ProofMode.md 16.1 KB
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Tactic overview
===============

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Many of the tactics below apply to more goals than described in this document
since the behavior of these tactics can be tuned via instances of the type
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classes in the file [proofmode/classes](proofmode/classes.v). Most notably, many
of the tactics can be applied when the connective to be introduced or to be eliminated
appears under a later, an update modality, or in the conclusion of a
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weakest precondition.
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Applying hypotheses and lemmas
------------------------------

- `iExact "H"`  : finish the goal if the conclusion matches the hypothesis `H`
- `iAssumption` : finish the goal if the conclusion matches any hypothesis
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- `iApply pm_trm` : match the conclusion of the current goal against the
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  conclusion of `pm_trm` and generates goals for the premises of `pm_trm`. See
  proof mode terms below.
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  If the applied term has more premises than given specialization patterns, the
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  pattern is extended with `[] ... []`.  As a consequence, all unused spatial
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  hypotheses move to the last premise.
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Context management
------------------

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- `iIntros (x1 ... xn) "ipat1 ... ipatn"` : introduce universal quantifiers
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  using Coq introduction patterns `x1 ... xn` and implications/wands using proof
  mode introduction patterns `ipat1 ... ipatn`.
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- `iClear (x1 ... xn) "selpat"` : clear the hypotheses given by the selection
  pattern `selpat` and the Coq level hypotheses/variables `x1 ... xn`.
- `iRevert (x1 ... xn) "selpat"` : revert the hypotheses given by the selection
  pattern `selpat` into wands, and the Coq level hypotheses/variables
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  `x1 ... xn` into universal quantifiers. Intuitionistic hypotheses are wrapped
  into the intuitionistic modality.
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- `iRename "H1" into "H2"` : rename the hypothesis `H1` into `H2`.
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- `iSpecialize pm_trm` : instantiate universal quantifiers and eliminate
  implications/wands of a hypothesis `pm_trm`. See proof mode terms below.
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- `iSpecialize pm_trm as #` : instantiate universal quantifiers and eliminate
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  implications/wands of a hypothesis `pm_trm` whose conclusion is persistent.
  All hypotheses can be used for proving the premises of `pm_trm`, as well as
  for the resulting main goal.
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- `iPoseProof pm_trm as (x1 ... xn) "ipat"` : put `pm_trm` into the context and
  eliminates it. This tactic is essentially the same as `iDestruct` with the
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  difference that when `pm_trm` is a non-universally quantified intuitionistic
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  hypothesis, it will not throw the hypothesis away.
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- `iAssert P with "spat" as "ipat"` : generates a new subgoal `P` and adds the
  hypothesis `P` to the current goal. The specialization pattern `spat`
  specifies which hypotheses will be consumed by proving `P`. The introduction
  pattern `ipat` specifies how to eliminate `P`.
  In case all branches of `ipat` start with a `#` (which causes `P` to be moved
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  to the intuitionistic context) or with an `%` (which causes `P` to be moved to
  the pure Coq context), then one can use all hypotheses for proving `P` as well
  as for proving the current goal.
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- `iAssert P as %cpat` : assert `P` and eliminate it using the Coq introduction
  pattern `cpat`. All hypotheses can be used for proving `P` as well as for
  proving the current goal.
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Introduction of logical connectives
-----------------------------------

- `iPureIntro` : turn a pure goal into a Coq goal. This tactic works for goals
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  of the shape `⌜φ⌝`, `a ≡ b` on discrete COFEs, and `✓ a` on discrete CMRAs.
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- `iLeft` : left introduction of disjunction.
- `iRight` : right introduction of disjunction.

- `iSplit` : introduction of a conjunction, or separating conjunction provided
  one of the operands is persistent.
- `iSplitL "H1 ... Hn"` : introduction of a separating conjunction. The
  hypotheses `H1 ... Hn` are used for the left conjunct, and the remaining ones
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  for the right conjunct. Intuitionistic hypotheses are ignored, since these do
  not need to be split.
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- `iSplitR "H0 ... Hn"` : symmetric version of the above.
- `iExist t1, .., tn` : introduction of an existential quantifier.

Elimination of logical connectives
----------------------------------

- `iExFalso` : Ex falso sequitur quod libet.
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- `iDestruct pm_trm as (x1 ... xn) "ipat"` : elimination of a series of
  existential quantifiers using Coq introduction patterns `x1 ... xn`, and
  elimination of an object level connective using the proof mode introduction
  pattern `ipat`.
  In case all branches of `ipat` start with a `#` (which causes the hypothesis
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  to be moved to the intuitionistic context) or with an `%` (which causes the
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  hypothesis to be moved to the pure Coq context), then one can use all
  hypotheses for proving the premises of `pm_trm`, as well as for proving the
  resulting goal. Note that in this case the hypotheses still need to be
  subdivided among the spatial premises.
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- `iDestruct pm_trm as %cpat` : elimination of a pure hypothesis using the Coq
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  introduction pattern `cpat`. When using this tactic, all hypotheses can be
  used for proving the premises of `pm_trm`, as well as for proving the
  resulting goal.
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Separation logic-specific tactics
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---------------------------------

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- `iFrame (t1 .. tn) "selpat"` : cancel the Coq terms (or Coq hypotheses)
  `t1 ... tn` and Iris hypotheses given by `selpat` in the goal. The constructs
  of the selection pattern have the following meaning:
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  + `%` : repeatedly frame hypotheses from the Coq context.
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  + `#` : repeatedly frame hypotheses from the intuitionistic context.
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  + `∗` : frame as much of the spatial context as possible. (N.B: this
          is the unicode symbol `∗`, not the regular asterisk `*`.)
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  Notice that framing spatial hypotheses makes them disappear, but framing Coq
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  or intuitionistic hypotheses does not make them disappear.
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  This tactic solves the goal if everything in the conclusion has been
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  framed.
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- `iCombine "H1" "H2" as "pat"` : combines `H1 : P1` and `H2 : P2` into
  `H: P1 ∗ P2`, then calls `iDestruct H as pat` on the combined hypothesis.
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- `iAccu` : solves a goal that is an evar by instantiating it with a all
  hypotheses from the spatial context joined together with a separating
  conjunction (or `emp` in case the spatial context is empty).
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Modalities
----------

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- `iModIntro mod` : introduction of a modality. The type class `FromModal` is
  used to specify which modalities this tactic should introduce. Instances of
  that type class include: later, except 0, basic update and fancy update,
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  intuitionistically, persistently, affinely, plainly, absorbingly, objectively,
  and subjectively. The optional argument `mod` can be used to specify what
  modality to introduce in case of ambiguity, e.g. `⎡|==> P⎤`.
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- `iAlways` : a deprecated alias of `iModIntro`.
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- `iNext n` : an alias of `iModIntro (▷^n P)`.
- `iNext` : an alias of `iModIntro (▷^1 P)`.
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- `iMod pm_trm as (x1 ... xn) "ipat"` : eliminate a modality `pm_trm` that is
  an instance of the `ElimModal` type class. Instances include: later, except 0,
  basic update and fancy update.

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Induction
---------
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- `iLöb as "IH" forall (x1 ... xn) "selpat"` : perform Löb induction by
  generating a hypothesis `IH : ▷ goal`. The tactic generalizes over the Coq
  level variables `x1 ... xn`, the hypotheses given by the selection pattern
  `selpat`, and the spatial context.
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- `iInduction x as cpat "IH" forall (x1 ... xn) "selpat"` : perform induction on
  the Coq term `x`. The Coq introduction pattern is used to name the introduced
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  variables. The induction hypotheses are inserted into the intuitionistic
  context and given fresh names prefixed `IH`. The tactic generalizes over the
  Coq level variables `x1 ... xn`, the hypotheses given by the selection pattern
  `selpat`, and the spatial context.
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Rewriting / simplification
--------------------------

- `iRewrite pm_trm` / `iRewrite pm_trm in "H"` : rewrite using an internal
  equality in the proof mode goal / hypothesis `H`.
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- `iRewrite -pm_trm` / `iRewrite -pm_trm in "H"` : rewrite in reverse direction
  using an internal equality in the proof mode goal / hypothesis `H`.
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- `iEval (tac)` / `iEval (tac) in "selpat"` : performs a tactic `tac`
  on the proof mode goal / hypotheses given by the selection pattern
  `selpat`. Using `%` as part of the selection pattern is unsupported.
  The tactic `tac` should be a reduction or rewriting tactic like
  `simpl`, `cbv`, `lazy`, `rewrite` or `setoid_rewrite`. The `iEval`
  tactic is implemented by running `tac` on `?evar ⊢ P` / `P ⊢ ?evar`
  where `P` is the proof goal / a hypothesis given by `selpat`. After
  running `tac`, `?evar` is unified with the resulting `P`, which in
  turn becomes the new proof mode goal / a hypothesis given by
  `selpat`. Note that parentheses around `tac` are needed.
- `iSimpl` / `iSimpl in "selpat"` : performs `simpl` on the proof mode
  goal / hypotheses given by the selection pattern `selpat`. This is a
  shorthand for `iEval (simpl)`.
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Iris
----

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- `iInv S with "selpat" as (x1 ... xn) "ipat" "Hclose"` : where `S` is either
   a namespace N or an identifier "H". Open the invariant indicated by S.  The
   selection pattern `selpat` is used for any auxiliary assertions needed to
   open the invariant (e.g. for cancelable or non-atomic invariants). The update
   for closing the invariant is put in a hypothesis named `Hclose`.
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Miscellaneous
-------------

- The tactic `done` is extended so that it also performs `iAssumption` and
  introduces pure connectives.
- The proof mode adds hints to the core `eauto` database so that `eauto`
  automatically introduces: conjunctions and disjunctions, universal and
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  existential quantifiers, implications and wand, plainness, persistence, later
  and update modalities, and pure connectives.
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Selection patterns
==================

Selection patterns are used to select hypotheses in the tactics `iRevert`,
`iClear`, `iFrame`, `iLöb` and `iInduction`. The proof mode supports the
following _selection patterns_:

- `H` : select the hypothesis named `H`.
- `%` : select the entire pure/Coq context.
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- `#` : select the entire intuitionistic context.
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- `∗` : select the entire spatial context. (N.B: this
        is the unicode symbol `∗`, not the regular asterisk `*`.)
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Introduction patterns
=====================

Introduction patterns are used to perform introductions and eliminations of
multiple connectives on the fly. The proof mode supports the following
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_introduction patterns_:
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- `H` : create a hypothesis named `H`.
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- `?` : create an anonymous hypothesis.
- `_` : remove the hypothesis.
- `$` : frame the hypothesis in the goal.
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- `[ipat1 ipat2]` : (separating) conjunction elimination. In order to eliminate
  conjunctions `P ∧ Q` in the spatial context, one of the following conditions
  should hold:
  + Either the proposition `P` or `Q` should be persistent.
  + Either `ipat1` or `ipat2` should be `_`, which results in one of the
    conjuncts to be thrown away.
- `[ipat1|ipat2]` : disjunction elimination.
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- `[]` : false elimination.
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- `%` : move the hypothesis to the pure Coq context (anonymously).
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- `->` and `<-` : rewrite using a pure Coq equality
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- `# ipat` : move the hypothesis into the intuitionistic context.
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- `> ipat` : eliminate a modality (if the goal permits).
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Apart from this, there are the following introduction patterns that can only
appear at the top level:

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- `{selpat}` : clear the hypotheses given by the selection pattern `selpat`.
  Items of the selection pattern can be prefixed with `$`, which cause them to
  be framed instead of cleared.
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- `!%` : introduce a pure goal (and leave the proof mode).
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- `!>` : introduce a modality by calling `iModIntro`.
- `!#` : introduce a modality by calling `iModIntro` (deprecated).
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- `/=` : perform `simpl`.
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- `//` : perform `try done` on all goals.
- `//=` : syntactic sugar for `/= //`
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- `*` : introduce all universal quantifiers.
- `**` : introduce all universal quantifiers, as well as all arrows and wands.

For example, given:
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        ∀ x, <affine> ⌜ x = 0 ⌝ ⊢
          □ (P → False ∨ □ (Q ∧ ▷ R) -∗
          P ∗ ▷ (R ∗ Q ∧ ⌜ x = pred 2 ⌝)).
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You can write
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        iIntros (x Hx) "!# $ [[] | #[HQ HR]] /= !>".
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which results in:
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        x : nat
        H : x = 0
        ______________________________________(1/1)
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        "HQ" : Q
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        "HR" : R
        --------------------------------------□
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        R ∗ Q ∧ x = 1
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Specialization patterns
=======================
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Since we are reasoning in a spatial logic, when eliminating a lemma or
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hypothesis of type ``P_0 -∗ ... -∗ P_n -∗ R``, one has to specify how the
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hypotheses are split between the premises. The proof mode supports the following
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_specification patterns_ to express splitting of hypotheses:
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- `H` : use the hypothesis `H` (it should match the premise exactly). If `H` is
  spatial, it will be consumed.
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- `(H spat1 .. spatn)` : first recursively specialize the hypothesis `H` using
  the specialization patterns `spat1 .. spatn`, and finally use the result of
  the specialization of `H` (it should match the premise exactly). If `H` is
  spatial, it will be consumed.

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- `[H1 .. Hn]` and `[H1 .. Hn //]` : generate a goal for the premise with the
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  (spatial) hypotheses `H1 ... Hn` and all intuitionistic hypotheses. The
  spatial hypotheses among `H1 ... Hn` will be consumed, and will not be
  available for subsequent goals. Hypotheses prefixed with a `$` will be framed
  in the goal for the premise. The pattern can be terminated with a `//`, which
  causes `done` to be called to close the goal (after framing).
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- `[-H1 ... Hn]` and `[-H1 ... Hn //]` : the negated forms of the above
  patterns, where the goal for the premise will have all spatial premises except
  `H1 .. Hn`.

- `[> H1 ... Hn]` and `[> H1 ... Hn //]` : like the above patterns, but these
  patterns can only be used if the goal is a modality `M`, in which case
  the goal for the premise will be wrapped in the modality `M`.
- `[> -H1 ... Hn]` and `[> -H1 ... Hn //]` : the negated forms of the above
  patterns.

- `[# $H1 .. $Hn]` and `[# $H1 .. $Hn //]` : generate a goal for a persistent
  premise in which all hypotheses are available. This pattern does not consume
  any hypotheses; all hypotheses are available in the goal for the premise, as
  well in the subsequent goal. The hypotheses `$H1 ... $Hn` will be framed in
  the goal for the premise. These patterns can be terminated with a `//`, which
  causes `done` to be called to close the goal (after framing).
- `[%]` and `[% //]` : generate a Coq goal for a pure premise. This pattern
  does not consume any hypotheses. The pattern can be terminated with a `//`,
  which causes `done` to be called to close the goal.

- `[$]` : solve the premise by framing. It will first repeatedly frame the
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  spatial hypotheses, and then repeatedly frame the intuitionistic hypotheses.
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  Spatial hypothesis that are not framed are carried over to the subsequent
  goal.
- `[> $]` : like the above pattern, but this pattern can only be used if the
  goal is a modality `M`, in which case the goal for the premise will be wrapped
  in the modality `M` before framing.
- `[# $]` : solve the persistent premise by framing. It will first repeatedly
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  frame the spatial hypotheses, and then repeatedly frame the intuitionistic
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  hypotheses. This pattern does not consume any hypotheses.
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For example, given:
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        H : □ P -∗ P 2 -∗ R -∗ x = 0 -∗ Q1 ∗ Q2

One can write:
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        iDestruct ("H" with "[#] [H1 $H2] [$] [% //]") as "[H4 H5]".
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Proof mode terms
================

Many of the proof mode tactics (such as `iDestruct`, `iApply`, `iRewrite`) can
take both hypotheses and lemmas, and allow one to instantiate universal
quantifiers and implications/wands of these hypotheses/lemmas on the fly.
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The syntax for the arguments of these tactics, called _proof mode terms_, is:
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        (H $! t1 ... tn with "spat1 .. spatn")
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Here, `H` can be either a hypothesis or a Coq lemma whose conclusion is
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of the shape `P ⊢ Q`. In the above, `t1 ... tn` are arbitrary Coq terms used
for instantiation of universal quantifiers, and `spat1 .. spatn` are
specialization patterns to eliminate implications and wands.
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Proof mode terms can be written down using the following shorthand syntaxes, too:
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        (H with "spat1 .. spatn")
        (H $! t1 ... tn)
        H
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HeapLang tactics
================

If you came here looking for the `wp_` tactics, those are described in the
[HeapLang documentation](HeapLang.md).