ProofMode.md 8.75 KB
Newer Older
1 2 3
Tactic overview
===============

4 5 6 7 8 9 10
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.

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 17
- `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
18 19 20 21 22
   proof mode terms below.

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

23
- `iIntros (x1 ... xn) "ipat1 ... ipatn"` : introduce universal quantifiers
24 25 26
  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
27 28
  be used to clear entire spatial context.
- `iRevert (x1 ... xn) "H1 ... Hn"` : revert the proof mode hypotheses
29 30 31 32
  `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`.
33 34 35 36
- `iSpecialize pm_trm` : instantiate universal quantifiers and eliminate
  implications/wands of a hypothesis `pm_trm`. See proof mode terms below.
- `iPoseProof pm_trm as "H"` : put `pm_trm` into the context as a new hypothesis
  `H`.
37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62
- `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.
63 64 65 66 67
- `iDestruct pm_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`.
- `iDestruct pm_trm as %cpat` : elimination of a pure hypothesis using the Coq
68
  introduction pattern `cpat`.
69 70 71 72

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

73 74 75 76
- `iFrame "H0 ... Hn"` : cancel the hypotheses `H0 ... Hn` in the goal. The
  symbol `★` can be used to frame as much of the spatial context as possible,
  and the symbol `#` can be used to repeatedly frame as much of the persistent
  context as possible. When without arguments, it attempts to frame all spatial
77
  hypotheses.
78 79 80 81 82 83
- `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.
84
- `iLöb (x1 ... xn) as "IH"` : perform Löb induction by generalizing over the
85 86 87 88 89
  Coq level variables `x1 ... xn` and the entire spatial context.

Rewriting
---------

90 91
- `iRewrite pm_trm` : rewrite an equality in the conclusion.
- `iRewrite pm_trm in "H"` : rewrite an equality in the hypothesis `H`.
92 93 94 95 96 97

Iris
----

- `iPvsIntro` : introduction of a primitive view shift. Generates a goal if
  the masks are not syntactically equal.
98
- `iPvs pm_trm as (x1 ... xn) "ipat"` : runs a primitive view shift `pm_trm`.
99 100
- `iInv N as (x1 ... xn) "ipat"` : open the invariant `N`.
- `iInv> N as (x1 ... xn) "ipat"` : open the invariant `N` and establish that
101 102
  it is timeless so no laters have to be added.
- `iTimeless "H"` : strip a later of a timeless hypotheses `H` in case the
103
  conclusion is a primitive view shifts or weakest precondition.
104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128

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.
Ralf Jung's avatar
Ralf Jung committed
129
- `[ipat|ipat]` : disjunction elimination.
130 131 132 133 134 135 136 137 138
- `[]` : 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 ... $Hn}` : frame `H1 ... Hn` (this pattern can be mixed with the
Robbert Krebbers's avatar
Robbert Krebbers committed
139
  previous pattern, e.g., `{$H1 H2 $H3}`).
140 141 142 143 144
- `/=` : 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
145

146
        ∀ x, x = 0 ⊢ □ (P → False ∨ □ (Q ∧ ▷ R) -★ P ★ ▷ (R ★ Q ∧ x = pred 2)).
Ralf Jung's avatar
Ralf Jung committed
147

148
You can write
Ralf Jung's avatar
Ralf Jung committed
149

150
        iIntros (x) "% ! $ [[] | #[HQ HR]] /= >".
151 152

which results in:
Ralf Jung's avatar
Ralf Jung committed
153

154 155 156
        x : nat
        H : x = 0
        ______________________________________(1/1)
Robbert Krebbers's avatar
Robbert Krebbers committed
157
        "HQ" : Q
158 159 160
        "HR" : R
        --------------------------------------□
        R ★ Q ∧ x = 1
Ralf Jung's avatar
Ralf Jung committed
161 162


163 164
Specialization patterns
=======================
Ralf Jung's avatar
Ralf Jung committed
165

166 167 168 169
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:
Ralf Jung's avatar
Ralf Jung committed
170

171 172 173 174 175 176
- `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
177 178
  is a primitive view shift, in which case the view shift will be kept in the
  goal of the premise too.
179 180 181 182 183 184
- `[#]` : 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.
Ralf Jung's avatar
Ralf Jung committed
185

186
For example, given:
Ralf Jung's avatar
Ralf Jung committed
187

188
        H : □ P -★ P 2 -★ x = 0 -★ Q1 ∗ Q2
Ralf Jung's avatar
Ralf Jung committed
189

190
You can write:
Ralf Jung's avatar
Ralf Jung committed
191

192
        iDestruct ("H" with "[#] [H1 H2] [%]") as "[H4 H5]".
Ralf Jung's avatar
Ralf Jung committed
193

194 195 196 197 198 199
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
200

201
The syntax for the arguments of these tactics, called _proof mode terms_, is:
Ralf Jung's avatar
Ralf Jung committed
202

203
        (H $! t1 ... tn with "spat1 .. spatn")
Ralf Jung's avatar
Ralf Jung committed
204

205 206 207 208
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
209

210
Proof mode terms can be written down using the following short hands too:
Ralf Jung's avatar
Ralf Jung committed
211

212 213 214
        (H with "spat1 .. spatn")
        (H $! t1 ... tn)
        H
Ralf Jung's avatar
Ralf Jung committed
215