ProofMode.md 8.06 KB
Newer Older
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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
------------------

16
- `iIntros (x1 ... xn) "ipat1 ... ipatn"` : introduce universal quantifiers
17 18 19
  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
20 21
  be used to clear entire spatial context.
- `iRevert (x1 ... xn) "H1 ... Hn"` : revert the proof mode hypotheses
22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54
  `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.
55
- `iDestruct trm as (x1 ... xn) "spat1 ... spatn"` : elimination of existential
56 57 58
  quantifiers using Coq introduction patterns `x1 ... xn` and elimination of
  object level connectives using the proof mode introduction patterns
  `ipat1 ... ipatn`.
59 60
- `iDestruct trm as %cpat` : elimination of a pure hypothesis using the Coq
  introduction pattern `cpat`.
61 62 63 64 65 66 67 68 69 70 71

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.
72
- `iLöb (x1 ... xn) as "IH"` : perform Löb induction by generalizing over the
73 74 75 76 77 78 79 80 81 82 83 84 85
  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.
86 87 88
- `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
89 90
  it is timeless so no laters have to be added.
- `iTimeless "H"` : strip a later of a timeless hypotheses `H` in case the
91
  conclusion is a primitive view shifts or weakest precondition.
92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116

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
117
- `[ipat|ipat]` : disjunction elimination.
118 119 120 121 122 123 124 125 126
- `[]` : 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 ... $Hn}` : frame `H1 ... Hn` (this pattern can be mixed with the
Robbert Krebbers's avatar
Robbert Krebbers committed
127
  previous pattern, e.g., `{$H1 H2 $H3}`).
128 129 130 131 132
- `/=` : perform `simpl`.
- `*` : introduce all universal quantifiers.
- `**` : introduce all universal quantifiers, as well as all arrows and wands.

For example, given:
133

134
        ∀ x, x = 0 ⊢ □ (P → False ∨ □ (Q ∧ ▷ R) -★ P ★ ▷ (R ★ Q ∧ x = pred 2)).
135

136
You can write
137

138
        iIntros (x) "% ! $ [[] | #[HQ HR]] /= >".
139 140

which results in:
141

142 143 144
        x : nat
        H : x = 0
        ______________________________________(1/1)
Robbert Krebbers's avatar
Robbert Krebbers committed
145
        "HQ" : Q
146 147 148
        "HR" : R
        --------------------------------------□
        R ★ Q ∧ x = 1
149 150


151 152
Specialization patterns
=======================
153

154 155 156 157
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:
158

159 160 161 162 163 164
- `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
165 166
  is a primitive view shift, in which case the view shift will be kept in the
  goal of the premise too.
167 168 169 170 171 172
- `[#]` : 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.
173

174
For example, given:
175

176
        H : □ P -★ P 2 -★ x = 0 -★ Q1 ∗ Q2
177

178
You can write:
179

180
        iDestruct ("H" with "[#] [H1 H2] [%]") as "[H4 H5]".
181

182 183 184 185 186 187
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.
188

Ralf Jung's avatar
Ralf Jung committed
189
The syntax for the arguments, called _proof mode terms_, of these tactics is:
190

191
        (H $! t1 ... tn with "spat1 .. spatn")
192

193 194 195 196
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.
197

198
Proof mode terms can be written down using the following short hands too:
199

200 201 202
        (H with "spat1 .. spatn")
        (H $! t1 ... tn)
        H
203