ProofMode.md

Tactic overview

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.

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 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 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 pm_trm : instantiate universal quantifiers and eliminate implications/wands of a hypothesis pm_trm. See proof mode terms below.
• 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.
• iPoseProof pm_trm as "H" : put pm_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. The introduction pattern ipat specifies how to eliminate P.
• iAssert P with "spat" as %cpat : assert P and eliminate it using the Coq introduction pattern cpat.

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. Persistent hypotheses are ignored, since these do not need to be split.

• 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 pm_trm as (x1 ... xn) "ipat" : elimination of existential quantifiers using Coq introduction patterns x1 ... xn and elimination of object level connectives using the proof mode introduction pattern ipat. In case all branches of ipat start with an # (moving the hypothesis to the persistent context) or % (moving the hypothesis to the pure Coq context), one can use all hypotheses for proving the premises of pm_trm, as well as for proving the resulting goal.
• iDestruct pm_trm as %cpat : elimination of a pure hypothesis using the Coq 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.

Separating logic specific tactics

• iFrame (t1 .. tn) "H0 ... Hn" : cancel the Coq terms (or Coq hypotheses) t1 ... tn and Iris hypotheses H0 ... Hn in the goal. Apart from hypotheses, the following symbols can be used:

  • % : repeatedly frame hypotheses from the Coq context.
  • # : repeatedly frame hypotheses from the persistent context.
  • ★ : frame as much of the spatial context as possible.

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

  This tactic finishes the goal in case everything in the conclusion has been framed.

• 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.

Induction

• iInduction x as cpat "IH" : perform induction on the Coq term x. The Coq introduction pattern is used to name the introduced variables. The induction hypotheses are inserted into the persistent context and given fresh names prefixed IH.

Rewriting

• iRewrite pm_trm : rewrite an equality in the conclusion.
• iRewrite pm_trm in "H" : rewrite an equality in the hypothesis H.

Iris

• iVsIntro : introduction of a raw or primitive view shift.
• iVs pm_trm as (x1 ... xn) "ipat" : run a raw or primitive view shift pm_trm (if the goal permits, i.e. it is a raw or primitive view shift, or a weakest precondition).
• iInv N as (x1 ... xn) "ipat" : open the invariant N.
• iTimeless "H" : strip a later of a timeless hypothesis H (if the goal permits, i.e. it is a later, True now, raw or primitive view shift, or a 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 ipat] : (separating) conjunction elimination.
• [ipat|ipat] : disjunction elimination.
• [] : false elimination.
• % : move the hypothesis to the pure Coq context (anonymously).
• # ipat : move the hypothesis to the persistent context.
• > ipat : remove a later of a timeless hypothesis (if the goal permits).
• ==> ipat : run a view shift (if the goal permits).

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

• {H1 ... Hn} : clear H1 ... Hn.
• {$H1 ... $Hn} : frame H1 ... Hn (this pattern can be mixed with the previous pattern, e.g., {$H1 H2 $H3}).
• !% : introduce a pure goal (and leave the proof mode).
• !# : introduce an always modality (given that the spatial context is empty).
• !> : introduce a later (which strips laters from all hypotheses).
• !==> : introduce a view shift.
• /= : 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 spatial hypotheses among H1 ... Hn will be consumed. Hypotheses may be prefixed with a $, which results in them being framed in the generated goal.
• [-H1 ... Hn] : negated form of the above pattern. This pattern does not accept hypotheses prefixed with a $.
• ==>[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 with P persistent. Using this pattern, all hypotheses are available in the goal for P, as well the remaining goal.
• [%] : 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 of these tactics, called proof mode terms, 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