ProofMode.md 8.06 KB
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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
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   conclusion of `trm` and generates goals for the premises of `trm`. See
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   proof mode terms below.

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

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- `iIntros (x1 ... xn) "ipat1 ... ipatn"` : introduce universal quantifiers
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  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
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  be used to clear entire spatial context.
- `iRevert (x1 ... xn) "H1 ... Hn"` : revert the proof mode hypotheses
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  `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.
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- `iDestruct trm as (x1 ... xn) "spat1 ... spatn"` : elimination of existential
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  quantifiers using Coq introduction patterns `x1 ... xn` and elimination of
  object level connectives using the proof mode introduction patterns
  `ipat1 ... ipatn`.
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- `iDestruct trm as %cpat` : elimination of a pure hypothesis using the Coq
  introduction pattern `cpat`.
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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.
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- `iLöb (x1 ... xn) as "IH"` : perform Löb induction by generalizing over the
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  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.
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- `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
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  it is timeless so no laters have to be added.
- `iTimeless "H"` : strip a later of a timeless hypotheses `H` in case the
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  conclusion is a primitive view shifts or weakest precondition.
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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.
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- `[ipat|ipat]` : disjunction elimination.
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- `[]` : 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 ... $Hn}` : frame `H1 ... Hn` (this pattern can be mixed with the
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  previous pattern, e.g., `{$H1 H2 $H3}`).
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- `/=` : perform `simpl`.
- `*` : introduce all universal quantifiers.
- `**` : introduce all universal quantifiers, as well as all arrows and wands.

For example, given:
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        ∀ x, x = 0 ⊢ □ (P → False ∨ □ (Q ∧ ▷ R) -★ P ★ ▷ (R ★ Q ∧ x = pred 2)).
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You can write
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        iIntros (x) "% ! $ [[] | #[HQ HR]] /= >".
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which results in:
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        x : nat
        H : x = 0
        ______________________________________(1/1)
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        "HQ" : Q
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        "HR" : R
        --------------------------------------□
        R ★ Q ∧ x = 1
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Specialization patterns
=======================
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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:
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- `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
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  is a primitive view shift, in which case the view shift will be kept in the
  goal of the premise too.
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- `[#]` : 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.
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For example, given:
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        H : □ P -★ P 2 -★ x = 0 -★ Q1 ∗ Q2
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You can write:
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        iDestruct ("H" with "[#] [H1 H2] [%]") as "[H4 H5]".
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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.
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The syntax for the arguments, called _proof mode terms_, of these tactics is:
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        (H $! t1 ... tn with "spat1 .. spatn")
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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.
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Proof mode terms can be written down using the following short hands too:
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        (H with "spat1 .. spatn")
        (H $! t1 ... tn)
        H
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