Tactics guide

intros

Moves things from the goal to the context. It works on quantified variables:

  • FORM: intros x y z
  • WHEN: goal looks like forall a b c, H
  • EFFECT: add x, y, and z to the context (bound to a, b, and c, respectively); goal becomes H
  • INFORMAL: "Let x, y, and z be given."

It also works on premises of implications:

  • FORM: intros H
  • WHEN: goal looks like H1 -> H2
  • EFFECT: add H1 to the context, goal becomes H2
  • INFORMAL: "Suppose H1; we must show H2."

The two forms can be combined, which leads to a canned phrase in informal proofs.

  • FORM: intros n H
  • WHEN goal looks like forall n, H -> H'
  • EFFECT: n and H added to the context; goal becomes H'
  • INFORMAL: "Let n be given such that H; we must show H'."

simpl

  • WHEN: whenever
  • EFFECT: does some reduction in the goal
  • INFORMAL: No real correlate, but it can be nice to show the steps of computation.

You can also simplify in a hypothesis.

  • FORM: simpl in H
  • WHEN: H is in the context
  • EFFECT: does some reduction in H
  • INFORMAL: As above.

reflexivity

  • WHEN: goal looks like 'e = e'
  • EFFECT: finishes the current case
  • INFORMAL: No real correlate, but it can be nice to show the steps of computation. Conclude proofs with appropriate language, like, "and we are done" or "and we have ... immediately".

rewrite

Rewriting using equalities.

  • FORM: rewrite -> H
  • WHEN: H : e1 = e2 is in the context and e1 appears in the goal
  • EFFECT: e1 is replaced with e2 in the goal
  • INFORMAL: "By H, we can replace e1 with e2 to find ...". Or do an algebraic proof, showing a series of equalities.

It's best to always give a direction when rewriting. The direction is in terms of the equation in your context: -> means find an occurrence of the thing on the left of the equality and replace it with the thing on the right; <- means the reverse.

  • FORM: rewrite <- H
  • WHEN: H : e1 = e2 is in the context and e2 appears in the goal
  • EFFECT: e2 is replaced with e1 in the goal
  • INFORMAL: As above.

You can rewrite in the context, too.

  • FORM: rewrite -> H1 in H2
  • WHEN H : e1 = e2 and e1 appears in H2
  • EFFECT: e1 is replaced with e2 in H2
  • INFORMAL: "Since we know e1 = e2, we can replace e1 with e2 in H to find..."

There's also a right-to-left form. You can also rewrite if-and-only-ifs.

apply

Apply quantifications and implications---backward reasoning.

  • FORM: apply H
  • WHEN: H : P -> Q is in the context and the goal is Q
  • EFFECT: goal is replaced with P

  • INFORMAL: "Since P->Q (by H), it suffices to show P."

  • FORM: apply H
  • WHEN: H : forall x, Q is in the context and the goal is Q
  • EFFECT: goal is solved

  • INFORMAL: "Since forall x, Q (by H), we are done."

You can use apply on things in the context, too---forward reasoning.

  • FORM: apply H1 in H2
  • WHEN: H1 : P -> Q and H2 : P are in the context
  • EFFECT: H2 is replaced with P
  • INFORMAL: "Since P->Q (by H1) and P (by H2), we can conclude that Q."

When some quantified variables can't be inferred, you can explicitly specify values for variables.

  • FORM: apply H with x:=e
  • WHEN: H : forall x, P e -> Q and the goal is Q and Q doesn't determine the value of e
  • EFFECT: the goal is replaced with P e
  • INFORMAL: "Since P->Q (by H) it suffices to show P for e."

When there is just one ambiguous variable, you can leave off the x:= part and just write apply H with e.

destruct

Performs case analysis. Its precise use depends on the inductive type being analyzed. Be certain to use -/+/* to nest your case analyses. Always write an as pattern.

  • FORM: destruct n as [| n']
  • WHEN: n : nat is in the context
  • EFFECT: proofs splits into two cases, where n=0 and n=S n' for some n'
  • INFORMAL: "By cases on n. - If n=0 then... - If n=S n', then..." If you're at the beginning of a proof, don't forget to "let n be given". It's often good to say what your goal is in each case.

Note that you can use destruct on compound expressions, as in destruct (beq_nat n m), which will do a case analysis on the boolean value of testing n and m for equality. If you need to remember the result of the case analysis, you can ask for an equation to be saved.

  • FORM: destruct (foo bar) eqn:H
  • WHEN: foo bar is of an inductive type
  • EFFECT: case analysis on the possible results of foo bar, where in each case of a possible value v, the hypothesis H : foo bar = v is added to the contexr
  • INFORMAL: As above.

You can combine intros and destruct in one go by replacing the variable name with the pattern.

  • FORM: intros []
  • WHEN: the goal is of the form forall (b : bool), H
  • EFFECT: the same as intros b. destruct b as [], i.e. the goal is split into two cases: H with true substituted for b and H with false substituted for b.
  • INFORMAL: "Let b be given---it could be either true or false; we consider both cases." Or, more tersely, "We go by cases on b."

You can also use destruct to explode False in the context.

  • FORM: destruct H
  • WHEN: H : False
  • EFFECT: solves the current goal
  • INFORMAL: "We have reached a contradiction!" Good paper proofs will explain the nature of the contradiction rather than talking about "False" explicitly.

inversion

Reason by injectivity and distinctness of constructors.

When the constructors are the same, inversion generates equalities:

  • FORM: inversion H
  • WHEN: H : c e1 ... en = c e1' ... en' where c is a constructor of an inductive type
  • EFFECT: adds hypothesis that e1 = e1' (or deeper, if they have obvious subparts), substituting appropriately in the goal and context
  • INFORMAL: "To have c e1 ... en = c e1' ... en', it must be the case that each ei = ei', since constructors are injective."

When constructors are different, inversion solves the goal:

  • FORM: inversion H
  • WHEN: H : c e1 ... en = d e1' ... en' where c and d are different constructors of an inductive type
  • EFFECT: solves the goal
  • INFORMAL: "But it is a contradiction to have c e1 ... en = d e1' ... en'."

You can use inversion on H : False to solve the goal, as well.

induction

Performs induction. Its precise use depends on the inductive type. Be certain to use -/+/* to nest your case analyses.

  • FORM: induction l as [|h t IHl']
  • WHEN: l : list X is in the context and the goal is H
  • EFFECT: proof splits into two cases:
    • The "base case", where l = nil. You must prove H where nil is substituted for l.
    • The "inductive case", where l = cons h t. You must prove H where cons h t is substituted for l. You are given an "inductive hypothesis" IHl', which is H where l' is substituted for l.
  • INFORMAL: "By induction on n. - If n=0 then... - If n=S n', then our IH is ... and we must show ... ." It's very important that you state the IH and the new goal in each case.

assert

Sets a new, subsidiary goal. Typically used to control rewriting or perform forward reasoning. Be certain to use { and } to mark your subsidiary proofs.

  • FORM: assert (H: e) or assert (e) as H
  • WHEN: at any time; all variables in e must be in your context
  • EFFECT: introduce a "local lemma" e and call it H
  • INFORMAL: "In order to ..., we first show that ... ."

generalize dependent

The opposite of intros: requantifies variables. Typically used before induction to make sure the IH is sufficiently general.

  • FORM: generalize dependent x
  • WHEN: x : T is in the context; the goal is Q
  • EFFECT: the goal changes to forall x:T, P1 -> ... -> Pn -> Q, where each of the Pi are those hypotheses in the context that mentioned x
  • INFORMAL: Just after you say you go by induction, add "leaving x general".

symmetry

Swaps the sides of an equality.

  • FORM: symmetry
  • WHEN: goal is of the form e1 = e2
  • EFFECT: goal changes to e2 = e1
  • INFORMAL: General algebraic/equational reasoning.

You can also work in the context.

  • FORM: symmetry in H
  • WHEN: H : e1 = e2 is in the context
  • EFFECT: H changes to e2 = e1
  • INFORMAL: General algebraic/equational reasoning.

unfold

Unfolds a definition.

  • FORM: unfold defn
  • WHEN: the definition defn appears in the goal
  • EFFECT: unfolds the definition defn in the goal
  • INFORMAL: General algebraic/equational reasoning; recalling definitions.

You can also unfold in the context.

  • FORM: unfold defn in H
  • WHEN: the definition defn appears in hypothesis H in the context
  • EFFECT: unfolds the definition defn in H
  • INFORMAL: General algebraic/equational reasoning; recalling definitions.