ProofMode.md

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

• 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 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.
• iDestruct 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 trm as %cpat : elimination of a pure hypothesis using the Coq introduction pattern cpat.

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.
• iLöb (x1 ... xn) as "IH" : perform Löb induction by generalizing over the 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.
• 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 it is timeless so no laters have to be added.
• iTimeless "H" : strip a later of a timeless hypotheses H in case the conclusion is a primitive view shifts or 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 : move the hypothesis to the persistent context.
• % : move the hypothesis to the pure Coq context (anonymously).
• [ipat ipat] : (separating) conjunction elimination.
• [ipat|ipat] : disjunction elimination.
• [] : 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 ... $Hn} : frame H1 ... Hn (this pattern can be mixed with the previous pattern, e.g., {$H1 H2 $H3}).
• /= : 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 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 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 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.

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, called proof mode terms, of these tactics 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