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# 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.

- Common elements of tactics
- Applying theorems
- Managing the local context
- Controlling the proof flow
- Classical tactics
- Performance-oriented tactic variants
- Reasoning with equalities
- Reasoning with inductive types

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