DISJOINT-ARRAY-CONSTANT-INDEX-SUBSLICE-FE");
 514                 Overlap::Disjoint
 515             }
 516         }
 517         (ProjectionElem::Subslice { .. }, ProjectionElem::Subslice { .. }) => {
 518             debug!("place_element_conflict: DISJOINT-OR-EQ-ARRAY-SUBSLICES");
 519              Overlap::EqualOrDisjoint
 520         }
 521         (ProjectionElem::Deref, _)
 522         | (ProjectionElem::Field(..), _)
 523         | (ProjectionElem::Index(..), _)
 524         | (ProjectionElem::ConstantIndex { .. }, _)
 525         | (ProjectionElem::Subslice { .. }, _)
 526         | (ProjectionElem::Downcast(..), _) => bug!(
 527             "mismatched projections in place_element_conflict: {:?} and {:?}",
 528             pi1,
 529             pi2
 530         ),
 531     }
 532 }