proof_mode.md

Tactic overview
===============

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
classes in the file [proofmode/classes](theories/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
weakest precondition.

Starting and stopping the proof mode
------------------------------------

- `iStartProof PROP` : start the proof mode by turning a Coq goal into a proof
  mode entailment. This tactic is performed implicitly by all proof mode tactics
  described in this file, and thus should generally not be used by hand. The
  optional argument `PROP` can be used to explicitly specify which BI logic
  `PROP : bi` should be used. This is useful to drop down in a layered logic,
  e.g. to drop down from `monPred PROP` to `PROP`.
- `iStopProof` to turn the proof mode entailment into an ordinary Coq goal
  `big star of context ⊢ proof mode goal`.

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 in
  either the proofmode or the Coq context. Only hypotheses in the Coq context
  that are _syntactically_ of the shape `⊢ P` are recognized by this tactic
  (this means that assumptions of the shape `P ⊢ Q` are not recognized).
- `iApply pm_trm` : match the conclusion of the current goal against the
  conclusion of `pm_trm` and generates goals for the premises of `pm_trm`. See
  proof mode terms below.
  If the applied term has more premises than given specialization patterns, the
  pattern is extended with `[] ... []`.  As a consequence, all unused spatial
  hypotheses move to the last premise.

Context management
------------------

- `iIntros (x1 ... xn) "ipat1 ... ipatn"` : introduce universal quantifiers
  using Coq introduction patterns `x1 ... xn` and implications/wands using proof
  mode introduction patterns `ipat1 ... ipatn`.
- `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
  `x1 ... xn` into universal quantifiers. Intuitionistic hypotheses are wrapped
  into the intuitionistic modality.
- `iRename "H1" into "H2"` : rename the hypothesis `H1` into `H2`.
- `iSpecialize pm_trm` : instantiate universal quantifiers and eliminate
  implications/wands of a hypothesis `pm_trm`. See proof mode terms below.
- `iSpecialize pm_trm as #` : instantiate universal quantifiers and eliminate
  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.
- `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
  difference that when `pm_trm` is a non-universally quantified intuitionistic
  hypothesis, it will not throw the hypothesis away.
- `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
  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.
- `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.

Introduction of logical connectives
-----------------------------------

- `iPureIntro` : turn a pure goal into a Coq goal. This tactic works for goals
  of the shape `⌜φ⌝`, `a ≡ b` on discrete OFEs, and `✓ a` on discrete cameras.
- `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
  for the right conjunct. Intuitionistic hypotheses are ignored, since these do
  not need to be split.
- `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.
- `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
  to be moved to the intuitionistic context) or with an `%` (which causes the
  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.
- `iDestruct pm_trm as %cpat` : elimination of a pure hypothesis using the Coq
  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.

Separation logic-specific tactics
---------------------------------

- `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:
  + `%` : repeatedly frame hypotheses from the Coq context.
  + `#` : repeatedly frame hypotheses from the intuitionistic context.
  + `∗` : frame as much of the spatial context as possible. (N.B: this
          is the unicode symbol `∗`, not the regular asterisk `*`.)
  Notice that framing spatial hypotheses makes them disappear, but framing Coq
  or intuitionistic hypotheses does not make them disappear.
  This tactic solves the goal if everything in the conclusion has been framed.
- `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.
- `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).

Modalities
----------

- `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,
  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⎤`.
- `iAlways` : a deprecated alias of `iModIntro`.
- `iNext n` : an alias of `iModIntro (▷^n P)`.
- `iNext` : an alias of `iModIntro (▷^1 P)`.
- `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.

Induction
---------

- `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.
- `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
  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.

Rewriting / simplification
--------------------------

- `iRewrite pm_trm` / `iRewrite pm_trm in "H"` : rewrite using an internal
  equality in the proof mode goal / hypothesis `H`.
- `iRewrite -pm_trm` / `iRewrite -pm_trm in "H"` : rewrite in reverse direction
  using an internal equality in the proof mode goal / hypothesis `H`.
- `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)`.

Iris
----

- `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`.

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
  existential quantifiers, implications and wand, plainness, persistence, later
  and update modalities, and pure connectives.

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.
- `#` : select the entire intuitionistic context.
- `∗` : select the entire spatial context. (N.B: this
        is the unicode symbol `∗`, not the regular asterisk `*`.)

Introduction patterns
=====================

Introduction patterns are used to perform introductions and eliminations of
multiple connectives on the fly. The proof mode supports the following
_introduction patterns_:

- `H` : create a hypothesis named `H`.
- `?` : create an anonymous hypothesis.
- `_` : remove the hypothesis.
- `$` : frame the hypothesis in the goal.
- `[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.
- `(pat1 & pat2 & ... & patn)` : syntactic sugar for `[pat1 [pat2 .. patn ..]]`
  to eliminate nested (separating) conjunctions.
- `[ipat1|ipat2]` : disjunction elimination.
- `[]` : false elimination.
- `%H` : move the hypothesis to the pure Coq context, and name it `H`. Support
  for the `%H` introduction pattern requires an implementation of the hook
  `string_to_ident`. Without an implementation of this hook, the `%H` pattern
  will fail. We provide an implementation of the hook using Ltac2, which works
  with Coq 8.11, and can be installed with opam; see
  [iris/string-ident](https://gitlab.mpi-sws.org/iris/string-ident) for details.
- `%` : move the hypothesis to the pure Coq context (anonymously). Note that if
  `%` is followed by a string, and not another token, a space is needed.
- `->` and `<-` : rewrite using a pure Coq equality
- `# ipat` : move the hypothesis into the intuitionistic context. The tactic
  will fail if the hypothesis is not intuitionistic. On success, the tactic will
  strip any number of intuitionistic and persistence modalities. If the
  hypothesis is already in the intuitionistic context, the tactic will still
  strip intuitionistic and persistence modalities (it is a no-op if the
  hypothesis does not contain such modalities).
- `-# ipat` : move the hypothesis from the intuitionistic context into the
  spatial context. If the hypothesis is already in the spatial context, the
  tactic is a no-op. If the hypothesis is not affine, an `<affine>` modality is
  added to the hypothesis.
- `> ipat` : eliminate a modality (if the goal permits).

Apart from this, there are the following introduction patterns that can only
appear at the top level:

- `{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.
- `!%` : introduce a pure goal (and leave the proof mode).
- `!>` : introduce a modality by calling `iModIntro`.
- `!#` : introduce a modality by calling `iModIntro` (deprecated).
- `/=` : perform `simpl`.
- `//` : perform `try done` on all goals.
- `//=` : syntactic sugar for `/= //`
- `*` : introduce all universal quantifiers.
- `**` : introduce all universal quantifiers, as well as all arrows and wands.

For example, given:

        ∀ x, <affine> ⌜ x = 0 ⌝ ⊢
          □ (P → False ∨ □ (Q ∧ ▷ R) -∗
          P ∗ ▷ (R ∗ Q ∧ ⌜ x = pred 2 ⌝)).

You can write

        iIntros (x Hx) "!> $ [[] | #[HQ HR]] /= !>".

which results in:

        x : nat
        Hx : x = 0
        ______________________________________(1/1)
        "HQ" : Q
        "HR" : R
        --------------------------------------□
        R ∗ Q ∧ x = 1


Specialization patterns
=======================

Since we are reasoning in a spatial logic, when eliminating a lemma or
hypothesis of type ``P_0 -∗ ... -∗ P_n -∗ R``, one has to specify how the
hypotheses are split between the premises. The proof mode supports the following
_specification patterns_ to express splitting of hypotheses:

- `H` : use the hypothesis `H` (it should match the premise exactly). If `H` is
  spatial, it will be consumed.
- `(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.
- `[H1 .. Hn]` and `[H1 .. Hn //]` : generate a goal for the premise with the
  (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).
- `[-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
  spatial hypotheses, and then repeatedly frame the intuitionistic hypotheses.
  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
  frame the spatial hypotheses, and then repeatedly frame the intuitionistic
  hypotheses. This pattern does not consume any hypotheses.

For example, given:

        H : □ P -∗ P 2 -∗ R -∗ x = 0 -∗ Q1 ∗ Q2

One can write:

        iDestruct ("H" with "[#] [H1 $H2] [$] [% //]") as "[H4 H5]".


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.

The syntax for the arguments of these tactics, called _proof mode terms_, is:

        (H $! t1 ... tn with "spat1 .. spatn")

Here, `H` can be either a hypothesis or a Coq lemma whose conclusion is
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.

Proof mode terms can be written down using the following shorthand syntaxes, too:

        (H with "spat1 .. spatn")
        (H $! t1 ... tn)
        H

HeapLang tactics
================

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