# Tactics and Theorem Proving¶

Idris supports interactive theorem proving, and the analyse of context
through holes. To list all unproven holes, use the command `:m`

.
This will display their qualified names and the expected types. To
interactively prove a holes, use the command `:p name`

where `name`

is the hole. Once the proof is complete, the command `:a`

will append
it to the current module.

Once in the interactive prover, the following commands are available:

## Basic commands¶

`:q`

- Quits the prover (gives up on proving current lemma).`abandon`

- Same as :q`state`

- Displays the current state of the proof.`term`

- Displays the current proof term complete with its yet-to-be-filled holes (is only really useful for debugging).`undo`

- Undoes the last tactic.`qed`

- Once the interactive theorem prover tells you “No more goals,” you get to type this in celebration! (Completes the proof and exits the prover)

## Commonly Used Tactics¶

### Compute¶

`compute`

- Normalises all terms in the goal (note: does not normalise assumptions)

```
---------- Goal: ----------
(Vect (S (S Z + (S Z) + (S n))) Nat) -> Vect (S (S (S (S n)))) Nat
-lemma> compute
---------- Goal: ----------
(Vect (S (S (S (S n)))) Nat) -> Vect (S (S (S (S n)))) Nat
-lemma>
```

### Exact¶

`exact`

- Provide a term of the goal type directly.

```
---------- Goal: ----------
Nat
-lemma> exact Z
lemma: No more goals.
-lemma>
```

### Refine¶

`refine`

- Use a name to refine the goal. If the name needs arguments, introduce them as new goals.

### Trivial¶

`trivial`

- Satisfies the goal using an assumption that matches its type.

```
---------- Assumptions: ----------
value : Nat
---------- Goal: ----------
Nat
-lemma> trivial
lemma: No more goals.
-lemma>
```

### Intro¶

`intro`

- If your goal is an arrow, turns the left term into an assumption.

```
---------- Goal: ----------
Nat -> Nat -> Nat
-lemma> intro
---------- Assumptions: ----------
n : Nat
---------- Goal: ----------
Nat -> Nat
-lemma>
```

You can also supply your own name for the assumption:

```
---------- Goal: ----------
Nat -> Nat -> Nat
-lemma> intro number
---------- Assumptions: ----------
number : Nat
---------- Goal: ----------
Nat -> Nat
```

### Intros¶

`intros`

- Exactly like intro, but it operates on all left terms at once.

```
---------- Goal: ----------
Nat -> Nat -> Nat
-lemma> intros
---------- Assumptions: ----------
n : Nat
m : Nat
---------- Goal: ----------
Nat
-lemma>
```

### let¶

`let`

- Introduces a new assumption; you may use current assumptions to define the new one.

```
---------- Assumptions: ----------
n : Nat
---------- Goal: ----------
BigInt
-lemma> let x = toIntegerNat n
---------- Assumptions: ----------
n : Nat
x = toIntegerNat n: BigInt
---------- Goal: ----------
BigInt
-lemma>
```

### rewrite¶

`rewrite`

- Takes an expression with an equality type (x = y), and replaces all instances of x in the goal with y. Is often useful in combination with ‘sym’.

```
---------- Assumptions: ----------
n : Nat
a : Type
value : Vect Z a
---------- Goal: ----------
Vect (mult n Z) a
-lemma> rewrite sym (multZeroRightZero n)
---------- Assumptions: ----------
n : Nat
a : Type
value : Vect Z a
---------- Goal: ----------
Vect Z a
-lemma>
```

### induction¶

`induction`

- (`Note that this is still experimental`

and you may get strange results and error messages. We are aware of these and will finish the implementation eventually!) Prove the goal by induction. Each constructor of the datatype becomes a goal. Constructors with recursive arguments become induction steps, while simple constructors become base cases. Note that this only works for datatypes that have eliminators: a datatype definition must have the`%elim`

modifier.

### sourceLocation¶

`sourceLocation`

- Solve the current goal with information about the location in the source code where the tactic was invoked. This is mostly for embedded DSLs and programmer tools like assertions that need to know where they are called. See`Language.Reflection.SourceLocation`

for more information.

## Less commonly-used tactics¶

`applyTactic`

- Apply a user-defined tactic. This should be a function of type`List (TTName, Binder TT) -> TT -> Tactic`

, where the first argument represents the proof context and the second represents the goal. If your tactic will produce a proof term directly, use the`Exact`

constructor from`Tactic`

.`attack`

- ?`equiv`

- Replaces the goal with a new one that is convertible with the old one`fill`

- ?`focus`

- ?`mrefine`

- Refining by matching against a type`reflect`

- ?`solve`

- Takes a guess with the correct type and fills a hole with it, closing a proof obligation. This happens automatically in the interactive prover, so`solve`

is really only relevant in tactic scripts used for helping implicit argument resolution.`try`

- ?