ProofMode.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. Most notable, many
of the tactics can be applied when the to be introduced or to be eliminated
connective appears under a later, an update modality, or in the conclusion of a
weakest precondition.

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

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. Persistent hypotheses are wrapped into
the always 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 whose conclusion is persistent. In this
case, all hypotheses can be used for proving the premises, as well as for
the resulting goal.

iPoseProof pm_trm as "H" : put pm_trm into the context as a new hypothesis
H.

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 persistent 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 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
for the right conjunct. Persistent 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 persistent 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.


Separating 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 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.
This tactic finishes the goal in case everything in the conclusion has been
framed.


iCombine "H1" "H2" as "H" : turns H1 : P1 and H2 : P2 into
H : P1 ∗ P2.


Modalities


iModIntro : introduction of a modality that is an instance of the
IntoModal type class. Instances include: later, except 0, basic update and
fancy update.

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.


The later modality


iNext n : introduce n laters by stripping that number of laters from all
hypotheses. If the argument n is not given, it strips one later if the
leftmost conjunct is of the shape ▷ P, or n laters if the leftmost
conjunct is of the shape ▷^n P.

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.


Induction


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 persistent 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


iRewrite pm_trm : rewrite an equality in the conclusion.

iRewrite pm_trm in "H" : rewrite an equality in the hypothesis H.


Iris


iInv N as (x1 ... xn) "ipat" "Hclose" : open the invariant N, 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, always, 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 persistent context.

∗ : select the entire spatial context.


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.

[ipat|ipat] : disjunction elimination.

[] : false elimination.

% : move the hypothesis to the pure Coq context (anonymously).

# ipat : move the hypothesis to the persistent context.

> ipat : eliminate a modality (if the goal permits).