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Robbert Krebbers authoredRobbert Krebbers authored
- Tactic overview
- Starting and stopping the proof mode
- Applying hypotheses and lemmas
- Context management
- Introduction of logical connectives
- Elimination of logical connectives
- Separation logic-specific tactics
- Modalities
- Induction
- Rewriting / simplification
- Iris
- Miscellaneous
- Selection patterns
- Introduction patterns
- Specialization patterns
- Proof mode terms
- HeapLang tactics
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 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 argumentPROP
can be used to explicitly specify which BI logicPROP : bi
should be used. This is useful to drop down in a layered logic, e.g. to drop down frommonPred PROP
toPROP
. -
iStopProof
to turn the proof mode entailment into an ordinary Coq goalbig star of context ⊢ proof mode goal
.
Applying hypotheses and lemmas
-
iExact "H"
: finish the goal if the conclusion matches the hypothesisH
-
iAssumption
: finish the goal if the conclusion matches any hypothesis -
iApply pm_trm
: match the conclusion of the current goal against the conclusion ofpm_trm
and generates goals for the premises ofpm_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 patternsx1 ... xn
and implications/wands using proof mode introduction patternsipat1 ... ipatn
. -
iClear (x1 ... xn) "selpat"
: clear the hypotheses given by the selection patternselpat
and the Coq level hypotheses/variablesx1 ... xn
. -
iRevert (x1 ... xn) "selpat"
: revert the hypotheses given by the selection patternselpat
into wands, and the Coq level hypotheses/variablesx1 ... xn
into universal quantifiers. Intuitionistic hypotheses are wrapped into the intuitionistic modality. -
iRename "H1" into "H2"
: rename the hypothesisH1
intoH2
. -
iSpecialize pm_trm
: instantiate universal quantifiers and eliminate implications/wands of a hypothesispm_trm
. See proof mode terms below. -
iSpecialize pm_trm as #
: instantiate universal quantifiers and eliminate implications/wands of a hypothesispm_trm
whose conclusion is persistent. All hypotheses can be used for proving the premises ofpm_trm
, as well as for the resulting main goal. -
iPoseProof pm_trm as (x1 ... xn) "ipat"
: putpm_trm
into the context and eliminates it. This tactic is essentially the same asiDestruct
with the difference that whenpm_trm
is a non-universally quantified intuitionistic hypothesis, it will not throw the hypothesis away. -
iAssert P with "spat" as "ipat"
: generates a new subgoalP
and adds the hypothesisP
to the current goal. The specialization patternspat
specifies which hypotheses will be consumed by provingP
. The introduction patternipat
specifies how to eliminateP
. In case all branches ofipat
start with a#
(which causesP
to be moved to the intuitionistic context) or with an%
(which causesP
to be moved to the pure Coq context), then one can use all hypotheses for provingP
as well as for proving the current goal. -
iAssert P as %cpat
: assertP
and eliminate it using the Coq introduction patterncpat
. All hypotheses can be used for provingP
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 hypothesesH1 ... 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 patternsx1 ... xn
, and elimination of an object level connective using the proof mode introduction patternipat
. In case all branches ofipat
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 ofpm_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 patterncpat
. When using this tactic, all hypotheses can be used for proving the premises ofpm_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 byselpat
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"
: combinesH1 : P1
andH2 : P2
intoH: P1 ∗ P2
, then callsiDestruct 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 (oremp
in case the spatial context is empty).
Modalities
-
iModIntro mod
: introduction of a modality. The type classFromModal
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 argumentmod
can be used to specify what modality to introduce in case of ambiguity, e.g.⎡|==> P⎤
. -
iAlways
: a deprecated alias ofiModIntro
. -
iNext n
: an alias ofiModIntro (▷^n P)
. -
iNext
: an alias ofiModIntro (▷^1 P)
. -
iMod pm_trm as (x1 ... xn) "ipat"
: eliminate a modalitypm_trm
that is an instance of theElimModal
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 hypothesisIH : ▷ goal
. The tactic generalizes over the Coq level variablesx1 ... xn
, the hypotheses given by the selection patternselpat
, and the spatial context. -
iInduction x as cpat "IH" forall (x1 ... xn) "selpat"
: perform induction on the Coq termx
. The Coq introduction pattern is used to name the introduced variables. The induction hypotheses are inserted into the intuitionistic context and given fresh names prefixedIH
. The tactic generalizes over the Coq level variablesx1 ... xn
, the hypotheses given by the selection patternselpat
, and the spatial context.
Rewriting / simplification
-
iRewrite pm_trm
/iRewrite pm_trm in "H"
: rewrite using an internal equality in the proof mode goal / hypothesisH
. -
iRewrite -pm_trm
/iRewrite -pm_trm in "H"
: rewrite in reverse direction using an internal equality in the proof mode goal / hypothesisH
. -
iEval (tac)
/iEval (tac) in "selpat"
: performs a tactictac
on the proof mode goal / hypotheses given by the selection patternselpat
. Using%
as part of the selection pattern is unsupported. The tactictac
should be a reduction or rewriting tactic likesimpl
,cbv
,lazy
,rewrite
orsetoid_rewrite
. TheiEval
tactic is implemented by runningtac
on?evar ⊢ P
/P ⊢ ?evar
whereP
is the proof goal / a hypothesis given byselpat
. After runningtac
,?evar
is unified with the resultingP
, which in turn becomes the new proof mode goal / a hypothesis given byselpat
. Note that parentheses aroundtac
are needed. -
iSimpl
/iSimpl in "selpat"
: performssimpl
on the proof mode goal / hypotheses given by the selection patternselpat
. This is a shorthand foriEval (simpl)
.
Iris
-
iInv S with "selpat" as (x1 ... xn) "ipat" "Hclose"
: whereS
is either a namespaceN
or an identifierH
. Open the invariant indicated byS
. The selection patternselpat
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 namedHclose
.
Miscellaneous
- The tactic
done
is extended so that it also performsiAssumption
and introduces pure connectives. - The proof mode adds hints to the core
eauto
database so thateauto
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 namedH
. -
%
: 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 namedH
. -
?
: create an anonymous hypothesis. -
_
: remove the hypothesis. -
$
: frame the hypothesis in the goal. -
[ipat1 ipat2]
: (separating) conjunction elimination. In order to eliminate conjunctionsP ∧ Q
in the spatial context, one of the following conditions should hold:- Either the proposition
P
orQ
should be persistent. - Either
ipat1
oripat2
should be_
, which results in one of the conjuncts to be thrown away.
- Either the proposition
-
(pat1 & pat2 & ... & patn)
: syntactic sugar for[pat1 [pat2 .. patn ..]]
to eliminate nested (separating) conjunctions. -
[ipat1|ipat2]
: disjunction elimination. -
[]
: false elimination. -
%
: move the hypothesis to the pure Coq context (anonymously). -
->
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
: 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 patternselpat
. 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 callingiModIntro
. -
!#
: introduce a modality by callingiModIntro
(deprecated). -
/=
: performsimpl
. -
//
: performtry 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 hypothesisH
(it should match the premise exactly). IfH
is spatial, it will be consumed. -
(H spat1 .. spatn)
: first recursively specialize the hypothesisH
using the specialization patternsspat1 .. spatn
, and finally use the result of the specialization ofH
(it should match the premise exactly). IfH
is spatial, it will be consumed. -
[H1 .. Hn]
and[H1 .. Hn //]
: generate a goal for the premise with the (spatial) hypothesesH1 ... Hn
and all intuitionistic hypotheses. The spatial hypotheses amongH1 ... 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 causesdone
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 exceptH1 .. Hn
. -
[> H1 ... Hn]
and[> H1 ... Hn //]
: like the above patterns, but these patterns can only be used if the goal is a modalityM
, in which case the goal for the premise will be wrapped in the modalityM
. -
[> -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 ... $Hn
will be framed in the goal for the premise. These patterns can be terminated with a//
, which causesdone
to be called to close the goal (after framing). -
[%]
and[% //]
: generate a Coq goal for a pure premise. This pattern does not consume any hypotheses. The pattern can be terminated with a//
, which causesdone
to be called to close the goal. -
[$]
: 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 modalityM
, in which case the goal for the premise will be wrapped in the modalityM
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.