bedrock.lang.cpp.logic.pred

(*
 * Copyright (c) 2020-2023 BedRock Systems, Inc.
 * This software is distributed under the terms of the BedRock Open-Source License.
 * See the LICENSE-BedRock file in the repository root for details.
 *)
This file defines the core logic (called mpred) that we use for C++.
Known issues:
  • currently the logic is sequentially consistent
  • the memory model is simplified from the standard C++ memory model.
Require Import elpi.apps.locker.locker.
Require Export bedrock.prelude.addr.

Require Export bedrock.lang.bi.prelude.
Require Export bedrock.lang.bi.observe.
Require Export bedrock.lang.bi.spec.knowledge.
Require Export bedrock.lang.cpp.reserved_notation.
Require Export bedrock.lang.cpp.logic.mpred.
Require Export bedrock.lang.cpp.logic.rep.
^^ Delicate; export types and canonical structures (CS) for monPred, mpred and Rep. Export order can affect CS inference.

Require Import bedrock.prelude.bytestring.

Require Export bedrock.lang.cpp.algebra.cfrac.
Require Export bedrock.lang.cpp.bi.cfractional.

Require Export iris.base_logic.lib.iprop.
(* TODO: ^^ only needed to export uPredI, should be removed. *)
Require Import iris.bi.lib.fractional.
Require Import bedrock.lang.proofmode.proofmode.

Require Import bedrock.lang.bi.na_invariants.
Require Import bedrock.lang.bi.cancelable_invariants.
Export ChargeNotation.
Require Import bedrock.lang.cpp.bi.cfractional.

Require Import bedrock.lang.cpp.syntax.
Require Import bedrock.lang.cpp.semantics.values.
Require Import bedrock.lang.cpp.semantics.subtyping.

#[local] Set Printing Coercions.

Variant validity_type : Set := Strict | Relaxed.

Implicit Types (vt : validity_type) (σ resolve : genv).
Implicit Types (n : N) (z : Z).

(* Namespace for the invariants of the C++ abstraction's ghost state. *)
Definition pred_ns : namespace := nroot .@@ "bedrock" .@@ "lang" .@@ "cpp_logic".

(* Used by has_type_prop_has_type_noptr. No theory. *)
Definition nonptr_prim_type {lang} (ty : type' lang) : bool :=
  match drop_qualifiers ty with
  | Tfloat_ _ | Tchar_ _ | Tvoid | Tbool | Tenum _ | Tnum _ _ => true
  | Tnullptr | Tptr _ | Tref _ | Trv_ref _ | _ => false
  end.

Lemma nonptr_prim_type_erase_qualifiers : forall {lang} (ty : type' lang),
    nonptr_prim_type ty = nonptr_prim_type (erase_qualifiers ty).
Proof. induction ty; simpl; eauto. Qed.

Module Type CPP_LOGIC
  (Import P : PTRS_INTF)
  (Import INTF : VALUES_INTF_FUNCTOR P)
  (Import CC : CPP_LOGIC_CLASS).

  Implicit Types (p : ptr).

  Section with_cpp.

_valid_ptr vt p is a persistent assertion that p is a valid pointer, that is:
  • p can point to a function or a (possibly dead) object o
  • if vt = Relaxed, p can be nullptr, or past-the-end of a (possibly dead) object o.
In particular, _valid_ptr vt p prevents producing p by incrementing past-the-end pointers into overflow territory.
Our definition of validity includes all cases in which a pointer is not an invalid pointer value in the sense of the standard (<https://eel.is/c++draft/basic.compound3.1>), except that our concept of validity survives deallocation; a pointer is only valid according to the standard (or "standard-valid") if it is _both_ valid ([_valid_ptr vt p]) and live ([live_ptr p]); we require both where needed (e.g. [eval_ptr_eq]). When the duration of a region of storage ends [note 1], contained objects [o] go from live to dead, and pointers to such objects become _dangling_, or _invalid pointer values_ (pointer zapping note 1. In our semantics, that only consumes the non-persistent predicate live_ptr p, not the persistent predicate _valid_ptr vt p.
Following Cerberus, live_alloc_id tracks liveness per allocation ID (see comments for ptr), and live_ptr is derived from it. Hence, a pointer p past-the-end of o also becomes dangling when o is deallocated.
It's implementation-defined whether invalid pointer values are (non-copyable) trap representations. Instead, we restrict to implementations where dangling pointers are not trap representations (which is allowed, since this choice is implementation-defined) and pointer zapping does not actually clear pointers.
Note 1. See https://eel.is/c++draft/basic.stc.general4 and https://eel.is/c++draft/basic.compound3.1 for C++, and http://www.open-std.org/jtc1/sc22/wg14/www/docs/n2369.pdf for discussion in the context of the C standard.
    Parameter _valid_ptr : forall `{!cpp_logic thread_info Σ}
      (vt : validity_type), ptr -> mpred.
    (* strict validity (not past-the-end) *)
    Notation strict_valid_ptr := (_valid_ptr Strict).
    (* validity (past-the-end allowed) *)
    Notation valid_ptr := (_valid_ptr Relaxed).

  Section with_cpp_logic.
  Context `{cpp_logic}.

    Axiom _valid_ptr_persistent : forall b p, Persistent (_valid_ptr b p).
    Axiom _valid_ptr_affine : forall b p, Affine (_valid_ptr b p).
    Axiom _valid_ptr_timeless : forall b p, Timeless (_valid_ptr b p).
    #[global] Existing Instances _valid_ptr_persistent _valid_ptr_affine _valid_ptr_timeless.

    Axiom valid_ptr_nullptr : |-- valid_ptr nullptr.
    Axiom not_strictly_valid_ptr_nullptr : strict_valid_ptr nullptr |-- False.
    Axiom strict_valid_valid : forall p,
      strict_valid_ptr p |-- valid_ptr p.

  End with_cpp_logic.

has_type v ty is an approximation in mpred of "v is an initialized value of type t." This is intended to be preserved by the C++ type system. TODO: include in primR. TODO: refine for which types https://bedrocksystems.atlassian.net/browse/FM-3760
    Parameter has_type : ∀ `{!cpp_logic thread_info Σ} {σ : genv}, val -> type -> mpred.

    #[global] Declare Instance has_type_mono `{cpp_logic} :
      Proper (genv_leq ==> eq ==> eq ==> (⊢)) (@has_type _ _ _).

reference_to ty p states that the location p stores a value of type ty. Being a reference is stronger than being a pointer because pointers allow past-the-end pointers while references do not.
This should be thought of as closely related to has_type (Vref p) (Tref ty) except that it holds on types that are not valid C++ types. For example, if struct C { int& r; } x; >> then reference_to (Tref Tint) (x ., ``::C::r``).
In the future, has_type (Vref (x ., ``::C::r``)) (Tref (Tref Tint)) may not hold since Tref (Tref Tint) is not a valid C++ type. The current blocker to this is that we use tptsto to represent ownership of references. This would need to change.
Note that the following code is (currently?) legal according to our semantics: { long x; *static_cast<int*>(&x); } >> because reference_to only guarantees alignment, not anything about the dynamic type of the object. Requiring a property of the dynamic type of the object would require us to carry type information in strict_valid_ptr (which would make it something like a pre-construction state of type_ptr).
    Parameter reference_to : forall `{!cpp_logic thread_info Σ} {σ : genv}, type -> ptr -> mpred.

    #[global] Declare Instance reference_to_mono `{cpp_logic} :
      Proper (genv_leq ==> eq ==> eq ==> (⊢)) (@reference_to _ _ _).

    Section with_genv.
      Context `{cpp_logic} {σ : genv}.

      #[global] Declare Instance has_type_knowledge : Knowledge2 has_type.
      #[global] Declare Instance has_type_timeless : Timeless2 has_type.

      Axiom has_type_has_type_prop : ∀ v ty, has_type v ty |-- [| has_type_prop v ty |].

      Axiom has_type_prop_has_type_noptr : ∀ v ty,
        nonptr_prim_type ty ->
        [| has_type_prop v ty |] |-- has_type v ty.

      (* has_type is independent of qualifiers.
         This allows casting betweeen const and non-const.
         *)
      Axiom has_type_erase_qualifiers : ∀ ty v,
        has_type v ty -|- has_type v (erase_qualifiers ty).

      (* Internal statements: *)
      Axiom has_type_nullptr' : ∀ p,
        has_type (Vptr p) Tnullptr -|- [| p = nullptr |].
      Axiom has_type_ptr' : ∀ p ty,
        has_type (Vptr p) (Tptr ty) -|-
        valid_ptr p ** [| aligned_ptr_ty ty p |].

      (* These two definitions are needed because of tptsto_has_type *)
      Axiom has_type_ref' : ∀ p ty,
        has_type (Vref p) (Tref ty) |-- reference_to ty p.
      Axiom has_type_rv_ref' : ∀ p ty,
        has_type (Vref p) (Trv_ref ty) |-- reference_to ty p.

      #[global] Declare Instance reference_to_knowledge : Knowledge2 reference_to.
      #[global] Declare Instance reference_to_timeless : Timeless2 reference_to.

      (* NOTE: this is almost derivable from reference_to_intro and
         reference_to_elim except for the issue with non-standard types. *)
      Axiom reference_to_erase : forall ty p,
          reference_to ty p -|- reference_to (erase_qualifiers ty) p.

The introduction and elimination forms are not the same to avoid using has_type as types that are not C++ types. In practice, the only way to prove a reference_to is to have a tptsto assertion.
**A note on 0-sized objects** 0-sized objects are problematic from the point of view of C++ (they are not permitted by the standard, but they are supported by standard compilers). The problem with 0-sized objects is that they do not have a unique identity but the BRiCk semantics will give them an identity using struct_padding/union_padding.
Mitigation: We currently address this by forbidding reasoning about constructors of 0-sized objects.
      Axiom reference_to_intro : forall ty p,
          strict_valid_ptr p |-- has_type (Vptr p) (Tptr ty) -* reference_to ty p.
      Axiom reference_to_elim : forall ty p,
          reference_to ty p |--
            [| aligned_ptr_ty ty p |] ** [| p <> nullptr |] **
            valid_ptr p ** if zero_sized_array ty then emp else strict_valid_ptr p.

    End with_genv.

    Parameter has_type_or_undef : forall `{!cpp_logic thread_info Σ} {σ : genv}, val -> type -> mpred.
    Axiom has_type_or_undef_unfold : forall `{cpp_logic},
        @has_type_or_undef _ _ _ = funI σ v ty => has_type v ty \\// [| v = Vundef |].

Formalizes the notion of "provides storage", http://eel.is/c++draft/intro.objectdef:provides_storage
    Parameter provides_storage :
      forall `{!cpp_logic thread_info Σ} (storage : ptr) (object : ptr) (object_type : Rtype), mpred.

  Section with_cpp_logic.
  Context `{cpp_logic}.

    Axiom provides_storage_persistent :
      forall storage_ptr obj_ptr ty,
      Persistent (provides_storage storage_ptr obj_ptr ty).
    Axiom provides_storage_affine :
      forall storage_ptr obj_ptr ty,
      Affine (provides_storage storage_ptr obj_ptr ty).
    Axiom provides_storage_timeless :
      forall storage_ptr obj_ptr ty,
      Timeless (provides_storage storage_ptr obj_ptr ty).
    #[global] Existing Instances provides_storage_persistent provides_storage_affine provides_storage_timeless.

  End with_cpp_logic.

Typed points-to predicate.

Fact tptsto t q p v asserts the following things: 1. Pointer p points to value v. 2. We have fractional ownership q (in the separation logic sense). 3. Pointer p points to a memory location with C++ type t. However: 1. Value v need not be initialized. 2. Hence, v might not satisfy has_type_prop t v.
tptsto is only used to represent memory cells that contain values representable as a single val. This includes:
  • value types (see is_value_type), and
  • reference types. Note that all gl-values, i.e. x-values (Trv_ref) and l-values (Tref), are represented using Tref.
These requirements justify the axiom tptsto_valid_type.
We use this predicate both for pointers to actual memory and for pointers to C++ locations that are not stored in memory (as an optimization).
    Parameter tptsto : forall `{!cpp_logic thread_info Σ} {σ:genv}
      (t : Rtype) (q : cQp.t) (a : ptr) (v : val), mpred.

  Section with_cpp_logic.
  Context `{cpp_logic}.

    #[global] Declare Instance tptsto_valid_type
      : forall {σ:genv} (t : Rtype) (q : cQp.t) (a : ptr) (v : val),
        Observe [| is_heap_type t |] (tptsto t q a v).

    Axiom tptsto_nonnull : forall {σ} ty q a,
      tptsto (σ:=σ) ty q nullptr a |-- False.

    #[global] Declare Instance tptsto_params : Params (@tptsto) 3.
    #[global] Declare Instance tptsto_proper :
      Proper (genv_eq ==> eq ==> eq ==> eq ==> eq ==> (≡)) (@tptsto _ _ _).
    #[global] Declare Instance tptsto_mono :
      Proper (genv_leq ==> eq ==> eq ==> eq ==> eq ==> (⊢)) (@tptsto _ _ _).

    #[global] Declare Instance tptsto_timeless : Timeless5 (@tptsto _ _ _).
    #[global] Declare Instance tptsto_cfractional {σ} ty : CFractional2 (tptsto ty).

    #[global] Declare Instance tptsto_cfrac_valid {σ} t : CFracValid2 (tptsto t).

    #[global] Declare Instance tptsto_welltyped : forall {σ} p ty q v,
      Observe (has_type_or_undef v ty) (tptsto ty q p v).

    #[global] Declare Instance tptsto_reference_to : forall {σ} p ty q v,
      Observe (reference_to ty p) (tptsto ty q p v).

NOTE: We'll eventually need the stronger tptsto_learn : ∀ {σ} ty q1 q2 p v1 v2, <absorb> (tptsto ty q1 p v1) //\\ <absorb> (tptsto ty q2 p v2) |-- [! v1 = v2 !] but setting up the proof in simple_pred now would be a bit of a digression.
    #[global] Declare Instance tptsto_agree : forall {σ} ty q1 q2 p v1 v2,
      Observe2 [| v1 = v2 |]
               (tptsto ty q1 p v1)
               (tptsto ty q2 p v2).

    End with_cpp_logic.

The allocation is alive. Neither persistent nor fractional. See https://eel.is/c++draft/basic.stc.general4 and https://eel.is/c++draft/basic.compound3.1.
    Parameter live_alloc_id : forall `{!cpp_logic thread_info Σ},
      alloc_id -> mpred.

  Section with_cpp_logic.
  Context `{cpp_logic}.

    Axiom live_alloc_id_timeless : forall aid, Timeless (live_alloc_id aid).
    #[global] Existing Instance live_alloc_id_timeless.

    Axiom valid_ptr_alloc_id : forall p,
      valid_ptr p |-- [| is_Some (ptr_alloc_id p) |].

This pointer is from a live allocation; this does not imply _valid_ptr, because even overflowing offsets preserve the allocation ID.
    Definition live_ptr (p : ptr) :=
      default False%I (live_alloc_id <$> ptr_alloc_id p).

We consider nullptr as live, following Krebbers, as a way to simplify stating rules for pointer comparison.
    Axiom nullptr_live : |-- live_ptr nullptr.

    Axiom tptsto_live : forall {σ} ty (q : cQp.t) p v,
      tptsto ty q p v |-- live_ptr p ** True.

  End with_cpp_logic.

mdc_path σ this mdc q p state that p is a pointer to a (live) object of type this that is part of an object that can be reached using the *path* mdc. The ownership mdc_path is only exposed to the program for an object of class C if C has virtual methods (or base classes, which are not currently supported).
  • if mdc = [] then this object mdc_path is not initialized yet, e.g. because its base classes are still being constructed.
  • otherwise, mdc is the *path* from the most derived class to this object. For example, suppose you have: struct A { virtual int f() { return 0; } }; struct B : public A { virtual int f() { return 1; } }; struct C : public A { }; struct D : public B, public C { };
    int doA(A* a) { return a->f(); } int test() { D d; return doA(static_cast<B*>(&d)) /* = 1 */ + doA(static_cast<C*>(&d)) /* = 0 */; } >> for a fully constructed object of type `D` (at pointer d), you would have:
              mdc_path "::B" ["::D","::B"] 1 (d ., _base "::B") **
              mdc_path "::A" ["::D","::B","::A"] 1 (d ,, _base "::B" ,, _base "::A") **
              mdc_path "::A" ["::D","::C","::A"] 1 (d ,, _base "::C" ,, _base "::A")
    in the partially constructed state, where "::D" has not yet been constructed but the base classes have been, you have the following:
              mdc_path "::B" ["::B"] 1 (d ., _base "::B") **
              mdc_path "::A" ["::B","::A"] 1 (d ,, _base "::B" ,, _base "::A") **
              mdc_path "::A" ["::C","::A"] 1 (d ,, _base "::C" ,, _base "::A")
    Note that you do not get mdc_path "::D" [] 1 d at this point, you get mdc_path "::D" ["::D"] 1 d when you update all the other paths (but not atomically).
mdc_path is primarily used to dispatch virtual function calls. Compilers can use the ownership here to represent dynamic dispatch tables.
    Parameter mdc_path : forall `{!cpp_logic thread_info Σ} {σ : genv}
        (this : globname) (most_derived : list globname),
        cQp.t -> ptr -> mpred.

  Section with_cpp_logic.
  Context `{cpp_logic}.
    #[global] Declare Instance mdc_path_cfractional σ this mdc : CFractional1 (mdc_path this mdc).
    #[global] Declare Instance mdc_path_cfrac_valid {σ} cls path : CFracValid1 (mdc_path cls path).
    #[global] Declare Instance mdc_path_timeless {σ} : Timeless4 (mdc_path).
    #[global] Declare Instance mdc_path_strict_valid {σ} this mdc q p : Observe (strict_valid_ptr p) (mdc_path this mdc q p).
    #[global] Declare Instance mdc_path_agree {σ} cls1 cls2 q1 q2 p mdc1 mdc2 :
      Observe2 [| mdc1 = mdc2 /\ cls1 = cls2 |] (mdc_path cls1 mdc1 q1 p) (mdc_path cls2 mdc2 q2 p).

this allows you to forget an object mdc_path, necessary for doing placement new over an existing object.
    Axiom mdc_path_forget : forall σ mdc this p,
        mdc_path (σ:=σ) this mdc (cQp.m 1) p |-- |={↑pred_ns}=> mdc_path (σ:=σ) this nil (cQp.m 1) p.

  End with_cpp_logic.

the pointer points to the code
note that in the presence of code-loading, function calls will require an extra side-condition that the code is loaded.
    Parameter code_at : forall `{!cpp_logic thread_info Σ},
      genv -> translation_unit -> Func -> ptr -> mpred.
    Parameter method_at : forall `{!cpp_logic thread_info Σ},
      genv -> translation_unit -> Method -> ptr -> mpred.
    Parameter ctor_at : forall `{!cpp_logic thread_info Σ},
      genv -> translation_unit -> Ctor -> ptr -> mpred.
    Parameter dtor_at : forall `{!cpp_logic thread_info Σ},
      genv -> translation_unit -> Dtor -> ptr -> mpred.
    #[global] Arguments code_at {_ _ _ σ} tu _ _.
    #[global] Arguments method_at {_ _ _ σ} tu _ _.
    #[global] Arguments ctor_at {_ _ _ σ} tu _ _.
    #[global] Arguments dtor_at {_ _ _ σ} tu _ _.

    Section with_genv.
      Context `{cpp_logic} {σ : genv} (tu : translation_unit).
      #[local] Notation code_at := (@code_at _ _ _ σ tu) (only parsing).
      #[local] Notation method_at := (@method_at _ _ _ σ tu) (only parsing).
      #[local] Notation ctor_at := (@ctor_at _ _ _ σ tu) (only parsing).
      #[local] Notation dtor_at := (@dtor_at _ _ _ σ tu) (only parsing).

      Axiom code_at_persistent : forall f p, Persistent (code_at f p).
      Axiom code_at_affine : forall f p, Affine (code_at f p).
      Axiom code_at_timeless : forall f p, Timeless (code_at f p).

      Axiom method_at_persistent : forall f p, Persistent (method_at f p).
      Axiom method_at_affine : forall f p, Affine (method_at f p).
      Axiom method_at_timeless : forall f p, Timeless (method_at f p).

      Axiom ctor_at_persistent : forall f p, Persistent (ctor_at f p).
      Axiom ctor_at_affine : forall f p, Affine (ctor_at f p).
      Axiom ctor_at_timeless : forall f p, Timeless (ctor_at f p).

      Axiom dtor_at_persistent : forall f p, Persistent (dtor_at f p).
      Axiom dtor_at_affine : forall f p, Affine (dtor_at f p).
      Axiom dtor_at_timeless : forall f p, Timeless (dtor_at f p).

      #[global] Existing Instances
        code_at_persistent code_at_affine code_at_timeless
        method_at_persistent method_at_affine method_at_timeless
        ctor_at_persistent ctor_at_affine ctor_at_timeless
        dtor_at_persistent dtor_at_affine dtor_at_timeless.

      Axiom code_at_live : forall f p, code_at f p |-- live_ptr p.
      Axiom method_at_live : forall f p, method_at f p |-- live_ptr p.
      Axiom ctor_at_live : forall f p, ctor_at f p |-- live_ptr p.
      Axiom dtor_at_live : forall f p, dtor_at f p |-- live_ptr p.

      Axiom code_at_strict_valid : forall f p, code_at f p |-- strict_valid_ptr p.
      Axiom method_at_strict_valid : forall f p, method_at f p |-- strict_valid_ptr p.
      Axiom ctor_at_strict_valid : forall f p, ctor_at f p |-- strict_valid_ptr p.
      Axiom dtor_at_strict_valid : forall f p, dtor_at f p |-- strict_valid_ptr p.

    End with_genv.

  Section with_cpp_logic.
  Context `{cpp_logic}.

    Axiom offset_pinned_ptr_pure : forall σ o z va p,
      eval_offset σ o = Some z ->
      ptr_vaddr p = Some va ->
      valid_ptr (p ,, o) |--
      [| 0 <= Z.of_N va + z |]%Z **
      [| ptr_vaddr (p ,, o) = Some (Z.to_N (Z.of_N va + z)) |].

    Axiom offset_inv_pinned_ptr_pure : forall σ o z va p,
      eval_offset σ o = Some z ->
      ptr_vaddr (p ,, o) = Some va ->
      valid_ptr (p ,, o) |--
      [| 0 <= Z.of_N va - z |]%Z **
      [| ptr_vaddr p = Some (Z.to_N (Z.of_N va - z)) |].

    Axiom provides_storage_same_address : forall storage_ptr obj_ptr ty,
      Observe [| same_address storage_ptr obj_ptr |] (provides_storage storage_ptr obj_ptr ty).

    Axiom provides_storage_valid_storage_ptr : forall storage_ptr obj_ptr aty,
      Observe (valid_ptr storage_ptr) (provides_storage storage_ptr obj_ptr aty).
    Axiom provides_storage_valid_obj_ptr : forall storage_ptr obj_ptr aty,
      Observe (valid_ptr obj_ptr) (provides_storage storage_ptr obj_ptr aty).

    #[global] Existing Instances provides_storage_same_address
      provides_storage_valid_storage_ptr provides_storage_valid_obj_ptr.

  End with_cpp_logic.

exposed_aid aid states that the storage instance identified by aid is "exposed" 1. This enables int2ptr casts to produce pointers into this storage instance.
1 We use "exposed" in the sense defined by the N2577 draft C standard (http://www.open-std.org/jtc1/sc22/wg14/www/docs/n2577.pdf). See https://dl.acm.org/doi/10.1145/3290380 for an introduction.
    Parameter exposed_aid : forall `{!cpp_logic thread_info Σ}, alloc_id -> mpred.

  Section with_cpp_logic.
  Context `{cpp_logic}.

    Axiom exposed_aid_persistent : forall aid, Persistent (exposed_aid aid).
    Axiom exposed_aid_affine : forall aid, Affine (exposed_aid aid).
    Axiom exposed_aid_timeless : forall aid, Timeless (exposed_aid aid).

    Axiom exposed_aid_null_alloc_id : |-- exposed_aid null_alloc_id.

    #[global] Existing Instances
      exposed_aid_persistent exposed_aid_affine exposed_aid_timeless.

  End with_cpp_logic.

type_ptr {resolve := resolve} ty p asserts that p points to a (possibly dead) object of type ty (in environment resolve), as defined by https://eel.is/c++draft/basic.compound3.1. This implies: - the pointer is strictly valid [type_ptr_strict_valid], and "p + 1" is also valid (while possibly past-the-end) [type_ptr_valid_plus_one]. - the pointer is not (an offset of) null pointers [type_ptr_off_nonnull, type_ptr_nonnull] - the pointer is properly aligned [type_ptr_aligned_pure] [type_ptr] is persistent and survives deallocation of the pointed-to object, like [_valid_ptr]. TODO: before a complete object is fully initialized, what [type_ptr] facts are available? For now, we only use [type_ptr] for fully initialized objects. Consider http://eel.is/c++draft/basic.memobjbasic.life, especially from http://eel.is/c++draft/basic.memobjbasic.life-1 to http://eel.is/c++draft/basic.memobjbasic.life-4.
    Parameter type_ptr : forall `{!cpp_logic thread_info Σ} {resolve : genv} (c: Rtype), ptr -> mpred.

  Section with_cpp_logic.
  Context `{cpp_logic}.

    Axiom type_ptr_persistent : forall {σ} p ty,
      Persistent (type_ptr ty p).
    Axiom type_ptr_affine : forall {σ} p ty,
      Affine (type_ptr ty p).
    Axiom type_ptr_timeless : forall {σ} p ty,
      Timeless (type_ptr ty p).
    #[global] Existing Instances type_ptr_persistent type_ptr_affine type_ptr_timeless.

    Axiom type_ptr_erase : forall {σ} ty p,
        type_ptr ty p -|- type_ptr (erase_qualifiers ty) p.

    Axiom type_ptr_aligned_pure : forall {σ} ty p,
      type_ptr ty p |-- [| aligned_ptr_ty ty p |].

    Axiom type_ptr_off_nonnull : forall {σ} ty p o,
      type_ptr ty (p ,, o) |-- [| p <> nullptr |].

    Axiom tptsto_type_ptr : forall {σ : genv} ty q p v,
      Observe (type_ptr ty p) (tptsto ty q p v).
    #[global] Existing Instance tptsto_type_ptr.

    (* All objects in the C++ abstract machine have a size

       NOTE to support un-sized objects, we can simply say that the sizeof operator
            in C++ is only a conservative approximation of the true size of an object.
     *)
    Axiom type_ptr_size : forall σ ty p,
        type_ptr ty p |-- [| is_Some (size_of σ ty) |].

Recall that type_ptr and strict_valid_ptr don't include past-the-end pointers...
    Axiom type_ptr_strict_valid : forall resolve ty p,
      type_ptr ty p |-- strict_valid_ptr p.
Hence they can be incremented into (possibly past-the-end) valid pointers.
    Axiom type_ptr_valid_plus_one : forall resolve ty p,
      type_ptr ty p |-- valid_ptr (p ,, o_sub resolve ty 1).

    (* When p is a ptr to a C++ object of type ty, p is /also/ a pointer
       to the "object representation" of ty - which consists of "the sequence
       of sizeof(ty) unsigned char objects taken up by the object of
       type ty" 1.

       Detailed Justification:
       a) From the comment above the axiomatization of type_ptr:
          | type_ptr ty p asserts that p points to a (possibly dead) object of type ty.
       b) In basic.types.general#4 1 the C++ Standard states that:
          | The object representation of an object of type T is the sequence of N unsigned
          | char objects taken up by the object of type T, where N equals sizeof(T).
       d) (NOTE: the C++ Standard lacks language regarding this point) the "object representation"
          for an object pointed to by ptr p is /also/ accessible via p.
       e) (a)+(b)+(d) implies that a (potentially dead) object representation for type ty
          exists at ptr p
       f) (a)+(c)+(e) implies that type_ptr Tbyte (p .[ Tbyte ! i ]) holds (regardless of whether
          not the object/"object representation" is alive)

       NOTE: There is no need to deal with past-the-end pointers explicitly since
       type_ptr explicitly excludes them; validity of past-the-end pointers
       can be established using type_ptr_valid_plus_one.

       1 <https://eel.is/c++draft/basic.types.general4> *)
    Section type_ptr_object_representation.
      (* This section is intended to axiomatize a /sufficient/ and /sound/
         set of transport rules for type_ptr facts which can be used to
         satisfy the preconditions required when using ptr_congP to
         transport other resources.
       *)

      (* The following Axiom reflects a trivially faithful encoding of
         the quote from the C++ standard above Section type_ptr_object_representation.

         NOTE: To practically use this Axiom, type_ptr must be Persistent;
         a reasonable alternative axiomatization which sidestepts this issue
         could produce /all/ of the type_ptr facts for the "object representation"
         at once:
         | ... ->
         |     type_ptr ty p
         | |-- ∗list i ∈ seqN 0 (sizeof ty), type_ptr Tbyte (p . Tbyte ! i )
       *)
      Section conservative.
        Axiom type_ptr_obj_repr_byte :
          forall (σ : genv) (ty : Rtype) (p : ptr) (i sz : N),
            size_of σ ty = Some sz -> (* 1) ty has some byte-size sz *)
            (i < sz)%N -> (* 2) by (1), sz is nonzero and i is a
                                            byte-offset into the object rooted at p ,, o

                                         NOTE: forall ty, size_of (Tarray ty 0) = Some 0,
                                         but zero-length arrays are not permitted by the Standard
                                         (cf. <https://eel.is/c++draft/dcl.arraydef:array,bound>). NOTE: if support for flexible array members is ever added, it will need to be carefully coordinated with these sorts of transport lemmas. *)
            (* 4) The existence of the "object representation" of an object of type ty -
               |  in conjunction with the premises - justifies "lowering" any
               |  type_ptr ty p fact to a collection of type_ptr Tbyte (p ,, .[Tbyte ! i])
               |  facts - where i is a byte-offset within the ty (0 <= i < sizeof(ty)).
               v *)
            type_ptr ty p |-- type_ptr Tbyte (p ,, .[ Tbyte ! i ]).
      End conservative.

      (* NOTE: This might be reasonable to axiomatize directly; cf. the NOTE above
         Section conservative.
       *)
      Section all_at_once.
        Lemma type_ptr_obj_repr :
          forall (σ : genv) (ty : Rtype) (p : ptr) (sz : N),
            size_of σ ty = Some sz ->
            type_ptr ty p |-- [∗list] i ∈ seqN 0 sz, type_ptr Tbyte (p .[ Tbyte ! Z.of_N i ]).
        Proof.
          intros * Hsz; iIntros "#tptr".
          iApply big_sepL_intro; iIntros "!>" (k n) "%Hn'".
          assert (lookup (K:=N) (N.of_nat k) (seqN 0%N sz) = Some n)
            as Hn
            by (unfold lookupN, list_lookupN; rewrite Nat2N.id //);
            clear Hn'.
          apply lookupN_seqN in Hn as [? ?].
          iDestruct (type_ptr_obj_repr_byte σ ty p n sz Hsz ltac:(lia) with "tptr") as "$".
        Qed.
      End all_at_once.
    End type_ptr_object_representation.

    (* offset_congP hoists offset_cong to mpred *)
    Definition offset_congP (σ : genv) (o1 o2 : offset) : mpred :=
      [| offset_cong σ o1 o2 |].

    (* ptr_congP σ p1 p2 is an mpred which quotients ptr_cong σ p1 p2
       by requiring that type_ptr Tbyte holds for both p1 /and/ p2. This property
       is intended to be sound and sufficient for transporting certain physical
       resources between p1 and p2 - and we hypothesize that it is also
       necessary.
     *)
    Definition ptr_congP (σ : genv) (p1 p2 : ptr) : mpred :=
      [| ptr_cong σ p1 p2 |] ** type_ptr Tbyte p1 ** type_ptr Tbyte p2.

    (* All tptsto Tbyte facts can be transported over ptr_congP ptrs.

       High level meaning:
       In the C++ object model, a single byte of storage can be accessed through different pointers,
       e.g. consider struct C { int x; int y; } c;. The first byte of the struct can be read through
       static_cast<byte*>(&c) (with pointer representation c) as well as static_cast<byte*>(&c.x)
       (with pointer representation c ,, _field "::C" "x"). To put an ownership discipline on this
       single byte, we build an equivalence relation on pointers that allows us to transport ownership
       of the byte between these different pointers. For example, half of the ownership could live at c
       and the other half of the ownership can live at c ,, _field "::C" "x".

       The standard justifies this as follows:
       1) (cf. tptsto comment) tptsto ty q p v ensures that p points to a memory
          location with C++ type ty and which has some value v.
       2) (cf. Section type_ptr_object_representation) type_ptr Tbyte holds for all of the
          bytes (i.e. the "object reprsentation") constituting well-typed C++ objects.
       3) NOTE (JH): the following isn't quite true yet, but we'll want this when we flesh
          out rawR/RAW_BYTES:
          a) all values v can be converted into (potentially many) raw_bytes -
             which capture its "object representation"
          b) all tptsto ty facts can be shattered into (potentially many)
             tptsto Tbyte _ _ (Vraw _) facts corresponding to its "object representation"
       4) tptsto Tbyte _ _ (Vraw _) can be transported over ptr_congP ptrs:
          a) tptso Tbyte _ _ (Vraw _) facts deal with the "object representation" directly
             and thus permit erasing the structure of pointers in favor of reasoning about
             relative byte offsets from a shared ptr-prefix.
          b) the ptrs are ptr_congP so we know that:
             i) they share a common base pointer p_base
             ii) the byte-offset values of the C++ offsets which reconstitute the src/dst from
                 p_base are equal
             iii) NOTE: (cf. valid_ptr_nonnull_nonzero/type_ptr_valid_ptr/type_ptr_nonnull below)
                  p_base has some vaddr, but we don't currently rely on this fact.
     *)
    (* TODO: improve our axiomatic support for raw values - including "shattering"
       non-raw values into their constituent raw pieces - to enable deriving
       tptsto_ptr_congP_transport from tptsto_raw_ptr_congP_transport.
     *)
    Axiom tptsto_ptr_congP_transport : forall {σ} q p1 p2 v,
      ptr_congP σ p1 p2 |-- tptsto (σ:=σ) Tbyte q p1 v -* tptsto (σ:=σ) Tbyte q p2 v.

Deducing pointer equalities

The following axioms, together with same_address_o_sub_eq, enable going from same_address (produced by C++ pointer equality) to actual pointer equalities.
Pointer equality with nullptr is easy, as long as your pointer is valid. Validity is necessary: the C++ expression (char * )p - (uintptr_t) p produces an invalid pointer with address 0, which is not nullptr because it preserves the provenance of p.
    Axiom same_address_eq_null : forall p tv,
      _valid_ptr tv p |--
      [| same_address p nullptr <-> p = nullptr |].

same_address_eq_type_ptr concludes that two pointers p1 and p2 are equal if they have the same address, point to live objects o1 and o2, and have the same (non-uchar) type ty with nonzero size.
Justifying this from the standard is tricky; here's a proof sketch.
  • Because ty has "nonzero size" (https://eel.is/c++draft/intro.object8), and we don't support bitfields, we apply the standard: > Unless it is a bit-field, an object with nonzero size shall occupy one or more bytes of storage, including every byte that is occupied in full or in part by any of its subobjects. - Because [o1] and [o2] are live, these objects share storage, hence they must coincide or one must be nested inside the other (https://eel.is/c++draft/intro.object4); if they coincide, our proof is done.
  • Because ty is not unsigned char, neither pointer can provide storage for the other (https://eel.is/c++draft/intro.object3); so one must be a subobject of the other. - Since [o1] and [o2] have type [ty], and type [ty] has "nonzero size", neither of [o1] and [o2] can be a subobject of the other; we conjecture is provable from the C++ type system. NOTE: we check "nonzero size" (https://eel.is/c++draft/basic.memobjintro.object-8) using size_of,
which might incorporate implementation-specific compiler decisions. TODO: handle std::byte like unsigned char.
    Axiom same_address_eq_type_ptr : forall resolve ty p1 p2 n,
      same_address p1 p2 ->
      size_of resolve ty = Some n ->
      (* if ty = Tuchar, one of these pointer could provide storage for the other. *)
      ty <> Tuchar ->
      (n > 0)%N ->
      type_ptr ty p1 ∧ type_ptr ty p2 ∧ live_ptr p1 ∧ live_ptr p2 ⊢
        |={↑pred_ns}=> [| p1 = p2 |].

  End with_cpp_logic.

Padding struct_padding and union_padding represent the object state of an aggregate. Ownership of this token implies that the object is a alive. They only given when an object is fully constructed, and are taken back at the end of the lifetime. This is related to the "construction state" in Tahina's work, i.e. <https://inria.hal.science/file/index/docid/674663/filename/cpp-construction.pdf>
    Axiom struct_padding : forall `{!cpp_logic thread_info Σ} {σ:genv},
      ptr -> globname -> cQp.t -> mpred.

  Section with_cpp_logic.
  Context `{cpp_logic}.

    #[global] Declare Instance struct_padding_timeless {σ:genv} : Timeless3 (struct_padding).
    #[global] Declare Instance struct_padding_fractional : forall {σ : genv} p cls, CFractional (struct_padding p cls).
    #[global] Declare Instance struct_padding_frac_valid : forall {σ : genv} p cls, CFracValid0 (struct_padding p cls).

    #[global] Declare Instance struct_padding_type_ptr_observe : forall {σ : genv} p cls q,
        Observe (type_ptr (Tnamed cls) p) (struct_padding p cls q).

  End with_cpp_logic.

union_padding cls q active_member is q ownership of the union padding for union cls for the active member active_member. When there is no active member the active_member is None, otherwise it is Some idx where idx is the numeric index of member field.
    Axiom union_padding : forall `{!cpp_logic thread_info Σ} {σ:genv},
      ptr -> globname -> cQp.t -> option nat -> mpred.

  Section with_cpp_logic.
  Context `{cpp_logic}.

    #[global] Declare Instance union_padding_timeless {σ:genv} : Timeless4 (union_padding).
    #[global] Declare Instance union_padding_fractional : forall {σ : genv} p cls, CFractional1 (union_padding p cls).
    #[global] Declare Instance union_padding_frac_valid : forall {σ : genv} p cls, CFracValid1 (union_padding p cls).

    #[global] Declare Instance union_padding_type_ptr_observe : forall {σ : genv} p cls q active,
        Observe (type_ptr (Tnamed cls) p) (union_padding p cls q active).
    #[global] Declare Instance union_padding_agree : forall {σ : genv} p cls q q' i i',
        Observe2 [| i = i' |] (union_padding p cls q i) (union_padding p cls q' i').

  End with_cpp_logic.

  End with_cpp.

  (* strict validity (not past-the-end) *)
  Notation strict_valid_ptr := (_valid_ptr Strict).
  (* validity (past-the-end allowed) *)
  Notation valid_ptr := (_valid_ptr Relaxed).
End CPP_LOGIC.

(* Pointer axioms. XXX Not modeled for now. *)
Module Type VALID_PTR_AXIOMS
  (Import P : PTRS_INTF)
  (Import INTF : VALUES_INTF_FUNCTOR P)
  (Import CC : CPP_LOGIC_CLASS)
  (Import CPP : CPP_LOGIC P INTF CC).

  Implicit Types (p : ptr).

  Section with_cpp.
    Context `{cpp_logic} {σ : genv}.

    Axiom invalid_ptr_invalid : forall vt,
      _valid_ptr vt invalid_ptr |-- False.

Justified by https://eel.is/c++draft/expr.add#4.1.
    Axiom _valid_ptr_nullptr_sub_false : forall vt ty (i : Z) (_ : i <> 0),
      _valid_ptr vt (nullptr ,, o_sub σ ty i) |-- False.
    (*
    TODO Controversial; if f is the first field, nullptr->f or casts relying on
    https://eel.is/c++draft/basic.compound4 might invalidate this. To make this valid, we could ensure our axiomatic semantics produces [nullptr] instead of [nullptr ., o_field]. *)
    (* Axiom _valid_ptr_nullptr_field_false : forall vt f,
      _valid_ptr vt (nullptr ,, o_field σ f) |-- False. *)

These axioms are named after the predicate in the conclusion.
TODO: The intended proof of strict_valid_ptr_sub assumes that, if p' normalizes to p ., [ ty ! i ], then valid_ptr p' is defined to imply validity of all pointers from p to p'.
Note that `arrR` exposes stronger reasoning principles, but this might still be useful.
    Axiom strict_valid_ptr_sub : ∀ (i j k : Z) p ty vt1 vt2,
      (i <= j < k)%Z ->
      _valid_ptr vt1 (p ,, o_sub σ ty i) |--
      _valid_ptr vt2 (p ,, o_sub σ ty k) -* strict_valid_ptr (p ,, o_sub σ ty j).

XXX: this axiom is convoluted but TODO: The intended proof of strict_valid_ptr_field_sub (and friends) is that (1) if p' normalizes to p'' ., [ ty ! i ], then valid_ptr p' implies valid_ptr p''. (2) p ,, o_field σ f ,, o_sub σ ty i will normalize to p ,, o_field σ f ,, o_sub σ ty i, without cancellation.
    Axiom strict_valid_ptr_field_sub : ∀ (p : ptr) ty (i : Z) f vt,
      (0 < i)%Z ->
      _valid_ptr vt (p ,, o_field σ f ,, o_sub σ ty i) |-- strict_valid_ptr (p ,, o_field σ f).

    (* TODO: can we deduce that p is strictly valid? *)
    Axiom _valid_ptr_field : ∀ p f vt,
      _valid_ptr vt (p ,, o_field σ f) |-- _valid_ptr vt p.
    (* TODO: Pointers to fields can't be past-the-end, right?
    Except 0-size arrays. *)
    (* Axiom strict_valid_ptr_field : ∀ p f,
      valid_ptr (p ,, o_field σ f) |--
      strict_valid_ptr (p ,, o_field σ f). *)
    (* TODO: if we add strict_valid_ptr_field, we can derive
    _valid_ptr_field from just strict_valid_ptr_field *)
    (* Axiom strict_valid_ptr_field : ∀ p f,
      strict_valid_ptr (p ,, o_field σ f) |-- strict_valid_ptr p. *)

    (* We're ignoring virtual inheritance here, since we have no plans to
    support it for now, but this might hold there too. *)
    Axiom o_base_directly_derives : forall p base derived,
      strict_valid_ptr (p ,, o_base σ derived base) |--
      [| directly_derives σ derived base |].

    Axiom o_derived_directly_derives : forall p base derived,
      strict_valid_ptr (p ,, o_derived σ base derived) |--
      [| directly_derives σ derived base |].

    (* TODO: maybe add a validity of offsets to allow stating this more generally. *)
    Axiom valid_o_sub_size : forall p ty i vt,
      _valid_ptr vt (p ,, o_sub σ ty i) |-- [| is_Some (size_of σ ty) |].

    Axiom type_ptr_o_base : forall derived base p,
      class_derives derived [base] ->
      type_ptr (Tnamed derived) p ⊢ type_ptr (Tnamed base) (p ,, _base derived base).

    (*
    Axiom type_ptr_o_field_type_ptr : forall p fld cls st,
      glob_def σ cls = Some (Gstruct st) ->
      fld ∈ s_fields st →
         type_ptr (Tnamed cls) p
      ⊢ type_ptr fld.(mem_type) (p ., Field cls fld.(mem_name)). (* TODO field names *)
      *)

    Axiom type_ptr_o_sub : forall p (m n : N) ty,
      (m < n)%N ->
      type_ptr (Tarray ty n) p ⊢ type_ptr ty (p ,, _sub ty m).
  End with_cpp.
End VALID_PTR_AXIOMS.

Declare Module L : CPP_LOGIC PTRS_INTF_AXIOM VALUES_INTF_AXIOM LC.
Export L.

Declare Module Export VALID_PTR : VALID_PTR_AXIOMS PTRS_INTF_AXIOM VALUES_INTF_AXIOM LC L.

Section valid_ptr_code.
  Context `{cpp_logic} {σ : genv} (tu : translation_unit).

  Lemma code_at_valid : forall f p, code_at tu f p |-- valid_ptr p.
  Proof. intros. rewrite code_at_strict_valid; apply strict_valid_valid. Qed.
  Lemma method_at_valid : forall f p, method_at tu f p |-- valid_ptr p.
  Proof. intros. rewrite method_at_strict_valid; apply strict_valid_valid. Qed.
  Lemma ctor_at_valid : forall f p, ctor_at tu f p |-- valid_ptr p.
  Proof. intros. rewrite ctor_at_strict_valid; apply strict_valid_valid. Qed.
  Lemma dtor_at_valid : forall f p, dtor_at tu f p |-- valid_ptr p.
  Proof. intros. rewrite dtor_at_strict_valid; apply strict_valid_valid. Qed.
End valid_ptr_code.

mlock Definition exposed_ptr `{cpp_logic} p : mpred :=
  valid_ptr p ** ∃ aid, [| ptr_alloc_id p = Some aid |] ** exposed_aid aid.
#[global] Arguments exposed_ptr {_ _ _} p : assert.

Physical representation of pointers. pinned_ptr va p states that the abstract pointer p is tied to a virtual address va. pinned_ptr will only hold on pointers that are associated to addresses, but other pointers exist.
mlock Definition pinned_ptr `{cpp_logic} (va : vaddr) (p : ptr) : mpred :=
  [| ptr_vaddr p = Some va |] ** exposed_ptr p.
#[global] Arguments pinned_ptr {_ _ _} va p : assert.

Notation pinned_ptr_Z va p :=
  ([| 0 <= va |]%Z ** pinned_ptr (Z.to_N va) p).

Section pinned_ptr_def.
  Context `{cpp_logic}.

  #[global] Instance exposed_ptr_persistent p : Persistent (exposed_ptr p).
  Proof. rewrite exposed_ptr.unlock. apply _. Qed.
  #[global] Instance exposed_ptr_affine p : Affine (exposed_ptr p).
  Proof. rewrite exposed_ptr.unlock. apply _. Qed.
  #[global] Instance exposed_ptr_timeless p : Timeless (exposed_ptr p).
  Proof. rewrite exposed_ptr.unlock. apply _. Qed.
  #[global] Instance exposed_ptr_valid p :
    Observe (valid_ptr p) (exposed_ptr p).
  Proof. rewrite exposed_ptr.unlock. apply _. Qed.

  Lemma exposed_ptr_nullptr : |-- exposed_ptr nullptr.
  Proof.
    rewrite exposed_ptr.unlock ptr_alloc_id_nullptr.
    iDestruct valid_ptr_nullptr as "$". iExists _.
    by iDestruct exposed_aid_null_alloc_id as "$".
  Qed.

  Lemma offset2_exposed_ptr p o1 o2 :
    valid_ptr (p ,, o2) |-- exposed_ptr (p ,, o1) -* exposed_ptr (p ,, o2).
  Proof.
    rewrite exposed_ptr.unlock.
    iIntros "#V2 #[V1 E]"; iFrame "V2".
    iDestruct (valid_ptr_alloc_id with "V1") as %?.
    iDestruct (valid_ptr_alloc_id with "V2") as %?.
    by rewrite ptr_alloc_id_offset // ptr_alloc_id_offset.
  Qed.

  Lemma offset_exposed_ptr p o :
    valid_ptr (p ,, o) |-- exposed_ptr p -* exposed_ptr (p ,, o).
  Proof. rewrite -{2}(offset_ptr_id p). apply offset2_exposed_ptr. Qed.

  Lemma offset_inv_exposed_ptr p o :
    valid_ptr p |-- exposed_ptr (p ,, o) -* exposed_ptr p.
  Proof. rewrite -{1 3}(offset_ptr_id p). apply offset2_exposed_ptr. Qed.

  Lemma offset_exposed_ptr_all p o :
    valid_ptr p ∗ valid_ptr (p ,, o) ⊢ exposed_ptr p ∗-∗ exposed_ptr (p ,, o).
  Proof.
    iIntros "#[V V']"; iSplit.
    by iApply offset_exposed_ptr.
    by iApply offset_inv_exposed_ptr.
  Qed.

  #[global] Instance pinned_ptr_persistent va p : Persistent (pinned_ptr va p).
  Proof. rewrite pinned_ptr.unlock. apply _. Qed.
  #[global] Instance pinned_ptr_affine va p : Affine (pinned_ptr va p).
  Proof. rewrite pinned_ptr.unlock. apply _. Qed.
  #[global] Instance pinned_ptr_timeless va p : Timeless (pinned_ptr va p).
  Proof. rewrite pinned_ptr.unlock. apply _. Qed.

  Lemma pinned_ptr_intro p va :
    ptr_vaddr p = Some va -> exposed_ptr p |-- pinned_ptr va p.
  Proof. rewrite pinned_ptr.unlock. by iIntros (?) "$". Qed.

  #[global] Instance pinned_ptr_ptr_vaddr va p :
    Observe [| ptr_vaddr p = Some va |] (pinned_ptr va p).
  Proof. rewrite pinned_ptr.unlock. apply _. Qed.

  Lemma pinned_ptr_change_va_eq (p : ptr) (va va' : vaddr)
    (Heq : ptr_vaddr p = Some va) :
    pinned_ptr va' p |-- [| va' = va |] ** pinned_ptr va p.
  Proof.
    iIntros "#P".
    iDestruct (observe_elim_pure (ptr_vaddr p = Some va') with "P") as %?.
    simplify_eq. auto.
  Qed.

  Lemma pinned_ptr_change_va p va va'
    (Heq : ptr_vaddr p = Some va) :
    pinned_ptr va' p |-- pinned_ptr va p.
  Proof. rewrite pinned_ptr_change_va_eq //. by iIntros "[_ $]". Qed.

  #[global] Instance pinned_ptr_agree va1 va2 p :
    Observe2 [| va1 = va2 |] (pinned_ptr va1 p) (pinned_ptr va2 p).
  Proof.
    iIntros "#P1 #P2 !>".
    iDestruct (observe_elim_pure (_ = _) with "P1") as %?.
    iDestruct (observe_elim_pure (_ = _) with "P2") as %?; simplify_eq.
    by [].
  Qed.

  #[global] Instance pinned_ptr_valid va p :
    Observe (valid_ptr p) (pinned_ptr va p).
  Proof. rewrite pinned_ptr.unlock. apply _. Qed.

Just a corollary of provides_storage_same_address in the style of provides_storage_pinned_ptr.
  Lemma provides_storage_pinned_ptr_pure {storage_ptr obj_ptr aty va} :
    ptr_vaddr storage_ptr = Some va ->
    provides_storage storage_ptr obj_ptr aty |-- [| ptr_vaddr obj_ptr = Some va |].
  Proof. rewrite provides_storage_same_address. by iIntros (HP <-). Qed.
End pinned_ptr_def.

Section with_cpp.
  Context `{cpp_logic} {σ : genv}.

  Lemma same_address_bool_null p tv :
    _valid_ptr tv p |--
    [| same_address_bool p nullptr = bool_decide (p = nullptr) |].
  Proof. rewrite same_address_eq_null; iIntros "!%". apply bool_decide_ext. Qed.

  Lemma valid_ptr_zero_null p :
    ptr_vaddr p = Some 0%N ->
    valid_ptr p |-- [| p = nullptr |].
  Proof.
    rewrite same_address_eq_null.
    iIntros (Haddr [Hsuff _]) "!%". apply: Hsuff.
    rewrite same_address_iff ptr_vaddr_nullptr; naive_solver.
  Qed.

  Lemma valid_ptr_nonnull_nonzero p :
    p <> nullptr ->
    valid_ptr p |-- [| ptr_vaddr p <> Some 0%N |].
  Proof.
    destruct (decide (ptr_vaddr p = Some 0%N)); last naive_solver.
    rewrite valid_ptr_zero_null; naive_solver.
  Qed.

  Lemma type_ptr_nonnull ty p :
    type_ptr ty p |-- [| p <> nullptr |].
  Proof. rewrite -{1}(offset_ptr_id p). apply type_ptr_off_nonnull. Qed.

  #[global] Instance type_ptr_observe_nonnull ty p :
    Observe [| p <> nullptr |] (type_ptr ty p).
  Proof. rewrite type_ptr_nonnull. refine _. Qed.

  #[global] Instance provides_storage_preserves_nullptr {storage_ptr obj_ptr aty} :
    Observe [| storage_ptr = nullptr <-> obj_ptr = nullptr |] (provides_storage storage_ptr obj_ptr aty).
  Proof.
    apply observe_intro_only_provable; iIntros "PS".
    iDestruct (provides_storage_same_address with "PS") as %Hsm.
    iDestruct (provides_storage_valid_obj_ptr with "PS") as "#VO".
    iDestruct (provides_storage_valid_storage_ptr with "PS") as "#VS {PS}".
    iDestruct (same_address_eq_null with "VO") as %[HeqO _].
    iDestruct (same_address_eq_null with "VS") as %[HeqS _].
    iIntros "!%"; split; intros ->.
    - apply HeqO, symmetry, Hsm.
    - apply HeqS, Hsm.
  Qed.

  #[global] Instance provides_storage_preserves_nonnull {storage_ptr obj_ptr aty} :
    Observe [| storage_ptr <> nullptr <-> obj_ptr <> nullptr |] (provides_storage storage_ptr obj_ptr aty).
  Proof.
    apply observe_intro_only_provable. iIntros "PS".
    by iDestruct (provides_storage_preserves_nullptr with "PS") as "#-> {PS}".
  Qed.

  #[global] Instance pinned_ptr_unique va va' p :
    Observe2 [| va = va' |] (pinned_ptr va p) (pinned_ptr va' p).
  Proof.
    rewrite pinned_ptr.unlock.
    iIntros "[%H1 _] [%H2 _] !> !%". congruence.
  Qed.

  Lemma offset_2_pinned_ptr_pure o1 o2 z1 z2 va p :
    eval_offset σ o1 = Some z1 ->
    eval_offset σ o2 = Some z2 ->
    ptr_vaddr (p ,, o1) = Some va ->
    valid_ptr p |-- valid_ptr (p ,, o1) -* valid_ptr (p ,, o2) -*
    [| 0 <= Z.of_N va - z1 + z2 |]%Z **
    [| ptr_vaddr (p ,, o2) = Some (Z.to_N (Z.of_N va - z1 + z2)) |].
  Proof.
    iIntros (He1 He2 Hpin1) "V V1 V2".
    iDestruct (offset_inv_pinned_ptr_pure with "V1") as %[Hle1 Hp]; [done..|].
    iDestruct (offset_pinned_ptr_pure with "V2") as %[Hle2 Hgoal]; [done..|].
    iIntros "!%". by rewrite Z2N.id in Hgoal, Hle2.
  Qed.

  Lemma pinned_ptr_null : |-- pinned_ptr 0 nullptr.
  Proof.
    rewrite pinned_ptr.unlock.
    iFrame (ptr_vaddr_nullptr).
    iApply exposed_ptr_nullptr.
  Qed.

  #[global] Instance pinned_ptr_zero_is_null (p : ptr) :
    Observe [| p = nullptr |] (pinned_ptr 0 p).
  Proof.
    rewrite pinned_ptr.unlock.
    iIntros "[%Heq #E]".
    rewrite -valid_ptr_zero_null //.
    by iApply (exposed_ptr_valid with "E").
  Qed.

  #[global] Instance pinned_ptr_null_is_zero addr :
    Observe [| addr = 0 |]%N (pinned_ptr addr nullptr).
  Proof.
    rewrite pinned_ptr.unlock ptr_vaddr_nullptr.
    apply: (observe_derive_only_provable (Some 0%N = Some addr)); naive_solver.
  Qed.

  Lemma pinned_ptr_same_address pp1 pp2 v :
    same_address pp1 pp2 ->
    exposed_ptr pp2 |-- pinned_ptr v pp1 -* pinned_ptr v pp2.
  Proof.
    rewrite pinned_ptr.unlock.
    by iIntros ((? & -> & ->)%same_address_iff) "$ [%Hp _] !%".
  Qed.

  Lemma offset_pinned_ptr_Z o z va p :
    eval_offset _ o = Some z ->
    valid_ptr (p ,, o) |--
    pinned_ptr va p -*
    pinned_ptr_Z (Z.of_N va + z) (p ,, o).
  Proof.
    rewrite pinned_ptr.unlock.
    iIntros (He) "#V' #(%P & E)".
    iDestruct (offset_pinned_ptr_pure with "V'") as "[$ $]"; [done..|].
    by iApply offset_exposed_ptr.
  Qed.

  Lemma offset_pinned_ptr o z va p :
    eval_offset _ o = Some z ->
    valid_ptr (p ,, o) |--
    pinned_ptr va p -*
    pinned_ptr (Z.to_N (Z.of_N va + z)) (p ,, o).
  Proof.
    iIntros (?) "V P".
    by iDestruct (offset_pinned_ptr_Z with "V P") as "[_ $]".
  Qed.

  Lemma offset_inv_pinned_ptr_Z o z va p :
    eval_offset _ o = Some z ->
    valid_ptr p |--
    pinned_ptr va (p ,, o) -*
    pinned_ptr_Z (Z.of_N va - z) p.
  Proof.
    rewrite pinned_ptr.unlock.
    iIntros (He) "#V #(%P & E)".
    iDestruct (offset_inv_pinned_ptr_pure with "[]") as "-#[$$]"; [done..| |].
    { by iApply (observe with "E"). }
    by iApply offset_inv_exposed_ptr.
  Qed.

  Lemma offset_2_pinned_ptr_Z o1 o2 z1 z2 va p :
    eval_offset σ o1 = Some z1 ->
    eval_offset σ o2 = Some z2 ->
    valid_ptr p |-- valid_ptr (p ,, o1) -* valid_ptr (p ,, o2) -*
    pinned_ptr va (p ,, o1) -*
    pinned_ptr_Z (Z.of_N va - z1 + z2) (p ,, o2).
  Proof.
    rewrite pinned_ptr.unlock.
    iIntros (He1 He2) "V V1 #V2 #(%P & E)".
    iDestruct (offset_2_pinned_ptr_pure with "V V1 V2") as "[$$]"; [done..|].
    by iApply offset2_exposed_ptr.
  Qed.

  Lemma pinned_ptr_aligned_divide va n p :
    pinned_ptr va p ⊢
    [| aligned_ptr n p <-> (n | va)%N |].
  Proof.
    rewrite pinned_ptr.unlock; iIntros "(%P & _) !%".
    exact: pinned_ptr_pure_aligned_divide.
  Qed.

  Lemma pinned_ptr_pure_type_divide_1 va n p ty
    (Hal : align_of ty = Some n) :
    type_ptr ty p ⊢ [| ptr_vaddr p = Some va |] -∗ [| (n | va)%N |].
  Proof.
    rewrite type_ptr_aligned_pure. iIntros "!%".
    exact: pinned_ptr_pure_divide_1.
  Qed.

  Lemma pinned_ptr_type_divide_1 va n p ty
    (Hal : align_of ty = Some n) :
    type_ptr ty p ⊢ pinned_ptr va p -∗ [| (n | va)%N |].
  Proof.
    rewrite pinned_ptr.unlock.
    iIntros "#? #(? & _)". by iApply pinned_ptr_pure_type_divide_1.
  Qed.

  Lemma shift_pinned_ptr_Z_sub ty z va (p1 p2 : ptr) o:
    p1 ,, o_sub _ ty z = p2 ->
    size_of σ ty = Some o ->
        valid_ptr p2 ** pinned_ptr va p1
    |-- pinned_ptr_Z (Z.of_N va + z * Z.of_N o) p2.
  Proof.
    move => <- o_eq.
    iIntros "[val pin1]".
    iApply (offset_pinned_ptr_Z with "val") => //.
    rewrite eval_o_sub o_eq /= Z.mul_comm //.
  Qed.

  Lemma shift_pinned_ptr_sub ty z va (p1 p2 : ptr) o:
    p1 ,, o_sub _ ty z = p2 ->
    size_of σ ty = Some o ->
        valid_ptr p2 ** pinned_ptr va p1
    |-- pinned_ptr (Z.to_N (Z.of_N va + z * Z.of_N o)) p2.
  Proof.
    iIntros (??) "VP".
    by iDestruct (shift_pinned_ptr_Z_sub with "VP") as "[_ $]".
  Qed.

  Lemma _valid_valid p vt : _valid_ptr vt p |-- valid_ptr p.
  Proof. case: vt => [|//]. exact: strict_valid_valid. Qed.

  Lemma valid_ptr_sub (i j k : Z) p ty vt
    (Hj : (i <= j <= k)%Z) :
    _valid_ptr vt (p ,, o_sub σ ty i) |--
    _valid_ptr vt (p ,, o_sub σ ty k) -* valid_ptr (p ,, o_sub σ ty j).
  Proof.
    destruct (decide (j = k)) as [->|Hne].
    { rewrite -_valid_valid. by iIntros "_ $". }
    rewrite -strict_valid_valid. apply strict_valid_ptr_sub. lia.
  Qed.

  Lemma _valid_ptr_field_sub (i : Z) (p : ptr) ty f vt (Hle : (0 <= i)%Z) :
    _valid_ptr vt (p ,, o_field σ f ,, o_sub σ ty i) |-- _valid_ptr vt (p ,, o_field σ f).
  Proof.
    iIntros "V". case: (decide (i = 0)%Z) Hle => [-> _|Hne Hle].
    - iDestruct (valid_o_sub_size with "V") as %?.
      by rewrite offset_ptr_sub_0.
    - rewrite strict_valid_ptr_field_sub; last by lia.
      case: vt => //. by rewrite strict_valid_valid.
  Qed.

  Lemma o_base_derived_strict p base derived :
    strict_valid_ptr (p ,, o_base σ derived base) |--
    [| p ,, o_base σ derived base ,, o_derived σ base derived = p |].
  Proof.
    rewrite o_base_directly_derives. f_equiv => ?. exact: o_base_derived.
  Qed.

  Lemma o_derived_base_strict p base derived :
    strict_valid_ptr (p ,, o_derived σ base derived) |--
    [| p ,, o_derived σ base derived ,, o_base σ derived base = p |].
  Proof.
    rewrite o_derived_directly_derives. f_equiv => ?. exact: o_derived_base.
  Qed.

  Lemma o_derived_base_type p base derived ty :
    type_ptr ty (p ,, o_derived σ base derived) |--
    [| p ,, o_derived σ base derived ,, o_base σ derived base = p |].
  Proof. rewrite type_ptr_strict_valid. apply (o_derived_base_strict p). Qed.

_inv because unlike type_ptr_o_base and type_ptr_o_field_type_ptr, this lemma fits the type_ptr _ (p ,, o) ⊢ type_ptr _ p schema instead of the converse.
  Lemma type_ptr_o_derived_inv derived base p :
    class_derives derived [base] ->
    type_ptr (Tnamed derived) (p ,, _derived base derived) |--
    type_ptr (Tnamed base) p.
  Proof.
    iIntros (Hcd) "T".
    iDestruct (o_derived_base_type with "T") as %Hp.
    by rewrite (type_ptr_o_base _ _ _ Hcd) Hp.
  Qed.

p is a valid pointer value in the sense of the standard, or "standard-valid" (https://eel.is/c++draft/basic.compound3.1), that is both valid (in our sense) and live. In particular, [p] is a valid pointer value even when accounting for pointer zapping.
  Definition _valid_live_ptr vt (p : ptr) : mpred :=
    _valid_ptr vt p ∗ live_ptr p.
  Definition valid_live_ptr p : mpred := _valid_live_ptr Relaxed p.
  Definition strict_valid_live_ptr p : mpred := _valid_live_ptr Strict p.

  #[global] Instance tptsto_flip_mono :
    Proper (flip genv_leq ==> eq ==> eq ==> eq ==> eq ==> flip (⊢))
      (@tptsto _ _ _).
  Proof. repeat intro. exact: tptsto_mono. Qed.

  #[global] Instance tptsto_as_cfractional ty : AsCFractional2 (tptsto ty).
  Proof. solve_as_cfrac. Qed.

  #[global] Instance mdc_path_as_cfractional this mdc :
    AsCFractional1 (mdc_path this mdc).
  Proof. solve_as_cfrac. Qed.

  #[global] Instance tptsto_observe_nonnull t q p v :
    Observe [| p <> nullptr |] (tptsto t q p v).
  Proof.
    apply: observe_intro.
    destruct (decide (p = nullptr)); subst; last by eauto.
    rewrite {1}tptsto_nonnull. exact: bi.False_elim.
  Qed.

  Lemma tptsto_disjoint ty c1 c2 q p v1 v2 :
    tptsto ty (cQp.mk c1 1) p v1 ** tptsto ty (cQp.mk c2 q) p v2 |-- False.
  Proof.
    iIntros "[T1 T2]".
    iDestruct (tptsto_agree with "T1 T2") as %->.
    iCombine "T1 T2" as "T".
    by iDestruct (cfrac_valid_2 with "T") as %?%Qp.not_add_le_l.
  Qed.

  Lemma offset_ptr_congP (p : ptr) o1 o2 :
    offset_congP σ o1 o2 |--
    type_ptr Tbyte (p ,, o1) -*
    type_ptr Tbyte (p ,, o2) -*
    ptr_congP σ (p ,, o1) (p ,, o2).
  Proof.
    iIntros "% T1 T2". rewrite /ptr_congP. iFrame "T1 T2".
    auto using offset_ptr_cong.
  Qed.

Just wrappers.

We can lift validity entailments through Observe (using Observe_mono. These are not instances, to avoid causing slowdowns in proof search.
  Lemma observe_strict_valid_valid
    `(Hobs : !Observe (strict_valid_ptr p) P) : Observe (valid_ptr p) P.
  Proof. by rewrite -strict_valid_valid. Qed.

  Lemma observe_type_ptr_strict_valid
    `(Hobs : !Observe (type_ptr ty p) P) : Observe (strict_valid_ptr p) P.
  Proof. by rewrite -type_ptr_strict_valid. Qed.

  Lemma observe_type_ptr_valid_plus_one
    `(Hobs : !Observe (type_ptr ty p) P) : Observe (valid_ptr (p ,, o_sub σ ty 1)) P.
  Proof. by rewrite -type_ptr_valid_plus_one. Qed.

  Lemma type_ptr_valid ty p : type_ptr ty p |-- valid_ptr p.
  Proof. by rewrite type_ptr_strict_valid strict_valid_valid. Qed.

  Lemma type_ptr__valid ty p vt : type_ptr ty p |-- _valid_ptr vt p.
  Proof.
    case: vt.
    apply type_ptr_strict_valid.
    apply type_ptr_valid.
  Qed.

  #[global] Instance type_ptr_size_observe ty p :
    Observe [| is_Some (size_of σ ty) |] (type_ptr ty p).
  Proof. rewrite type_ptr_size. apply _. Qed.

  #[global] Instance valid_ptr_sub_0 (p : ptr) ty :
    HasSize ty ->
    Observe (valid_ptr (p ,, o_sub σ ty 0)) (valid_ptr p).
  Proof. intros. rewrite o_sub_0 // offset_ptr_id. refine _. Qed.
  #[global] Instance type_ptr_sub_0 (p : ptr) ty :
    HasSize ty ->
    Observe (valid_ptr (p ,, o_sub σ ty 0)) (type_ptr ty p).
  Proof.
    intros. by rewrite type_ptr_valid; apply valid_ptr_sub_0.
  Qed.
  #[global] Instance type_ptr_valid_ptr_next (p : ptr) ty (m n : Z) :
    (m = n + 1)%Z ->
    Observe (valid_ptr (p ,, o_sub σ ty m)) (type_ptr ty (p ,, o_sub σ ty n)).
  Proof.
    intros; subst.
    iIntros "X".
    iDestruct (observe (valid_ptr (p ,, o_sub _ _ _ ,, o_sub _ _ _)) with "X") as "z".
    apply observe_type_ptr_valid_plus_one. refine _.
    by rewrite o_sub_sub.
  Qed.

  Lemma same_alloc_refl p : valid_ptr p ⊢ [| same_alloc p p |].
  Proof.
    rewrite valid_ptr_alloc_id same_alloc_iff. iIntros "!%". case; naive_solver.
  Qed.

  Lemma live_has_alloc_id p :
    live_ptr p ⊢ ∃ aid, [| ptr_alloc_id p = Some aid |] ∗ live_alloc_id aid.
  Proof. rewrite /live_ptr; iIntros. case: (ptr_alloc_id p) => /= [aid|]; eauto. Qed.

  #[global] Instance struct_padding_as_fractional p cls : AsCFractional0 (struct_padding p cls).
  Proof. solve_as_cfrac. Qed.
  #[global] Instance struct_padding_valid_observe p q cls : Observe (strict_valid_ptr p) (struct_padding p cls q).
  Proof. rewrite -type_ptr_strict_valid; apply _. Qed.
  #[global] Instance struct_paddingR_valid_observe p q cls : Observe (valid_ptr p) (struct_padding p cls q).
  Proof. rewrite -strict_valid_valid; apply _. Qed.

  #[global] Instance union_padding_as_fractional p cls : AsCFractional1 (union_padding p cls).
  Proof. solve_as_cfrac. Qed.

  #[global] Instance union_padding_strict_valid_observe p q cls i : Observe (strict_valid_ptr p) (union_padding p cls q i).
  Proof. rewrite -type_ptr_strict_valid; apply _. Qed.
  #[global] Instance union_padding_valid_observe p q cls i : Observe (valid_ptr p) (union_padding p cls q i).
  Proof. rewrite -strict_valid_valid; apply _. Qed.

End with_cpp.

Section has_type.
  Context `{cpp_logic}.

has_type

  #[global] Instance: Params (@has_type) 3 := {}.
  #[global] Instance has_type_proper :
    Proper (genv_eq ==> eq ==> eq ==> (≡)) (@has_type _ _ _).
  Proof. intros ?? Heq ??? ???; split'; apply has_type_mono => //; apply Heq. Qed.

  #[global] Instance has_type_flip_mono :
    Proper (flip genv_leq ==> eq ==> eq ==> flip bi_entails) (@has_type _ _ _).
  Proof. by move=> ??+ ??-> ??-> => ->. Qed.

  #[global] Instance has_type_knowledge {σ} : Knowledge2 has_type_or_undef.
  Proof. rewrite has_type_or_undef_unfold; split; apply _. Qed.
  #[global] Instance has_type_timeless {σ} : Timeless2 has_type_or_undef.
  Proof. rewrite has_type_or_undef_unfold; apply _. Qed.

  Lemma has_type_nonptr {σ} v ty :
    nonptr_prim_type ty ->
    has_type v ty -|- [| has_type_prop v ty |].
  Proof.
    intros. split'.
    - by rewrite -has_type_has_type_prop.
    - by rewrite -has_type_prop_has_type_noptr.
  Qed.

  Lemma has_type_qual_iff {σ} t q x :
    has_type x t -|- has_type x (Tqualified q t).
  Proof.
    by rewrite (has_type_erase_qualifiers t)
      (has_type_erase_qualifiers (Tqualified _ _)).
  Qed.

  Lemma has_type_drop_qualifiers {σ} v ty :
    has_type v ty -|- has_type v (drop_qualifiers ty).
  Proof.
    rewrite (has_type_erase_qualifiers (drop_qualifiers _)) erase_drop_qualifiers.
    by rewrite -has_type_erase_qualifiers.
  Qed.

  Lemma has_type_val_related {σ} ty v1 v2 :
    val_related ty v1 v2 ->
    has_type v1 ty -|- has_type v2 ty.
  Proof.
    induction 1.
    { done. }
    { by rewrite -!has_type_qual_iff. }
    all: by rewrite !has_type_nonptr ?has_type_prop_raw_bytes_of_val.
  Qed.

has_type_or_undef

  Definition is_undef (v : val) : bool :=
    if v is Vundef then true else false.

  #[global] Instance: Params (@has_type_or_undef) 3 := {}.
  #[global] Instance has_type_or_undef_proper :
    Proper (genv_eq ==> eq ==> eq ==> (≡)) (@has_type_or_undef _ _ _).
  Proof. rewrite has_type_or_undef_unfold; solve_proper. Qed.
  #[global] Instance has_type_or_undef_mono :
    Proper (genv_leq ==> eq ==> eq ==> (⊢)) (@has_type_or_undef _ _ _).
  Proof. rewrite has_type_or_undef_unfold; solve_proper. Qed.

  Lemma has_type_or_undef_undef {σ} t : |-- has_type_or_undef Vundef t.
  Proof. rewrite has_type_or_undef_unfold. by iRight. Qed.

  Lemma has_type_or_undef_nonundef {σ} t v :
    ~~ is_undef v -> has_type_or_undef v t -|- has_type v t.
  Proof.
    intros. rewrite has_type_or_undef_unfold. split'.
    - iIntros "[$ | %]". by destruct v.
    - iIntros "?". by iLeft.
  Qed.

  Lemma has_type_or_undef_qual_iff {σ} t q v :
    has_type_or_undef v t -|- has_type_or_undef v (Tqualified q t).
  Proof.
    by rewrite !has_type_or_undef_unfold -!has_type_qual_iff.
  Qed.

  Lemma has_type_or_undef_val_related {σ} ty v1 v2 :
    val_related ty v1 v2 ->
    has_type_or_undef v1 ty |-- has_type_or_undef v2 ty.
  Proof.
    induction 1.
    { done. }
    { by rewrite -!has_type_or_undef_qual_iff. }
    all: rewrite !has_type_or_undef_nonundef//.
    all: by rewrite !has_type_nonptr// has_type_prop_raw_bytes_of_val.
  Qed.
End has_type.

#[global] Notation "'validP<' ty > v" := (has_type v ty%cpp_type)
  (at level 30, ty at level 20, v at level 1, format "validP< ty > v") : bi_scope.