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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 notable, many
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of the tactics can be applied when the to be introduced or to be eliminated
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connective appears under a later, an update modality, or in the conclusion of a
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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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
  `x1 ... xn` into universal quantifiers. Persistent hypotheses are wrapped into
  the always 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
  implications/wands of a hypothesis whose conclusion is persistent. In this
  case, all hypotheses can be used for proving the premises, as well as for
  the resulting goal.
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- `iPoseProof pm_trm as "H"` : put `pm_trm` into the context as a new hypothesis
  `H`.
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- `iAssert P with "spat" as "ipat"` : create a new goal with conclusion `P` and
  put `P` in the context of the original goal. The specialization pattern
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  `spat` specifies which hypotheses will be consumed by proving `P`. The
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  introduction pattern `ipat` specifies how to eliminate `P`.
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- `iAssert P with "spat" as %cpat` : assert `P` and eliminate it using the Coq
  introduction pattern `cpat`.
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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 COFEs, and `✓ a` on discrete CMRAs.

- `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. Persistent 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 existential
  quantifiers using Coq introduction patterns `x1 ... xn` and elimination of
  object level connectives using the proof mode introduction pattern `ipat`.
  In case all branches of `ipat` start with an `#` (moving the hypothesis to the
  persistent context) or `%` (moving the hypothesis to the pure Coq context),
  one can use all hypotheses for proving the premises of `pm_trm`, as well as
  for proving the resulting goal.
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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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Separating logic specific tactics
---------------------------------

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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.
  + `#` : repeatedly frame hypotheses from the persistent context.
  + `★` : frame as much of the spatial context as possible.

  Notice that framing spatial hypotheses makes them disappear, but framing Coq
  or persistent hypotheses does not make them disappear.
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  This tactic finishes the goal in case everything in the conclusion has been
  framed.
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- `iCombine "H1" "H2" as "H"` : turns `H1 : P1` and `H2 : P2` into
  `H : P1 ★ P2`.

The later modality
------------------
- `iNext` : introduce a later by stripping laters from all hypotheses.
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- `iLöb as "IH" forall (x1 ... xn)` : 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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Induction
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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 persistent context
  and given fresh names prefixed `IH`. The tactic generalizes over the Coq level
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  variables `x1 ... xn`, the hypotheses given by the selection pattern `selpat`,
  and the spatial context.
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Rewriting
---------

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- `iRewrite pm_trm` : rewrite an equality in the conclusion.
- `iRewrite pm_trm in "H"` : rewrite an equality in the hypothesis `H`.
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Iris
----

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- `iUpdIntro` : introduction of an update modality.
- `iUpd pm_trm as (x1 ... xn) "ipat"` : run an update modality `pm_trm` (if the
  goal permits, i.e. it can be expanded to an update modality.
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- `iInv N as (x1 ... xn) "ipat"` : open the invariant `N`.
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- `iTimeless "H"` : strip a later of a timeless hypothesis `H` (if the goal
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  permits, i.e. it is a later, True now, update modality, or a weakest
  precondition).
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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, always, 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.
- `#` : select the entire persistent context.
- `★` : select the entire spatial context.

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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.
- `[ipat ipat]` : (separating) conjunction elimination.
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- `[ipat|ipat]` : disjunction elimination.
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- `[]` : false elimination.
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- `%` : move the hypothesis to the pure Coq context (anonymously).
- `# ipat` : move the hypothesis to the persistent context.
- `> ipat` : remove a later of a timeless hypothesis (if the goal permits).
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- `==> ipat` : run an update 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:

- `{H1 ... Hn}` : clear `H1 ... Hn`.
- `{$H1 ... $Hn}` : frame `H1 ... Hn` (this pattern can be mixed with the
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  previous pattern, e.g., `{$H1 H2 $H3}`).
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- `!%` : introduce a pure goal (and leave the proof mode).
- `!#` : introduce an always modality (given that the spatial context is empty).
- `!>` : introduce a later (which strips laters from all hypotheses).
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- `!==>` : introduce an update modality
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- `/=` : perform `simpl`.
- `*` : introduce all universal quantifiers.
- `**` : introduce all universal quantifiers, as well as all arrows and wands.

For example, given:
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        ∀ x, 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) "% !# $ [[] | #[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
        --------------------------------------□
        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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- `[H1 ... Hn]` : generate a goal with the (spatial) hypotheses `H1 ... Hn` and
  all persistent hypotheses. The spatial hypotheses among `H1 ... Hn` will be
  consumed. Hypotheses may be prefixed with a `$`, which results in them being
  framed in the generated goal.
- `[-H1 ... Hn]`  : negated form of the above pattern. This pattern does not
  accept hypotheses prefixed with a `$`.
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- `==>[H1 ... Hn]` : same as the above pattern, but can only be used if the goal
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  is an update modality, in which case the update modality will be kept in the
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  goal of the premise too.
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- `[#]` : This pattern can be used when eliminating `P -★ Q` with `P`
  persistent. Using this pattern, all hypotheses are available in the goal for
  `P`, as well the remaining goal.
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- `[%]` : This pattern can be used when eliminating `P -★ Q` when `P` is pure.
  It will generate a Coq goal for `P` and does not consume any hypotheses.
- `*` : instantiate all top-level universal quantifiers with meta variables.
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For example, given:
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        H : □ P -★ P 2 -★ x = 0 -★ Q1 ∗ Q2
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You 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 both a hypothesis, as well as 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.
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Proof mode terms can be written down using the following short hands too:
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        (H with "spat1 .. spatn")
        (H $! t1 ... tn)
        H
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