Tactic overview
===============
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 trm` : match the conclusion of the current goal against the
conclusion of `tetrmrm` and generates goals for the premises of `trm`. See
proof mode terms below.
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 "H1 ... Hn"` : clear the hypothesis `H1 ... Hn`. The symbol `★` can
be used to clear entire spatial context.
- `iRevert {x1 ... xn} "H1 ... Hn"` : revert the proof mode hypotheses
`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`.
- `iSpecialize trm` : instantiate universal quantifiers and eliminate
implications/wands of a hypothesis `trm`. See proof mode terms below.
- `iPoseProof trm as "H"` : put `trm` into the context as a new hypothesis `H`.
- `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
`spat` specifies which hypotheses will be consumed by proving `P` and the
introduction pattern `ipat` specifies how to eliminate `P`.
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.
- `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 trm as {x1 ... xn} "spat1 ... spatn"` : elimination of existential
quantifiers using Coq introduction patterns `x1 ... xn` and elimination of
object level connectives using the proof mode introduction patterns
`ipat1 ... ipatn`.
Separating logic specific tactics
---------------------------------
- `iFrame "H0 ... Hn"` : cancel the hypotheses `H0 ... Hn` in the goal.
- `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.
- `iLöb {x1 ... xn} as "IH"` : perform Löb induction by generalizing over the
Coq level variables `x1 ... xn` and the entire spatial context.
Rewriting
---------
- `iRewrite trm` : rewrite an equality in the conclusion.
- `iRewrite trm in "H"` : rewrite an equality in the hypothesis `H`.
Iris
----
- `iPvsIntro` : introduction of a primitive view shift. Generates a goal if
the masks are not syntactically equal.
- `iPvs trm as {x1 ... xn} "ipat"` : runs a primitive view shift `trm`.
- `iInv N as {x1 ... xn} "ipat"` : open the invariant `N`.
- `iInv> N as {x1 ... xn} "ipat"` : open the invariant `N` and establish that
it is timeless so no laters have to be added.
- `iTimeless "H"` : strip a later of a timeless hypotheses `H` in case the
conclusion is a primitive view shifts or weakest precondition.
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.
- `$` : frame the hypothesis in the goal.
- `# ipat` : move the hypothesis to the persistent context.
- `%` : move the hypothesis to the pure Coq context (anonymously).
- `[ipat ipat]` : (separating) conjunction elimination.
- `|ipat|ipat]` : disjunction elimination.
- `[]` : false elimination.
Apart from this, there are the following introduction patterns that can only
appear at the top level:
- `!` : introduce a box (provided that the spatial context is empty).
- `>` : introduce a later (which strips laters from all hypotheses).
- `{H1 ... Hn}` : clear `H1 ... Hn`.
- `{$H1 ... $Hn}` : frame `H1 ... Hn` (this pattern can be mixed with the
previous pattern, e.g., `{$H1 H2 #H3}`).
- `/=` : perform `simpl`.
- `*` : introduce all universal quantifiers.
- `**` : introduce all universal quantifiers, as well as all arrows and wands.
For example, given:
∀ x, x = 0 ⊢ □ (P → False ∨ □ (Q ∧ ▷ R) -★ P ★ ▷ (R ★ Q ∧ x = pred 2)).
You can write
iIntros {x} "% ! $ [[] | #[HQ HR]] /= >".
which results in:
x : nat
H : 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
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:
- `H` : use the hypothesis `H` (it should match the premise exactly). If `H` is
spatial, it will be consumed.
- `[H1 ... Hn]` : generate a goal with the spatial hypotheses `H1 ... Hn` and
all persistent hypotheses. The hypotheses `H1 ... Hn` will be consumed.
- `[-H1 ... Hn]` : negated form of the above pattern
- `=>[H1 ... Hn]` : same as the above pattern, but can only be used if the goal
is a primitive view shift, in which case the view shift will be kept in the
goal of the premise too.
- `[#]` : This pattern can be used when eliminating `P -★ Q` when either `P` or
`Q` is persistent. In this case, all hypotheses are available in the goal for
the premise as none will be consumed.
- `[%]` : 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.
For example, given:
H : □ P -★ P 2 -★ x = 0 -★ Q1 ∗ Q2
You 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, called _proof mode terms_ of these tactics is:
(H $! t1 ... tn with "spat1 .. spatn")
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.
Proof mode terms can be written down using the following short hands too:
(H with "spat1 .. spatn")
(H $! t1 ... tn)
H