ProofMode.md 10.6 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
classes in the file `proofmode/classes`. Most notable, many of the tactics can
be applied when the to be introduced or to be eliminated connective appears
under a later, a primitive view shift, or in the conclusion of a weakest
precondition connective.

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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
   conclusion of `pm_trm` and generates goals for the premises of `pm_trm`. See
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   proof mode terms below.

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) "H1 ... Hn"` : clear the hypothesis `H1 ... Hn` as well as
  the Coq level hypotheses/variables `x1 ... xn`. The symbol `★` can be used to
  clear entire spatial context.
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- `iRevert (x1 ... xn) "H1 ... Hn"` : revert the proof mode hypotheses
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  `H1 ... Hn` into wands, as well as the Coq level hypotheses/variables
  `x1 ... xn` into universal quantifiers. The symbol `★` can be used to revert
  the entire spatial context.
- `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) "H0 ... Hn"` : cancel the Coq terms (or Coq hypotheses)
  `t1 ... tn` and Iris hypotheses `H0 ... Hn` in the goal. Apart from
  hypotheses, the following symbols can be used:

  + `%` : 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 while generalizing
  over the Coq level variables `x1 ... xn` and the entire spatial context.
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Induction
---------
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- `iInduction x as cpat "IH" forall (x1 ... xn)` : 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 persistent context
  and given fresh names prefixed `IH`. The tactic generalizes over the Coq level
  variables `x1 ... xn` and the entire 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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- `iVsIntro` : introduction of a raw or primitive view shift.
- `iVs pm_trm as (x1 ... xn) "ipat"` : run a raw or primitive view shift
  `pm_trm` (if the goal permits, i.e. it is a raw or primitive view shift, or
   a weakest precondition).
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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
   permits, i.e. it is a later, True now, raw or primitive view shift, 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
  existential quantifiers, implications and wand, always and later modalities,
  primitive view shifts, and pure connectives.

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.
- `[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).
- `==> ipat` : run a view shift (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).
- `!==>` : introduce a view shift.
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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
hypotheses 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
so called specification patterns to express this splitting:
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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 a primitive view shift, in which case the view shift will be kept in the
  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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