ProofMode.md 12 KB
Newer Older
1 2 3
Tactic overview
===============

4 5
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
Robbert Krebbers's avatar
Robbert Krebbers committed
6
classes in the file [proofmode/classes](proofmode/classes.v). Most notable, many
Robbert Krebbers's avatar
Robbert Krebbers committed
7
of the tactics can be applied when the to be introduced or to be eliminated
8 9
connective appears under a later, an update modality, or in the conclusion of a
weakest precondition.
10

11 12 13 14 15
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
16
- `iApply pm_trm` : match the conclusion of the current goal against the
17 18
  conclusion of `pm_trm` and generates goals for the premises of `pm_trm`. See
  proof mode terms below.
Ralf Jung's avatar
Ralf Jung committed
19
  If the applied term has more premises than given specialization patterns, the
20
  pattern is extended with `[] ... []`.  As a consequence, all unused spatial
Ralf Jung's avatar
Ralf Jung committed
21
  hypotheses move to the last premise.
22 23 24 25

Context management
------------------

26
- `iIntros (x1 ... xn) "ipat1 ... ipatn"` : introduce universal quantifiers
27 28
  using Coq introduction patterns `x1 ... xn` and implications/wands using proof
  mode introduction patterns `ipat1 ... ipatn`.
29 30 31 32 33 34
- `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.
35
- `iRename "H1" into "H2"` : rename the hypothesis `H1` into `H2`.
36 37
- `iSpecialize pm_trm` : instantiate universal quantifiers and eliminate
  implications/wands of a hypothesis `pm_trm`. See proof mode terms below.
38 39 40 41
- `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.
42 43
- `iPoseProof pm_trm as "H"` : put `pm_trm` into the context as a new hypothesis
  `H`.
Robbert Krebbers's avatar
Robbert Krebbers committed
44 45 46 47 48 49 50 51 52 53 54
- `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.
55 56 57 58 59 60 61 62 63 64 65 66 67 68

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
69 70
  for the right conjunct. Persistent hypotheses are ignored, since these do not
  need to be split.
71 72 73 74 75 76 77
- `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.
Robbert Krebbers's avatar
Robbert Krebbers committed
78 79 80 81 82 83 84 85 86 87
- `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.
88
- `iDestruct pm_trm as %cpat` : elimination of a pure hypothesis using the Coq
89 90 91
  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.
92 93 94 95

Separating logic specific tactics
---------------------------------

96 97 98
- `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:
99 100 101

  + `%` : repeatedly frame hypotheses from the Coq context.
  + `#` : repeatedly frame hypotheses from the persistent context.
102
  + `∗` : frame as much of the spatial context as possible.
103 104 105

  Notice that framing spatial hypotheses makes them disappear, but framing Coq
  or persistent hypotheses does not make them disappear.
106 107 108

  This tactic finishes the goal in case everything in the conclusion has been
  framed.
109
- `iCombine "H1" "H2" as "H"` : turns `H1 : P1` and `H2 : P2` into
110
  `H : P1 ∗ P2`.
111

112 113 114 115 116 117 118 119 120 121
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.

122 123
The later modality
------------------
124

125 126
- `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
Robbert Krebbers's avatar
Robbert Krebbers committed
127 128
  leftmost conjunct is of the shape `▷ P`, or `n` laters if the leftmost
  conjunct is of the shape `▷^n P`.
129 130 131 132
- `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.
133

Robbert Krebbers's avatar
Robbert Krebbers committed
134 135
Induction
---------
136

137 138
- `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
139 140
  variables. The induction hypotheses are inserted into the persistent context
  and given fresh names prefixed `IH`. The tactic generalizes over the Coq level
141 142
  variables `x1 ... xn`, the hypotheses given by the selection pattern `selpat`,
  and the spatial context.
Robbert Krebbers's avatar
Robbert Krebbers committed
143

144 145 146
Rewriting
---------

147 148
- `iRewrite pm_trm` : rewrite an equality in the conclusion.
- `iRewrite pm_trm in "H"` : rewrite an equality in the hypothesis `H`.
149 150 151 152

Iris
----

153 154
- `iInv N as (x1 ... xn) "ipat" "Hclose"` : open the invariant `N`, the update
  for closing the invariant is put in a hypothesis named `Hclose`.
155 156 157 158 159 160 161 162

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
163 164
  existential quantifiers, implications and wand, always, later and update
  modalities, and pure connectives.
165

166 167 168 169 170 171 172 173 174 175
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.
176
- `∗` : select the entire spatial context.
177

178 179 180 181 182
Introduction patterns
=====================

Introduction patterns are used to perform introductions and eliminations of
multiple connectives on the fly. The proof mode supports the following
183
_introduction patterns_:
184

185
- `H` : create a hypothesis named `H`.
186 187 188 189
- `?` : create an anonymous hypothesis.
- `_` : remove the hypothesis.
- `$` : frame the hypothesis in the goal.
- `[ipat ipat]` : (separating) conjunction elimination.
Ralf Jung's avatar
Ralf Jung committed
190
- `[ipat|ipat]` : disjunction elimination.
191
- `[]` : false elimination.
192 193
- `%` : move the hypothesis to the pure Coq context (anonymously).
- `# ipat` : move the hypothesis to the persistent context.
194
- `> ipat` : eliminate a modality (if the goal permits).
195 196 197 198 199 200

Apart from this, there are the following introduction patterns that can only
appear at the top level:

- `{H1 ... Hn}` : clear `H1 ... Hn`.
- `{$H1 ... $Hn}` : frame `H1 ... Hn` (this pattern can be mixed with the
Robbert Krebbers's avatar
Robbert Krebbers committed
201
  previous pattern, e.g., `{$H1 H2 $H3}`).
202 203
- `!%` : introduce a pure goal (and leave the proof mode).
- `!#` : introduce an always modality (given that the spatial context is empty).
204
- `!>` : introduce a modality.
205 206 207 208 209
- `/=` : perform `simpl`.
- `*` : introduce all universal quantifiers.
- `**` : introduce all universal quantifiers, as well as all arrows and wands.

For example, given:
Ralf Jung's avatar
Ralf Jung committed
210

211
        ∀ x, x = 0 ⊢ □ (P → False ∨ □ (Q ∧ ▷ R) -∗ P ∗ ▷ (R ∗ Q ∧ x = pred 2)).
Ralf Jung's avatar
Ralf Jung committed
212

213
You can write
Ralf Jung's avatar
Ralf Jung committed
214

215
        iIntros (x) "% !# $ [[] | #[HQ HR]] /= !>".
216 217

which results in:
Ralf Jung's avatar
Ralf Jung committed
218

219 220 221
        x : nat
        H : x = 0
        ______________________________________(1/1)
Robbert Krebbers's avatar
Robbert Krebbers committed
222
        "HQ" : Q
223 224
        "HR" : R
        --------------------------------------□
225
        R ∗ Q ∧ x = 1
Ralf Jung's avatar
Ralf Jung committed
226 227


228 229
Specialization patterns
=======================
Ralf Jung's avatar
Ralf Jung committed
230

231
Since we are reasoning in a spatial logic, when eliminating a lemma or
232
hypothesis of type ``P_0 -∗ ... -∗ P_n -∗ R``, one has to specify how the
233
hypotheses are split between the premises. The proof mode supports the following
234
_specification patterns_ to express splitting of hypotheses:
Ralf Jung's avatar
Ralf Jung committed
235

236 237
- `H` : use the hypothesis `H` (it should match the premise exactly). If `H` is
  spatial, it will be consumed.
238 239 240 241
- `[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.
242
- `[-H1 ... Hn]` : negated form of the above pattern.
243 244 245
- `>[H1 ... Hn]` : same as the above pattern, but can only be used if the goal
  is a modality, in which case the modality will be kept in the generated goal
  for the premise will be wrapped into the modality.
246 247 248
- `>[-H1 ... Hn]`  : negated form of the above pattern.
- `>` : shorthand for `>[-]` (typically used for the last premise of an applied
  lemma).
249
- `[#]` : This pattern can be used when eliminating `P -∗ Q` with `P`
250 251
  persistent. Using this pattern, all hypotheses are available in the goal for
  `P`, as well the remaining goal.
252
- `[%]` : This pattern can be used when eliminating `P -∗ Q` when `P` is pure.
253
  It will generate a Coq goal for `P` and does not consume any hypotheses.
Ralf Jung's avatar
Ralf Jung committed
254

255
For example, given:
Ralf Jung's avatar
Ralf Jung committed
256

257
        H : □ P -∗ P 2 -∗ x = 0 -∗ Q1 ∗ Q2
Ralf Jung's avatar
Ralf Jung committed
258

259
You can write:
Ralf Jung's avatar
Ralf Jung committed
260

261
        iDestruct ("H" with "[#] [H1 H2] [%]") as "[H4 H5]".
Ralf Jung's avatar
Ralf Jung committed
262

263 264 265 266 267 268
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.
Ralf Jung's avatar
Ralf Jung committed
269

270
The syntax for the arguments of these tactics, called _proof mode terms_, is:
Ralf Jung's avatar
Ralf Jung committed
271

272
        (H $! t1 ... tn with "spat1 .. spatn")
Ralf Jung's avatar
Ralf Jung committed
273

274 275 276 277
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.
Ralf Jung's avatar
Ralf Jung committed
278

279
Proof mode terms can be written down using the following short hands too:
Ralf Jung's avatar
Ralf Jung committed
280

281 282 283
        (H with "spat1 .. spatn")
        (H $! t1 ... tn)
        H
Ralf Jung's avatar
Ralf Jung committed
284