\[\begin{split}\newcommand{\as}{\kw{as}}
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\end{split}\]
Basic tactics
Tactics specify how to transform the proof state of an
incomplete proof to eventually generate a complete proof.
This chapter introduces the basic tactics that are available in Rocq.
An alternative set of tactics is provided by the SSReflect proof language,
which is described in the The SSReflect proof language chapter.
Additional tactics are documented in the Automatic solvers and programmable tactics chapter.
Proofs can be developed in two basic ways: In forward reasoning,
the proof begins by proving simple statements that are then combined to prove the
theorem statement as the last step of the proof. With forward reasoning,
for example,
the proof of A /\ B would begin with proofs of A and B, which are
then used to prove A /\ B. Forward reasoning is probably the most common
approach in human-generated proofs.
In backward reasoning, the proof begins with the theorem statement
as the goal, which is then gradually transformed until every subgoal generated
along the way has been proven. In this case, the proof of A /\ B begins
with that formula as the goal. This can be transformed into two subgoals,
A and B, followed by the proofs of A and B. Rocq and its tactics
primarily use backward reasoning.
A tactic may fully prove a goal, in which case the goal is removed
from the proof state.
More commonly, a tactic replaces a goal with one or more subgoals.
(We say that a tactic reduces a goal to its subgoals.)
Most tactics require specific elements or preconditions to reduce a goal;
they display error messages if they can't be applied to the goal.
A few tactics, such as auto, don't fail even if the proof state
is unchanged.
Goals are identified by number (and optionally given names). The current goal is number
1. Tactics are applied to the current goal by default. (The
default can be changed with the Default Goal Selector
option.) They can
be applied to another goal or to multiple goals with a
goal selector such as 2: tactic.
This chapter describes many of the most common built-in tactics.
Built-in tactics can be combined to form tactic expressions, which are
described in the Ltac chapter. Since tactic expressions can
be used anywhere that a built-in tactic can be used, "tactic" may
refer to both built-in tactics and tactic expressions.