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tutorial/index
updates/updates
proofs/index
faq/faq
reference/index

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In order to understand how to write proofs in Idris I think its worth clarifying some fundamentals, such as,
- Propositions and judgments
- Boolean and constructive logic
- Curry-Howard correspondence
- Definitional and propositional equalities
- Axiomatic and constructive approaches
Propositions and Judgments
==========================
Propositions are the subject of our proofs, before the proof then we can't formally say if they are true or not. If the proof is successful then the result is a 'judgment'.
For instance, if the ``proposition`` is,
+-------+
| 1+1=2 |
+-------+
When we prove it, the ``judgment`` is,
+------------+
| 1+1=2 true |
+------------+
Or if the ``proposition`` is,
+-------+
| 1+1=3 |
+-------+
Obviously we can't prove it is true, but it is still a valid proposition and perhaps we can prove it is false so the ``judgment`` is,
+-------------+
| 1+1=3 false |
+-------------+
This may seem a bit pedantic but it is important to be careful, in mathematics not every proposition is true or false for instance, a proposition may be unproven or even unprovable.
So the logic here is different from the logic that comes from boolean algebra. In that case what is not true is false and what is not false is true. The logic we are using here does not have this 'law of excluded middle' so we have to be careful not to use it.
A false proposition is taken to be a contradiction and if we have a contradiction then we can prove anything, so we need to avoid this. Some languages, used in proof assistants, prevent contradictions but such languages cannot be Turing complete, so Idris does not prevent contradictions.
The logic we are using is called constructive (or sometimes intuitional) because we are constructing a 'database' of judgments.
There are also many other types of logic, another important type of logic for Idris programmers is '``linear logic``' but that's not discussed on this page.
Curry-Howard correspondence
===========================
So how to we relate these proofs to Idris programs? It turns out that there is a correspondence between constructive logic and type theory. They are the same structure and we can switch backward and forward between the two notations because they are the same thing.
The way that this works is that a proposition is a type so this,
.. code-block:: idris
Idris> 1+1=2
2 = 2 : Type
is a proposition and it is also a type. This is built into Idris so when an '=' equals sign appears in a function type an equality type is generated. The following will also produce an equality type:
.. code-block:: idris
Idris> 1+1=3
2 = 3 : Type
Both of these are valid propositions so both are valid equality types. But how do we represent true judgment, that is, how do we denote 1+1=2 is true but not 1+1=3.
A type that is true is inhabited, that is, it can be constructed. An equality type has only one constructor 'Refl' so a proof of 1+1=2 is
.. code-block:: idris
onePlusOne : 1+1=2
onePlusOne = Refl
So how can Refl, which is a constructor without any parameters, construct an equality type? If we type it on its own then it can't:
.. code-block:: idris
Idris> Refl
(input):Can't infer argument A to Refl, Can't infer argument x to Refl
So it must pattern match on its return type:
.. code-block:: idris
Idris> the (1=1) Refl
Refl : 1 = 1
So now that we can represent propositions as types other aspects of propositional logic can also be translated to types as follows:
+----------+-------------------+--------------------------+
| | propositions | example of possible type |
+----------+-------------------+--------------------------+
| A | x=y | |
+----------+-------------------+--------------------------+
| B | y=z | |
+----------+-------------------+--------------------------+
| and | A /\ B | Pair(x=y,y=z) |
+----------+-------------------+--------------------------+
| or | A \/ B | Either(x=y,y=z) |
+----------+-------------------+--------------------------+
| implies | A -> B | (x=y) -> (y=x) |
+----------+-------------------+--------------------------+
| for all | y=z | |
+----------+-------------------+--------------------------+
| exists | y=z | |
+----------+-------------------+--------------------------+
And (conjunction)
-----------------
We can have a type which corresponds to conjunction:
.. code-block:: idris
AndIntro : a -> b -> A a b
There is a built in type called 'Pair'.
Or (disjunction)
----------------
We can have a type which corresponds to disjunction:
.. code-block:: idris
data Or : Type -> Type -> Type where
OrIntroLeft : a -> A a b
OrIntroRight : b -> A a b
There is a built in type called 'Either'.
Definitional and Propositional Equalities
=========================================
We have seen that we can 'prove' a type by finding a way to construct a term. In the case of equality types there is only one constructor which is 'Refl'.
We have also seen that each side of the equation does not have to be identical like '2=2'. It is enough that both sides are ``definitionaly equal`` like this:
.. code-block:: idris
onePlusOne : 1+1=2
onePlusOne = Refl
So both sides of this equation nomalise to 2 and so Refl will type match and the proposition is proved.
We don't have to stick to terms, can also use symbolic parameters so the following will compile:
.. code-block:: idris
varIdentity : m = m
varIdentity = Refl
If a proposition/equality type is not definitionaly equal but is still true then it is ``propositionaly equal``. In this case we may still be able to prove it but some steps in the proof may require us to add something into the terms or at least to take some sideways steps to get to a proof.
Especially when working with equalities containing variable terms (inside functions) it can be hard to know which equality types are definitially equal, in this example plusReducesL is '``definitially equal``' but plusReducesR is not (although it is '``propositionaly equal``'). The only difference between them is the order of the operands.
.. code-block:: idris
plusReducesL : (n:Nat) -> plus Z n = n
plusReducesL n = Refl
plusReducesR : (n:Nat) -> plus n Z = n
plusReducesR n = Refl
plusReducesR gives the following error:
.. code-block:: idris
- + Errors (1)
`-- proof.idr line 6 col 17:
When checking right hand side of plusReducesR with expected type
plus n 0 = n
Type mismatch between
n = n (Type of Refl)
and
plus n 0 = n (Expected type)
Specifically:
Type mismatch between
n
and
plus n 0
So why is 'Refl' able to prove some equality types but not others?
The first answer is that 'plus' is defined in such a way that it splits on its first argument so it is simple to prove when 0 is the first argument but not the second. So what is the general way to know if Refl will work?
If an equality type can be proved/constructed by using Refl alone it is known as a ``definitional equality``. In order to be definitinally equal both sides of the equation must normalise to unique values. That is, each step in the proof must reduce the term so each step is effectively forced.
So when we type 1+1 in Idris it is immediately converted to 2 because definitional equality is built in.
.. code-block:: idris
Idris> 1+1
2 : Integer
In the following pages we discuss how to resolve propositionaly equalies.
Axiomatic and Constructive Approaches
=====================================
How should we define types so that we can do proofs on them? In the natural numbers with plus example we could have started by treating it as a group based on the plus operator. So we have axioms:
- for all x,y : ``x+y=y+x``
- for all x: ``x + 0 = x = 0 + x``
- for all x,y,z: ``(x + y) + z = x + (x + z)``
Then we can implement '+' so that it respects these axioms (presumably implemented in hardware).
These are axioms, that is a propositions/types that are asserted to be true without proof. In Idris we can use the 'postulate' keyword
.. code-block:: idris
commutePlus ``postulate``: x -> y -> plus x y = plus y x
Alternatively we could define the natural numbers based on Zero and Successor. The axioms above then become derived rules and we also gain the ability to do inductive proofs.
As we know, Idris uses both of these approaches with automatic coercion between them which gives the best of both worlds.
So what can we learn from this to implement out own types:
- Should we try to implement both approaches?
- Should we define our types by constructing up from primitive types?
Proof theory affects these design decisions.

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.. _proofs-index:
###############
Theorem Proving
###############
A tutorial on theorem proving in Idris.
NOT YET UPDATED FOR IDRIS 2!
.. note::
The documentation for Idris has been published under the Creative
Commons CC0 License. As such to the extent possible under law, *The
Idris Community* has waived all copyright and related or neighboring
rights to Documentation for Idris.
More information concerning the CC0 can be found online at: http://creativecommons.org/publicdomain/zero/1.0/
.. toctree::
:maxdepth: 1
definitional
pluscomm
inductive
patterns
propositional
interactive
interactiveOld

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.. _sect-inductive:
****************
Inductive Proofs
****************
Before embarking on proving ``plus_commutes`` in Idris itself, let us
consider the overall structure of a proof of some property of natural
numbers. Recall that they are defined recursively, as follows:
.. code-block:: idris
data Nat : Type where
Z : Nat
S : Nat -> Nat
A *total* function over natural numbers must both terminate, and cover
all possible inputs. Idris checks functions for totality by checking that
all inputs are covered, and that all recursive calls are on
*structurally smaller* values (so recursion will always reach a base
case). Recalling ``plus``:
.. code-block:: idris
plus : Nat -> Nat -> Nat
plus Z m = m
plus (S k) m = S (plus k m)
This is total because it covers all possible inputs (the first argument
can only be ``Z`` or ``S k`` for some ``k``, and the second argument
``m`` covers all possible ``Nat``) and in the recursive call, ``k``
is structurally smaller than ``S k`` so the first argument will always
reach the base case ``Z`` in any sequence of recursive calls.
In some sense, this resembles a mathematical proof by induction (and
this is no coincidence!). For some property ``P`` of a natural number
``x``, we can show that ``P`` holds for all ``x`` if:
- ``P`` holds for zero (the base case).
- Assuming that ``P`` holds for ``k``, we can show ``P`` also holds for
``S k`` (the inductive step).
In ``plus``, the property we are trying to show is somewhat trivial (for
all natural numbers ``x``, there is a ``Nat`` which need not have any
relation to ``x``). However, it still takes the form of a base case and
an inductive step. In the base case, we show that there is a ``Nat``
arising from ``plus n m`` when ``n = Z``, and in the inductive step we
show that there is a ``Nat`` arising when ``n = S k`` and we know we can
get a ``Nat`` inductively from ``plus k m``. We could even write a
function capturing all such inductive definitions:
.. code-block:: idris
nat_induction : (P : Nat -> Type) -> -- Property to show
(P Z) -> -- Base case
((k : Nat) -> P k -> P (S k)) -> -- Inductive step
(x : Nat) -> -- Show for all x
P x
nat_induction P p_Z p_S Z = p_Z
nat_induction P p_Z p_S (S k) = p_S k (nat_induction P p_Z p_S k)
Using ``nat_induction``, we can implement an equivalent inductive
version of ``plus``:
.. code-block:: idris
plus_ind : Nat -> Nat -> Nat
plus_ind n m
= nat_induction (\x => Nat)
m -- Base case, plus_ind Z m
(\k, k_rec => S k_rec) -- Inductive step plus_ind (S k) m
-- where k_rec = plus_ind k m
n
To prove that ``plus n m = plus m n`` for all natural numbers ``n`` and
``m``, we can also use induction. Either we can fix ``m`` and perform
induction on ``n``, or vice versa. We can sketch an outline of a proof;
performing induction on ``n``, we have:
- Property ``P`` is ``\x => plus x m = plus m x``.
- Show that ``P`` holds in the base case and inductive step:
- | Base case: ``P Z``, i.e.
| ``plus Z m = plus m Z``, which reduces to
| ``m = plus m Z`` due to the definition of ``plus``.
- | Inductive step: Inductively, we know that ``P k`` holds for a specific, fixed ``k``, i.e.
| ``plus k m = plus m k`` (the induction hypothesis). Given this, show ``P (S k)``, i.e.
| ``plus (S k) m = plus m (S k)``, which reduces to
| ``S (plus k m) = plus m (S k)``. From the induction hypothesis, we can rewrite this to
| ``S (plus m k) = plus m (S k)``.
To complete the proof we therefore need to show that ``m = plus m Z``
for all natural numbers ``m``, and that ``S (plus m k) = plus m (S k)``
for all natural numbers ``m`` and ``k``. Each of these can also be
proved by induction, this time on ``m``.
We are now ready to embark on a proof of commutativity of ``plus``
formally in Idris.

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***************************
Interactive Theorem Proving
***************************
Idris supports interactive theorem proving via elaborator reflection.
:ref:`elaborator-reflection` is also used to convert high-level Idris code into
the core language and for customising the language. Here we show how to use it
to interactively construct proofs.
The primary purpose of the elaboration mechanism is to elaborate Idris and so it
is not optimised to work as a proof assistant, however it can interactively
construct proofs as described on this page.
Elab and Pruviloj Libraries
===========================
Elaborator reflection is defined in prelude/Language/Reflection/Elab.idr
and pruviloj is defined in Idris-dev/libs/pruviloj/
``Elab`` defines the basics such as: solve, attack, intro, compute,
rewriteWith and others.
``pruviloj`` defines some more advanced derived commands such as:
reflexivity and others.
To use ``pruviloj`` call Idris with the "-p pruviloj" option and add:
.. code-block:: idris
import Pruviloj
import Pruviloj.Induction
to the top of your file.
It is useful to get the docs at the REPL by using the ``:doc`` command, and
search the docstrings using ``:apropos``. So to introduce the functions from
Elab and Pruviloj, that we will need for the following example, here are
their docstrings:
.. code-block:: idris
*plusReducesZ> :doc solve
Language.Reflection.Elab.Tactics.solve : Elab ()
Substitute a guess into a hole.
.. code-block:: idris
*plusReducesZ> :doc attack
Language.Reflection.Elab.Tactics.attack : Elab ()
Convert a hole to make it suitable for bindings - that is, transform
it such that it has the form ?h : t . h as opposed to ?h : t . f h.
The binding tactics require that a hole be directly under its binding,
or else the scopes of the generated terms won't make sense. This
tactic creates a new hole of the proper form, and points the old hole
at it.
.. code-block:: idris
*plusReducesZ> :doc intro
Language.Reflection.Elab.Tactics.intro : (n : TTName) -> Elab ()
Introduce a lambda binding around the current hole and focus on the
body. Requires that the hole be in binding form (use attack).
Arguments:
n : TTName -- the name to use for the argument
.. code-block:: idris
*plusReducesZ> :doc compute
Language.Reflection.Elab.Tactics.compute : Elab ()
Normalise the goal.
.. code-block:: idris
*plusReducesZ> :doc rewriteWith
Language.Reflection.Elab.Tactics.rewriteWith : Raw -> Elab ()
Attempt to rewrite the goal using an equality.
The tactic searches the goal for applicable subterms, and constructs a
context for replace using them. In some cases, this is not possible,
and replace must be called manually with an appropriate context.
Because this tactic internally introduces a let binding, it requires
that the hole be immediately under its binder (use attack if it might
not be).
Here is the command from pruviloj that we will need for the example on
this page:
.. code-block:: idris
*plusReducesZ> :doc reflexivity
Pruviloj.Core.reflexivity : Elab ()
A special-purpose tactic that attempts to solve a goal using Refl.
This is useful for ensuring that goals in fact are trivial when
developing or testing other tactics; otherwise, consider using search.
Interactive Example: plusReduces
================================
One way to write proofs interactively is to write the general *structure* of
the proof, and use the interactive mode to complete the details.
Consider the following definition, proved in :ref:`sect-theorems`:
.. code-block:: idris
plusReduces : (n:Nat) -> plus Z n = n
We’ll be constructing the proof by *induction*, so we write the cases for ``Z``
and ``S``, with a recursive call in the ``S`` case giving the inductive
hypothesis, and insert *holes* for the rest of the definition:
.. code-block:: idris
import Pruviloj
import Pruviloj.Induction
plusReducesZ' : (n:Nat) -> n = plus n Z
plusReducesZ' Z = ?plusredZ_Z
plusReducesZ' (S k) = let ih = plusReducesZ' k in
?plusredZ_S
On running , two global names are created, ``plusredZ_Z`` and
``plusredZ_S``, with no definition.
.. code-block:: idris
*theorems> : idris plusReducesZ.idr -p pruviloj
. / _/___/ /____(_)____
/ // __ / ___/ / ___/ Version 1.2.0
_/ // /_/ / / / (__ ) http://www.idris-lang.org/
/___/\__,_/_/ /_/____/ Type :? for help
Idris is free software with ABSOLUTELY NO WARRANTY.
For details type :warranty.
Holes: Main.plusredZ_S, Main.plusredZ_Z
This tells us that we have two holes Main.plusredZ_S and Main.plusredZ_Z. We can solve
these separately, ``plusredZ_Z`` is the simplest so we will do that first.
The ``:elab plusredZ_Z`` command enters interactive elaboration mode, which can be used to
complete the missing definition for plusredZ_Z.
.. code-block:: idris
*plusReducesZ> :elab plusredZ_Z
---------- Goal: ----------
{hole_0} : 0 = 0
This has been normalised to ``0 = 0`` so now we have to prove that ``0`` equals ``0``, which
is easy to prove by reflexivity from the pruviloj library:
.. code-block:: idris
-Main.plusredZ_Z> reflexivity
plusredZ_Z: No more goals.
This tells us that the proof is complete. We can now leave the interactive mode which
we entered with the ``:elab`` command. We do this with the ``:qed`` command:
.. code-block:: idris
-Main.plusredZ_Z> :qed
Proof completed!
Main.plusredZ_Z = %runElab (do reflexivity)
Holes: Main.plusredZ_S
This gives us a trace of the proof which is ``plusredZ_Z = %runElab (do reflexivity)``. We
can cut & paste this into the hole in the original file. This also tells us that we
have another hole ``Main.plusredZ_S`` remaining.
This remaining proof is bit more complicated, the following diagram gives an overview:
|image|
We approach this remaining proof in the same way by using the ``:elab`` command:
.. code-block:: idris
*plusReducesZ> :elab plusredZ_S
---------- Goal: ----------
{hole_0} : (k : Nat) -> (k = plus k 0) -> S k = S (plus k 0)
In this case, the goal is a function type, using ``k`` (the argument
accessible by pattern matching) and ``ih`` — the local variable
containing the result of the recursive call. We can introduce these as
assumptions using the ``intro`` tactic twice. The parameter is entered as
a constant of type ``TTName`` which is entered as a backtick with double
braces \`{{ih}}. This gives:
.. code-block:: idris
-Main.plusredZ_S> intro `{{k}}
---------- Assumptions: ----------
k : Nat
---------- Goal: ----------
{hole_0} : (k = plus k 0) -> S k = S (plus k 0)
-Main.plusredZ_S> intro `{{ih}}
---------- Assumptions: ----------
k : Nat
ih : k = plus k 0
---------- Goal: ----------
{hole_0} : S k = S (plus k 0)
We know, from the type of ``ih``, that ``k = plus k 0``, so we would
like to use this knowledge to replace ``plus k 0`` in the goal with
``k``. We can achieve this with the ``rewriteWith`` tactic:
.. code-block:: idris
-Main.plusredZ_S> rewriteWith (Var `{{ih}})
---------- Assumptions: ----------
k : Nat
ih : k = plus k 0
---------- Goal: ----------
{hole_0} : S k = S k
The ``rewriteWith`` tactic takes an equality proof as an argument, and tries
to rewrite the goal using that proof. The ih value is entered as a constant
of type ``TTName`` which is entered as a backtick with double braces `{{ih}} but
``rewriteWith`` requires an expression of type ``Raw``, rather than just a name,
so the Var constructor is used to make a variable. Here, it results in an equality
which is trivially provable using reflexivity:
.. code-block:: idris
-Main.plusredZ_S> reflexivity
plusredZ_S: No more goals.
-Main.plusredZ_S> :qed
Proof completed!
Main.plusredZ_S = %runElab (do intro `{{k}}
intro `{{ih}}
rewriteWith (Var `{{ih}})
reflexivity)
We can't just cut & paste this into the hole in the original file like this:
.. code-block:: idris
import Pruviloj
import Pruviloj.Induction
%language ElabReflection
plusReducesZ' : (n:Nat) -> n = plus n Z
plusReducesZ' Z = %runElab (do reflexivity)
plusReducesZ' (S k) = let ih = plusReducesZ' k in
(%runElab (do intro `{{k}}
intro `{{ih}}
rewriteWith (Var `{{ih}})
reflexivity)
)
because this gives the following error:
.. code-block:: idris
Idris> :load elabInteractiveEx2.idr
elabInteractiveEx2.idr:10:32:
|
10 | intro `{{ih}}
| ^
unexpected "in"
expecting dependent type signature
However if we put the proof into a separate function like this:
.. code-block:: idris
import Pruviloj
import Pruviloj.Induction
%language ElabReflection
plusredZ_S : (k : Nat) -> (ih:(k = plus k Z)) -> (S k = S (plus k Z))
plusredZ_S = %runElab (do intro `{{k}}
intro `{{ih}}
rewriteWith (Var `{{ih}})
reflexivity)
plusReducesZ' : (n:Nat) -> n = plus n Z
plusReducesZ' Z = %runElab (do reflexivity)
plusReducesZ' (S k) = let ih = plusReducesZ' k in plusredZ_S k ih
This then loads [#f1]_ .
.. [#f1] https://github.com/idris-lang/Idris-dev/issues/4556
.. |image| image:: ../image/plusReducesProof.png

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**************************************************************
DEPRECATED: Interactive Theorem Proving Using Old Tactics Code
**************************************************************
.. warning::
The interactive theorem-proving interface documented here has been
deprecated in favor of :ref:`elaborator-reflection`.
Idris also supports interactive theorem proving via tactics. This
is generally not recommended to be used directly, but rather used as a
mechanism for building proof automation which is beyond the scope of this
tutorial. In this section, we briefly discus tactics.
One way to write proofs interactively is to write the general *structure* of
the proof, and use the interactive mode to complete the details.
Consider the following definition, proved in :ref:`sect-theorems`:
.. code-block:: idris
plusReduces : (n:Nat) -> plus Z n = n
We’ll be constructing the proof by *induction*, so we write the cases for ``Z``
and ``S``, with a recursive call in the ``S`` case giving the inductive
hypothesis, and insert *holes* for the rest of the definition:
.. code-block:: idris
plusReducesZ' : (n:Nat) -> n = plus n Z
plusReducesZ' Z = ?plusredZ_Z
plusReducesZ' (S k) = let ih = plusReducesZ' k in
?plusredZ_S
On running , two global names are created, ``plusredZ_Z`` and
``plusredZ_S``, with no definition. We can use the ``:m`` command at the
prompt to find out which holes are still to be solved (or, more
precisely, which functions exist but have no definitions), then the
``:t`` command to see their types:
.. code-block:: idris
*theorems> :m
Global holes:
[plusredZ_S,plusredZ_Z]
.. code-block:: idris
*theorems> :t plusredZ_Z
plusredZ_Z : Z = plus Z Z
*theorems> :t plusredZ_S
plusredZ_S : (k : Nat) -> (k = plus k Z) -> S k = plus (S k) Z
The ``:p`` command enters interactive proof mode, which can be used to
complete the missing definitions.
.. code-block:: idris
*theorems> :p plusredZ_Z
---------------------------------- (plusredZ_Z) --------
{hole0} : Z = plus Z Z
This gives us a list of premises (above the line; there are none here)
and the current goal (below the line; named ``{hole0}`` here). At the
prompt we can enter tactics to direct the construction of the proof. In
this case, we can normalise the goal with the ``compute`` tactic:
.. code-block:: idris
-plusredZ_Z> compute
---------------------------------- (plusredZ_Z) --------
{hole0} : Z = Z
Now we have to prove that ``Z`` equals ``Z``, which is easy to prove by
``Refl``. To apply a function, such as ``Refl``, we use ``refine`` which
introduces subgoals for each of the function’s explicit arguments
(``Refl`` has none):
.. code-block:: idris
-plusredZ_Z> refine Refl
plusredZ_Z: no more goals
Here, we could also have used the ``trivial`` tactic, which tries to
refine by ``Refl``, and if that fails, tries to refine by each name in
the local context. When a proof is complete, we use the ``qed`` tactic
to add the proof to the global context, and remove the hole from the
unsolved holes list. This also outputs a trace of the proof:
.. code-block:: idris
-plusredZ_Z> qed
plusredZ_Z = proof
compute
refine Refl
.. code-block:: idris
*theorems> :m
Global holes:
[plusredZ_S]
The ``:addproof`` command, at the interactive prompt, will add the proof
to the source file (effectively in an appendix). Let us now prove the
other required lemma, ``plusredZ_S``:
.. code-block:: idris
*theorems> :p plusredZ_S
---------------------------------- (plusredZ_S) --------
{hole0} : (k : Nat) -> (k = plus k Z) -> S k = plus (S k) Z
In this case, the goal is a function type, using ``k`` (the argument
accessible by pattern matching) and ``ih`` — the local variable
containing the result of the recursive call. We can introduce these as
premises using the ``intro`` tactic twice (or ``intros``, which
introduces all arguments as premises). This gives:
.. code-block:: idris
k : Nat
ih : k = plus k Z
---------------------------------- (plusredZ_S) --------
{hole2} : S k = plus (S k) Z
Since plus is defined by recursion on its first argument, the term
``plus (S k) Z`` in the goal can be simplified, so we use ``compute``.
.. code-block:: idris
k : Nat
ih : k = plus k Z
---------------------------------- (plusredZ_S) --------
{hole2} : S k = S (plus k Z)
We know, from the type of ``ih``, that ``k = plus k Z``, so we would
like to use this knowledge to replace ``plus k Z`` in the goal with
``k``. We can achieve this with the ``rewrite`` tactic:
.. code-block:: idris
-plusredZ_S> rewrite ih
k : Nat
ih : k = plus k Z
---------------------------------- (plusredZ_S) --------
{hole3} : S k = S k
-plusredZ_S>
The ``rewrite`` tactic takes an equality proof as an argument, and tries
to rewrite the goal using that proof. Here, it results in an equality
which is trivially provable:
.. code-block:: idris
-plusredZ_S> trivial
plusredZ_S: no more goals
-plusredZ_S> qed
plusredZ_S = proof {
intros;
rewrite ih;
trivial;
}
Again, we can add this proof to the end of our source file using the
``:addproof`` command at the interactive prompt.

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***********************
Pattern Matching Proofs
***********************
In this section, we will provide a proof of ``plus_commutes`` directly,
by writing a pattern matching definition. We will use interactive
editing features extensively, since it is significantly easier to
produce a proof when the machine can give the types of intermediate
values and construct components of the proof itself. The commands we
will use are summarised below. Where we refer to commands
directly, we will use the Vim version, but these commands have a direct
mapping to Emacs commands.
+---------------------+-----------------+---------------+--------------------------------------------------------------------------------------------+
|Command | Vim binding | Emacs binding | Explanation |
+---------------------+-----------------+---------------+--------------------------------------------------------------------------------------------+
| Check type | ``\t`` | ``C-c C-t`` | Show type of identifier or hole under the cursor. |
+---------------------+-----------------+---------------+--------------------------------------------------------------------------------------------+
| Proof search | ``\o`` | ``C-c C-a`` | Attempt to solve hole under the cursor by applying simple proof search. |
+---------------------+-----------------+---------------+--------------------------------------------------------------------------------------------+
| Make new definition | ``\d`` | ``C-c C-s`` | Add a template definition for the type defined under the cursor. |
+---------------------+-----------------+---------------+--------------------------------------------------------------------------------------------+
| Make lemma | ``\l`` | ``C-c C-e`` | Add a top level function with a type which solves the hole under the cursor. |
+---------------------+-----------------+---------------+--------------------------------------------------------------------------------------------+
| Split cases | ``\c`` | ``C-c C-c`` | Create new constructor patterns for each possible case of the variable under the cursor. |
+---------------------+-----------------+---------------+--------------------------------------------------------------------------------------------+
Creating a Definition
=====================
To begin, create a file ``pluscomm.idr`` containing the following type
declaration:
.. code-block:: idris
plus_commutes : (n : Nat) -> (m : Nat) -> n + m = m + n
To create a template definition for the proof, press ``\d`` (or the
equivalent in your editor of choice) on the line with the type
declaration. You should see:
.. code-block:: idris
plus_commutes : (n : Nat) -> (m : Nat) -> n + m = m + n
plus_commutes n m = ?plus_commutes_rhs
To prove this by induction on ``n``, as we sketched in Section
:ref:`sect-inductive`, we begin with a case split on ``n`` (press
``\c`` with the cursor over the ``n`` in the definition.) You
should see:
.. code-block:: idris
plus_commutes : (n : Nat) -> (m : Nat) -> n + m = m + n
plus_commutes Z m = ?plus_commutes_rhs_1
plus_commutes (S k) m = ?plus_commutes_rhs_2
If we inspect the types of the newly created holes,
``plus_commutes_rhs_1`` and ``plus_commutes_rhs_2``, we see that the
type of each reflects that ``n`` has been refined to ``Z`` and ``S k``
in each respective case. Pressing ``\t`` over
``plus_commutes_rhs_1`` shows:
.. code-block:: idris
m : Nat
--------------------------------------
plus_commutes_rhs_1 : m = plus m 0
Note that ``Z`` renders as ``0`` because the pretty printer renders
natural numbers as integer literals for readability. Similarly, for
``plus_commutes_rhs_2``:
.. code-block:: idris
k : Nat
m : Nat
--------------------------------------
plus_commutes_rhs_2 : S (plus k m) = plus m (S k)
It is a good idea to give these slightly more meaningful names:
.. code-block:: idris
plus_commutes : (n : Nat) -> (m : Nat) -> n + m = m + n
plus_commutes Z m = ?plus_commutes_Z
plus_commutes (S k) m = ?plus_commutes_S
Base Case
=========
We can create a separate lemma for the base case interactively, by
pressing ``\l`` with the cursor over ``plus_commutes_Z``. This
yields:
.. code-block:: idris
plus_commutes_Z : m = plus m 0
plus_commutes : (n : Nat) -> (m : Nat) -> n + m = m + n
plus_commutes Z m = plus_commutes_Z
plus_commutes (S k) m = ?plus_commutes_S
That is, the hole has been filled with a call to a top level
function ``plus_commutes_Z``. The argument ``m`` has been made implicit
because it can be inferred from context when it is applied.
Unfortunately, we cannot prove this lemma directly, since ``plus`` is
defined by matching on its *first* argument, and here ``plus m 0`` has a
specific value for its *second argument* (in fact, the left hand side of
the equality has been reduced from ``plus 0 m``.) Again, we can prove
this by induction, this time on ``m``.
First, create a template definition with ``\d``:
.. code-block:: idris
plus_commutes_Z : m = plus m 0
plus_commutes_Z = ?plus_commutes_Z_rhs
Since we are going to write this by induction on ``m``, which is
implicit, we will need to bring ``m`` into scope manually:
.. code-block:: idris
plus_commutes_Z : m = plus m 0
plus_commutes_Z {m} = ?plus_commutes_Z_rhs
Now, case split on ``m`` with ``\c``:
.. code-block:: idris
plus_commutes_Z : m = plus m 0
plus_commutes_Z {m = Z} = ?plus_commutes_Z_rhs_1
plus_commutes_Z {m = (S k)} = ?plus_commutes_Z_rhs_2
Checking the type of ``plus_commutes_Z_rhs_1`` shows the following,
which is easily proved by reflection:
.. code-block:: idris
--------------------------------------
plus_commutes_Z_rhs_1 : 0 = 0
For such trivial proofs, we can let write the proof automatically by
pressing ``\o`` with the cursor over ``plus_commutes_Z_rhs_1``.
This yields:
.. code-block:: idris
plus_commutes_Z : m = plus m 0
plus_commutes_Z {m = Z} = Refl
plus_commutes_Z {m = (S k)} = ?plus_commutes_Z_rhs_2
For ``plus_commutes_Z_rhs_2``, we are not so lucky:
.. code-block:: idris
k : Nat
--------------------------------------
plus_commutes_Z_rhs_2 : S k = S (plus k 0)
Inductively, we should know that ``k = plus k 0``, and we can get access
to this inductive hypothesis by making a recursive call on k, as
follows:
.. code-block:: idris
plus_commutes_Z : m = plus m 0
plus_commutes_Z {m = Z} = Refl
plus_commutes_Z {m = (S k)} = let rec = plus_commutes_Z {m=k} in
?plus_commutes_Z_rhs_2
For ``plus_commutes_Z_rhs_2``, we now see:
.. code-block:: idris
k : Nat
rec : k = plus k (fromInteger 0)
--------------------------------------
plus_commutes_Z_rhs_2 : S k = S (plus k 0)
Again, the ``fromInteger 0`` is merely due to ``Nat`` having an implementation
of the ``Num`` interface. So we know that ``k = plus k 0``, but how do
we use this to update the goal to ``S k = S k``?
To achieve this, Idris provides a ``replace`` function as part of the
prelude:
.. code-block:: idris
*pluscomm> :t replace
replace : (x = y) -> P x -> P y
Given a proof that ``x = y``, and a property ``P`` which holds for
``x``, we can get a proof of the same property for ``y``, because we
know ``x`` and ``y`` must be the same. In practice, this function can be
a little tricky to use because in general the implicit argument ``P``
can be hard to infer by unification, so Idris provides a high level
syntax which calculates the property and applies ``replace``:
.. code-block:: idris
rewrite prf in expr
If we have ``prf : x = y``, and the required type for ``expr`` is some
property of ``x``, the ``rewrite ... in`` syntax will search for ``x``
in the required type of ``expr`` and replace it with ``y``. Concretely,
in our example, we can say:
.. code-block:: idris
plus_commutes_Z {m = (S k)} = let rec = plus_commutes_Z {m=k} in
rewrite rec in ?plus_commutes_Z_rhs_2
Checking the type of ``plus_commutes_Z_rhs_2`` now gives:
.. code-block:: idris
k : Nat
rec : k = plus k (fromInteger 0)
_rewrite_rule : plus k 0 = k
--------------------------------------
plus_commutes_Z_rhs_2 : S (plus k 0) = S (plus k 0)
Using the rewrite rule ``rec`` (which we can see in the context here as
``_rewrite_rule``\ [1]_, the goal type has been updated with ``k``
replaced by ``plus k 0``.
Alternatively, we could have applied the rewrite in the other direction
using the ``sym`` function:
.. code-block:: idris
*pluscomm> :t sym
sym : (l = r) -> r = l
.. code-block:: idris
plus_commutes_Z {m = (S k)} = let rec = plus_commutes_Z {m=k} in
rewrite sym rec in ?plus_commutes_Z_rhs_2
In this case, inspecting the type of the hole gives:
.. code-block:: idris
k : Nat
rec : k = plus k (fromInteger 0)
_rewrite_rule : k = plus k 0
--------------------------------------
plus_commutes_Z_rhs_2 : S k = S k
Either way, we can use proof search (``\o``) to complete the
proof, giving:
.. code-block:: idris
plus_commutes_Z : m = plus m 0
plus_commutes_Z {m = Z} = Refl
plus_commutes_Z {m = (S k)} = let rec = plus_commutes_Z {m=k} in
rewrite rec in Refl
The base case is now complete.
Inductive Step
==============
Our main theorem, ``plus_commutes`` should currently be in the following
state:
.. code-block:: idris
plus_commutes : (n : Nat) -> (m : Nat) -> n + m = m + n
plus_commutes Z m = plus_commutes_Z
plus_commutes (S k) m = ?plus_commutes_S
Looking again at the type of ``plus_commutes_S``, we have:
.. code-block:: idris
k : Nat
m : Nat
--------------------------------------
plus_commutes_S : S (plus k m) = plus m (S k)
Conveniently, by induction we can immediately tell that
``plus k m = plus m k``, so let us rewrite directly by making a
recursive call to ``plus_commutes``. We add this directly, by hand, as
follows:
.. code-block:: idris
plus_commutes : (n : Nat) -> (m : Nat) -> n + m = m + n
plus_commutes Z m = plus_commutes_Z
plus_commutes (S k) m = rewrite plus_commutes k m in ?plus_commutes_S
Checking the type of ``plus_commutes_S`` now gives:
.. code-block:: idris
k : Nat
m : Nat
_rewrite_rule : plus m k = plus k m
--------------------------------------
plus_commutes_S : S (plus m k) = plus m (S k)
The good news is that ``m`` and ``k`` now appear in the correct order.
However, we still have to show that the successor symbol ``S`` can be
moved to the front in the right hand side of this equality. This
remaining lemma takes a similar form to the ``plus_commutes_Z``; we
begin by making a new top level lemma with ``\l``. This gives:
.. code-block:: idris
plus_commutes_S : (k : Nat) -> (m : Nat) -> S (plus m k) = plus m (S k)
Unlike the previous case, ``k`` and ``m`` are not made implicit because
we cannot in general infer arguments to a function from its result.
Again, we make a template definition with ``\d``:
.. code-block:: idris
plus_commutes_S : (k : Nat) -> (m : Nat) -> S (plus m k) = plus m (S k)
plus_commutes_S k m = ?plus_commutes_S_rhs
Again, this is defined by induction over ``m``, since ``plus`` is
defined by matching on its first argument. The complete definition is:
.. code-block:: idris
total
plus_commutes_S : (k : Nat) -> (m : Nat) -> S (plus m k) = plus m (S k)
plus_commutes_S k Z = Refl
plus_commutes_S k (S j) = rewrite plus_commutes_S k j in Refl
All holes have now been solved.
The ``total`` annotation means that we require the final function to
pass the totality checker; i.e. it will terminate on all possible
well-typed inputs. This is important for proofs, since it provides a
guarantee that the proof is valid in *all* cases, not just those for
which it happens to be well-defined.
Now that ``plus_commutes`` has a ``total`` annotation, we have completed the
proof of commutativity of addition on natural numbers.
.. [1]
Note that the left and right hand sides of the equality have been
swapped, because ``replace`` takes a proof of ``x=y`` and the
property for ``x``, not ``y``.

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********************************************
Running example: Addition of Natural Numbers
********************************************
Throughout this tutorial, we will be working with the following
function, defined in the Idris prelude, which defines addition on
natural numbers:
.. code-block:: idris
plus : Nat -> Nat -> Nat
plus Z m = m
plus (S k) m = S (plus k m)
It is defined by the above equations, meaning that we have for free the
properties that adding ``m`` to zero always results in ``m``, and that
adding ``m`` to any non-zero number ``S k`` always results in
``S (plus k m)``. We can see this by evaluation at the Idris REPL (i.e.
the prompt, the read-eval-print loop):
.. code-block:: idris
Idris> \m => plus Z m
\m => m : Nat -> Nat
Idris> \k,m => plus (S k) m
\k => \m => S (plus k m) : Nat -> Nat -> Nat
Note that unlike many other language REPLs, the Idris REPL performs
evaluation on *open* terms, meaning that it can reduce terms which
appear inside lambda bindings, like those above. Therefore, we can
introduce unknowns ``k`` and ``m`` as lambda bindings and see how
``plus`` reduces.
The ``plus`` function has a number of other useful properties, for
example:
- It is *commutative*, that is for all ``Nat`` inputs ``n`` and ``m``,
we know that ``plus n m = plus m n``.
- It is *associative*, that is for all ``Nat`` inputs ``n``, ``m`` and
``p``, we know that ``plus n (plus m p) = plus (plus m n) p``.
We can use these properties in an Idris program, but in order to do so
we must *prove* them.
Equality Proofs
===============
Idris has a built-in propositional equality type, conceptually defined
as follows:
.. code-block:: idris
data (=) : a -> b -> Type where
Refl : x = x
Note that this must be built-in, rather than defined in the library,
because ``=`` is a reserved operator — you cannot define this directly
in your own code.
It is *propositional* equality, where the type states that any two
values in different types ``a`` and ``b`` may be proposed to be equal.
There is only one way to *prove* equality, however, which is by
reflexivity (``Refl``).
We have a *type* for propositional equality here, and correspondingly a
*program* inhabiting an instance of this type can be seen as a proof of
the corresponding proposition [1]_. So, trivially, we can prove that
``4`` equals ``4``:
.. code-block:: idris
four_eq : 4 = 4
four_eq = Refl
However, trying to prove that ``4 = 5`` results in failure:
.. code-block:: idris
four_eq_five : 4 = 5
four_eq_five = Refl
The type ``4 = 5`` is a perfectly valid type, but is uninhabited, so
when trying to type check this definition, Idris gives the following
error:
::
When elaborating right hand side of four_eq_five:
Type mismatch between
x = x (Type of Refl)
and
4 = 5 (Expected type)
Type checking equality proofs
-----------------------------
An important step in type checking Idris programs is *unification*,
which attempts to resolve implicit arguments such as the implicit
argument ``x`` in ``Refl``. As far as our understanding of type checking
proofs is concerned, it suffices to know that unifying two terms
involves reducing both to normal form then trying to find an assignment
to implicit arguments which will make those normal forms equal.
When type checking ``Refl``, Idris requires that the type is of the form
``x = x``, as we see from the type of ``Refl``. In the case of
``four_eq_five``, Idris will try to unify the expected type ``4 = 5``
with the type of ``Refl``, ``x = x``, notice that a solution requires
that ``x`` be both ``4`` and ``5``, and therefore fail.
Since type checking involves reduction to normal form, we can write the
following equalities directly:
.. code-block:: idris
twoplustwo_eq_four : 2 + 2 = 4
twoplustwo_eq_four = Refl
plus_reduces_Z : (m : Nat) -> plus Z m = m
plus_reduces_Z m = Refl
plus_reduces_Sk : (k, m : Nat) -> plus (S k) m = S (plus k m)
plus_reduces_Sk k m = Refl
Heterogeneous Equality
======================
Equality in Idris is *heterogeneous*, meaning that we can even propose
equalities between values in different types:
.. code-block:: idris
idris_not_php : 2 = "2"
Obviously, in Idris the type ``2 = "2"`` is uninhabited, and one might
wonder why it is useful to be able to propose equalities between values
in different types. However, with dependent types, such equalities can
arise naturally. For example, if two vectors are equal, their lengths
must be equal:
.. code-block:: idris
vect_eq_length : (xs : Vect n a) -> (ys : Vect m a) ->
(xs = ys) -> n = m
In the above declaration, ``xs`` and ``ys`` have different types because
their lengths are different, but we would still like to draw a
conclusion about the lengths if they happen to be equal. We can define
``vect_eq_length`` as follows:
.. code-block:: idris
vect_eq_length xs xs Refl = Refl
By matching on ``Refl`` for the third argument, we know that the only
valid value for ``ys`` is ``xs``, because they must be equal, and
therefore their types must be equal, so the lengths must be equal.
Alternatively, we can put an underscore for the second ``xs``, since
there is only one value which will type check:
.. code-block:: idris
vect_eq_length xs _ Refl = Refl
Properties of ``plus``
======================
Using the ``(=)`` type, we can now state the properties of ``plus``
given above as Idris type declarations:
.. code-block:: idris
plus_commutes : (n, m : Nat) -> plus n m = plus m n
plus_assoc : (n, m, p : Nat) -> plus n (plus m p) = plus (plus n m) p
Both of these properties (and many others) are proved for natural number
addition in the Idris standard library, using ``(+)`` from the ``Num``
interface rather than using ``plus`` directly. They have the names
``plusCommutative`` and ``plusAssociative`` respectively.
In the remainder of this tutorial, we will explore several different
ways of proving ``plus_commutes`` (or, to put it another way, writing
the function.) We will also discuss how to use such equality proofs, and
see where the need for them arises in practice.
.. [1]
This is known as the Curry-Howard correspondence.

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This page attempts to explain some of the techniques used in Idris to prove propositional equalities.
Proving Propositional Equality
==============================
We have seen that definitional equalities can be proved using Refl since they always normalise to unique values that can be compared directly.
However with propositional equalities we are using symbolic variables they do not always normalse.
So to take the previous example:
plusReducesR : (n:Nat) -> plus n Z = n
In this case 'plus n Z' does not normalise to n. Even though both sides are equal we cannot pattern match Refl.
If the pattern match cannot match for all 'n' then the way around this is to separately match all possible values of 'n'. In the case of natural numbers we do this by induction.
So here:
.. code-block:: idris
plusReducesR : n = plus n 0
plusReducesR {n = Z} = Refl
plusReducesR {n = (S k)} = let rec = plus_commutes_Z {n=k} in
rewrite rec in Refl
we don't call Refl to match on 'n = plus n 0' forall 'n' we call it for every number separately. So, in the second line, the pattern matcher knows to substitute Z for n in the type being matched. This uses 'rewrite' which is explained below.
Replace
=======
This implements the 'indiscernability of identicals' principle, if two terms are equal then they have the same properties. In other words, if x=y, then we can substitute y for x in any expression. In our proofs we can express this as:
if x=y
then P x = P y
where P is a pure function representing the property. In the examples below P is an expression in some variable with a type like this: P: n -> Type
So if n is a natural number variable then P could be something like 2*n + 3.
To use this in our proofs there is the following function in the prelude:
.. code-block:: idris
||| Perform substitution in a term according to some equality.
replace : {a:_} -> {x:_} -> {y:_} -> {P : a -> Type} -> x = y -> P x -> P y
replace Refl prf = prf
Removing the implicits, if we supply an equality (x=y) and a proof of a property of x (P x) then we get a proof of a property of y (P y)
.. code-block:: idris
> :t replace
replace : (x = y) -> P x -> P y
So, in the following example, if we supply p1 x which is a proof that x=2 and the equality x=y then we get a proof that y=2.
.. code-block:: idris
p1: Nat -> Type
p1 n = (n=2)
testReplace: (x=y) -> (p1 x) -> (p1 y)
testReplace a b = replace a b
Rewrite
=======
Similar to 'replace' above but Idris provides a nicer syntax which makes 'rewrite' easier to use in examples like plusReducesR above.
.. code-block:: idris
rewrite__impl : (P : a -> Type) -> x = y -> P y -> P x
rewrite__impl p Refl prf = prf
The difference from 'replace' above is nicer syntax and the property p1 is explicitly supplied and it goes in the opposite direction (input and output reversed).
Example: again we supply p1 which is a proof that x=2 and the equality x=y then we get a proof that y=2.
.. code-block:: idris
p1: Nat -> Type
p1 x = (x=2)
testRewrite2: (x=y) -> (p1 y) -> (p1 x)
testRewrite2 a b = rewrite a in b
We can think of rewrite doing this:
* Start with a equation x=y and a property P: x -> Type
* Searches y in P
* Replaces all occurrences of y with x in P.
That is, we are doing a substitution.
Symmetry and Transitivity
=========================
In addition to 'reflexivity' equality also obeys 'symmetry' and 'transitivity' and these are also included in the prelude:
.. code-block:: idris
||| Symmetry of propositional equality
sym : {left:a} -> {right:b} -> left = right -> right = left
sym Refl = Refl
||| Transitivity of propositional equality
trans : {a:x} -> {b:y} -> {c:z} -> a = b -> b = c -> a = c
trans Refl Refl = Refl
Heterogeneous Equality
======================
Also included in the prelude:
.. code-block:: idris
||| Explicit heterogeneous ("John Major") equality. Use this when Idris
||| incorrectly chooses homogeneous equality for `(=)`.
||| @ a the type of the left side
||| @ b the type of the right side
||| @ x the left side
||| @ y the right side
(~=~) : (x : a) -> (y : b) -> Type
(~=~) x y = (x = y)

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interp
views
theorems
provisional
interactive
syntax
miscellany

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.. _sect-provisional:
***********************
Provisional Definitions
***********************
[NOT UPDATED FOR IDRIS 2 YET, AND PROBABLY TO BE DELETED]
Sometimes when programming with dependent types, the type required by
the type checker and the type of the program we have written will be
different (in that they do not have the same normal form), but
nevertheless provably equal. For example, recall the ``parity``
function:
.. code-block:: idris
data Parity : Nat -> Type where
Even : Parity (n + n)
Odd : Parity (S (n + n))
We’d like to implement this as follows:
.. code-block:: idris
parity : (n:Nat) -> Parity n
parity Z = Even {n=Z}
parity (S Z) = Odd {n=Z}
parity (S (S k)) with (parity k)
parity (S (S (j + j))) | Even = Even {n=S j}
parity (S (S (S (j + j)))) | Odd = Odd {n=S j}
This simply states that zero is even, one is odd, and recursively, the
parity of ``k+2`` is the same as the parity of ``k``. Explicitly marking
the value of ``n`` is even and odd is necessary to help type inference.
Unfortunately, the type checker rejects this:
::
viewsbroken.idr:12:10:When elaborating right hand side of ViewsBroken.parity:
Type mismatch between
Parity (plus (S j) (S j))
and
Parity (S (S (plus j j)))
Specifically:
Type mismatch between
plus (S j) (S j)
and
S (S (plus j j))
The type checker is telling us that ``(j+1)+(j+1)`` and ``2+j+j`` do not
normalise to the same value. This is because ``plus`` is defined by
recursion on its first argument, and in the second value, there is a
successor symbol on the second argument, so this will not help with
reduction. These values are obviously equal — how can we rewrite the
program to fix this problem?
Provisional definitions
=======================
*Provisional definitions* help with this problem by allowing us to defer
the proof details until a later point. There are two main reasons why
they are useful.
- When *prototyping*, it is useful to be able to test programs before
finishing all the details of proofs.
- When *reading* a program, it is often much clearer to defer the proof
details so that they do not distract the reader from the underlying
algorithm.
Provisional definitions are written in the same way as ordinary
definitions, except that they introduce the right hand side with a
``?=`` rather than ``=``. We define ``parity`` as follows:
.. code-block:: idris
parity : (n:Nat) -> Parity n
parity Z = Even {n=Z}
parity (S Z) = Odd {n=Z}
parity (S (S k)) with (parity k)
parity (S (S (j + j))) | Even ?= Even {n=S j}
parity (S (S (S (j + j)))) | Odd ?= Odd {n=S j}
When written in this form, instead of reporting a type error, Idris
will insert a hole standing for a theorem which will correct the
type error. Idris tells us we have two proof obligations, with names
generated from the module and function names:
.. code-block:: idris
*views> :m
Global holes:
[views.parity_lemma_2,views.parity_lemma_1]
The first of these has the following type:
.. code-block:: idris
*views> :p views.parity_lemma_1
---------------------------------- (views.parity_lemma_1) --------
{hole0} : (j : Nat) -> (Parity (plus (S j) (S j))) -> Parity (S (S (plus j j)))
-views.parity_lemma_1>
The two arguments are ``j``, the variable in scope from the pattern
match, and ``value``, which is the value we gave in the right hand side
of the provisional definition. Our goal is to rewrite the type so that
we can use this value. We can achieve this using the following theorem
from the prelude:
.. code-block:: idris
plusSuccRightSucc : (left : Nat) -> (right : Nat) ->
S (left + right) = left + (S right)
We need to use ``compute`` again to unfold the definition of ``plus``:
.. code-block:: idris
-views.parity_lemma_1> compute
---------------------------------- (views.parity_lemma_1) --------
{hole0} : (j : Nat) -> (Parity (S (plus j (S j)))) -> Parity (S (S (plus j j)))
After applying ``intros`` we have:
.. code-block:: idris
-views.parity_lemma_1> intros
j : Nat
value : Parity (S (plus j (S j)))
---------------------------------- (views.parity_lemma_1) --------
{hole2} : Parity (S (S (plus j j)))
Then we apply the ``plusSuccRightSucc`` rewrite rule, symmetrically, to
``j`` and ``j``, giving:
.. code-block:: idris
-views.parity_lemma_1> rewrite sym (plusSuccRightSucc j j)
j : Nat
value : Parity (S (plus j (S j)))
---------------------------------- (views.parity_lemma_1) --------
{hole3} : Parity (S (plus j (S j)))
``sym`` is a function, defined in the library, which reverses the order
of the rewrite:
.. code-block:: idris
sym : l = r -> r = l
sym Refl = Refl
We can complete this proof using the ``trivial`` tactic, which finds
``value`` in the premises. The proof of the second lemma proceeds in
exactly the same way.
We can now test the ``natToBin`` function from Section :ref:`sect-nattobin`
at the prompt. The number 42 is 101010 in binary. The binary digits are
reversed:
.. code-block:: idris
*views> show (natToBin 42)
"[False, True, False, True, False, True]" : String
Suspension of Disbelief
=======================
Idris requires that proofs be complete before compiling programs
(although evaluation at the prompt is possible without proof details).
Sometimes, especially when prototyping, it is easier not to have to do
this. It might even be beneficial to test programs before attempting to
prove things about them — if testing finds an error, you know you had
better not waste your time proving something!
Therefore, Idris provides a built-in coercion function, which allows
you to use a value of the incorrect types:
.. code-block:: idris
believe_me : a -> b
Obviously, this should be used with extreme caution. It is useful when
prototyping, and can also be appropriate when asserting properties of
external code (perhaps in an external C library). The “proof” of
``views.parity_lemma_1`` using this is:
.. code-block:: idris
views.parity_lemma_2 = proof {
intro;
intro;
exact believe_me value;
}
The ``exact`` tactic allows us to provide an exact value for the proof.
In this case, we assert that the value we gave was correct.
Example: Binary numbers
=======================
Previously, we implemented conversion to binary numbers using the
``Parity`` view. Here, we show how to use the same view to implement a
verified conversion to binary. We begin by indexing binary numbers over
their ``Nat`` equivalent. This is a common pattern, linking a
representation (in this case ``Binary``) with a meaning (in this case
``Nat``):
.. code-block:: idris
data Binary : Nat -> Type where
BEnd : Binary Z
BO : Binary n -> Binary (n + n)
BI : Binary n -> Binary (S (n + n))
``BO`` and ``BI`` take a binary number as an argument and effectively
shift it one bit left, adding either a zero or one as the new least
significant bit. The index, ``n + n`` or ``S (n + n)`` states the result
that this left shift then add will have to the meaning of the number.
This will result in a representation with the least significant bit at
the front.
Now a function which converts a Nat to binary will state, in the type,
that the resulting binary number is a faithful representation of the
original Nat:
.. code-block:: idris
natToBin : (n:Nat) -> Binary n
The ``Parity`` view makes the definition fairly simple — halving the
number is effectively a right shift after all — although we need to use
a provisional definition in the Odd case:
.. code-block:: idris
natToBin : (n:Nat) -> Binary n
natToBin Z = BEnd
natToBin (S k) with (parity k)
natToBin (S (j + j)) | Even = BI (natToBin j)
natToBin (S (S (j + j))) | Odd ?= BO (natToBin (S j))
The problem with the Odd case is the same as in the definition of
``parity``, and the proof proceeds in the same way:
.. code-block:: idris
natToBin_lemma_1 = proof {
intro;
intro;
rewrite sym (plusSuccRightSucc j j);
trivial;
}
To finish, we’ll implement a main program which reads an integer from
the user and outputs it in binary.
.. code-block:: idris
main : IO ()
main = do putStr "Enter a number: "
x <- getLine
print (natToBin (fromInteger (cast x)))
For this to work, of course, we need a ``Show`` implementation for
``Binary n``:
.. code-block:: idris
Show (Binary n) where
show (BO x) = show x ++ "0"
show (BI x) = show x ++ "1"
show BEnd = ""

60
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@ -98,14 +98,14 @@ by recursion on the first argument to ``plus``, namely ``n``.
plusReducesZ : (n:Nat) -> n = plus n Z
plusReducesZ Z = Refl
plusReducesZ (S k) = cong (plusReducesZ k)
plusReducesZ (S k) = cong S (plusReducesZ k)
``cong`` is a function defined in the library which states that equality
respects function application:
.. code-block:: idris
cong : {f : t -> u} -> a = b -> f a = f b
cong : (f : t -> u) -> a = b -> f a = f b
We can do the same for the reduction behaviour of plus on successors:
@ -113,9 +113,9 @@ We can do the same for the reduction behaviour of plus on successors:
plusReducesS : (n:Nat) -> (m:Nat) -> S (plus n m) = plus n (S m)
plusReducesS Z m = Refl
plusReducesS (S k) m = cong (plusReducesS k m)
plusReducesS (S k) m = cong S (plusReducesS k m)
Even for trivial theorems like these, the proofs are a little tricky to
Even for small theorems like these, the proofs are a little tricky to
construct in one go. When things get even slightly more complicated, it
becomes too much to think about to construct proofs in this “batch
mode”.
@ -137,8 +137,8 @@ previously (:ref:`sec-views`) we implemented ``natToBin`` using a function
parity : (n:Nat) -> Parity n
However, we didn't provide a definition for ``parity``. We might expect it
to look something like the following:
We provided a definition for ``parity``, but without explanation. We might
hope that it would look something like the following:
.. code-block:: idris
@ -153,13 +153,11 @@ Unfortunately, this fails with a type error:
::
When checking right hand side of with block in views.parity with expected type
Parity (S (S (j + j)))
Type mismatch between
Parity (S j + S j) (Type of Even)
With.idr:26:17--27:3:While processing right hand side of Main.with block in 2419 at With.idr:24:3--27:3:
Can't solve constraint between:
plus j (S j)
and
Parity (S (S (plus j j))) (Expected type)
S (plus j j)
The problem is that normalising ``S j + S j``, in the type of ``Even``
doesn't result in what we need for the type of the right hand side of
@ -236,10 +234,10 @@ the type of the equation we just used (as the type of ``_rewrite_rule``):
.. code-block:: idris
j : Nat
p : Parity (S (plus j (S j)))
_rewrite_rule : S (plus j j) = plus j (S j)
--------------------------------------
Main> :t helpEven_rhs
j : Nat
p : Parity (S (plus j (S j)))
-------------------------------------
helpEven_rhs : Parity (S (plus j (S j)))
Using ``rewrite`` and another helper for the ``Odd`` case, we can complete
@ -292,35 +290,29 @@ definitions is total:
::
*Theorems> :total empty1
possibly not total due to: empty1#hd
not total as there are missing cases
*Theorems> :total empty2
possibly not total due to recursive path empty2
Void> :total empty1
Void.empty1 is not covering due to call to function empty1:hd
Void> :total empty2
Void.empty2 is possibly not terminating due to recursive path Void.empty2
Note the use of the word “possibly” — a totality check can, of course,
never be certain due to the undecidability of the halting problem. The
check is, therefore, conservative. It is also possible (and indeed
advisable, in the case of proofs) to mark functions as total so that it
will be a compile time error for the totality check to fail:
Note the use of the word “possibly” — a totality check can never be certain due
to the undecidability of the halting problem. The check is, therefore,
conservative. It is also possible (and indeed advisable, in the case of proofs)
to mark functions as total so that it will be a compile time error for the
totality check to fail:
.. code-block:: idris
total empty2 : Void
empty2 = empty2
::
Type checking ./theorems.idr
theorems.idr:25:empty2 is possibly not total due to recursive path empty2
Reassuringly, our proof in Section :ref:`sect-empty` that the zero and
successor constructors are disjoint is total:
.. code-block:: idris
*theorems> :total disjoint
Total
Main> :total disjoint
Main.disjoint is Total
The totality check is, necessarily, conservative. To be recorded as
total, a function ``f`` must:
@ -338,6 +330,8 @@ total, a function ``f`` must:
Directives and Compiler Flags for Totality
------------------------------------------
[NOTE: Not all of this is implemented yet for Idris 2]
By default, Idris allows all well-typed definitions, whether total or not.
However, it is desirable for functions to be total as far as possible, as this
provides a guarantee that they provide a result for all possible inputs, in

44
docs/tutorial/views.rst
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@ -65,8 +65,8 @@ extraneous and may be omitted:
.. code-block:: idris
foo : Int -> Int -> Bool
foo n m with (succ n)
foo _ m | 2 with (succ m)
foo n m with (n + 1)
foo _ m | 2 with (m + 1)
foo _ _ | 2 | 3 = True
foo _ _ | 2 | _ = False
foo _ _ | _ = False
@ -81,12 +81,13 @@ Otherwise, it is the sum of two equal ``Nat`` plus one:
.. code-block:: idris
data Parity : Nat -> Type where
Even : Parity (n + n)
Odd : Parity (S (n + n))
Even : {n : _} -> Parity (n + n)
Odd : {n : _} -> Parity (S (n + n))
We say ``Parity`` is a *view* of ``Nat``. It has a *covering function*
which tests whether it is even or odd and constructs the predicate
accordingly.
accordingly. Note that we're going to need access to ``n`` at run time, so
although it's an implicit argument, it has unrestricted multiplicity.
.. code-block:: idris
@ -129,29 +130,26 @@ Note that there is a function in the patterns (``+``) and repeated
occurrences of ``j`` - this is allowed because another argument has
determined the form of these patterns.
We will return to this function in the next section :ref:`sect-parity` to
complete the definition of ``parity``.
Defining ``parity``
===================
With and proofs
===============
To use a dependent pattern match for theorem proving, it is sometimes necessary
to explicitly construct the proof resulting from the pattern match.
To do this, you can postfix the with clause with ``proof p`` and the proof
generated by the pattern match will be in scope and named ``p``. For example:
The definition of ``parity`` is a little tricky, and requires some knowledge of
theorem proving (see Section :ref:`sect-theorems`), but for completeness, here
it is:
.. code-block:: idris
data Foo = FInt Int | FBool Bool
optional : Foo -> Maybe Int
optional (FInt x) = Just x
optional (FBool b) = Nothing
isFInt : (foo:Foo) -> Maybe (x : Int ** (optional foo = Just x))
isFInt foo with (optional foo) proof p
isFInt foo | Nothing = Nothing -- here, p : Nothing = optional foo
isFInt foo | (Just x) = Just (x ** Refl) -- here, p : Just x = optional foo
parity : (n : Nat) -> Parity n
parity Z = Even {n = Z}
parity (S Z) = Odd {n = Z}
parity (S (S k)) with (parity k)
parity (S (S (j + j))) | Even
= rewrite plusSuccRightSucc j j in Even {n = S j}
parity (S (S (S (j + j)))) | Odd
= rewrite plusSuccRightSucc j j in Odd {n = S j}
For full details on ``rewrite`` in particular, please refer to the theorem
proving tutorial, in Section :ref:`proofs-index`.
.. [1] Conor McBride and James McKinna. 2004. The view from the
left. J. Funct. Program. 14, 1 (January 2004),

23
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@ -0,0 +1,23 @@
fiveIsFive : 5 = 5
fiveIsFive = Refl
twoPlusTwo : 2 + 2 = 4
twoPlusTwo = Refl
disjoint : (n : Nat) -> Z = S n -> Void
disjoint n prf = replace {p = disjointTy} prf ()
where
disjointTy : Nat -> Type
disjointTy Z = ()
disjointTy (S k) = Void
plusReduces : (n:Nat) -> plus Z n = n
plusReduces n = Refl
plusReducesZ : (n:Nat) -> n = plus n Z
plusReducesZ Z = Refl
plusReducesZ (S k) = cong S (plusReducesZ k)
plusReducesS : (n:Nat) -> (m:Nat) -> S (plus n m) = plus n (S m)
plusReducesS Z m = Refl
plusReducesS (S k) m = cong S (plusReducesS k m)

11
samples/Void.idr Normal file
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@ -0,0 +1,11 @@
module Void
-- making use of 'hd' being partially defined
empty1 : Void
empty1 = hd [] where
hd : List a -> a
hd (x :: xs) = x
-- not terminating
empty2 : Void
empty2 = empty2

47
samples/With.idr Normal file
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@ -0,0 +1,47 @@
import Data.Vect
import Data.Nat
my_filter : (a -> Bool) -> Vect n a -> (p ** Vect p a)
my_filter p [] = ( _ ** [] )
my_filter p (x :: xs) with (filter p xs)
my_filter p (x :: xs) | ( _ ** xs' ) = if (p x) then ( _ ** x :: xs' ) else ( _ ** xs' )
foo : Int -> Int -> Bool
foo n m with (n + 1)
foo _ m | 2 with (m + 1)
foo _ _ | 2 | 3 = True
foo _ _ | 2 | _ = False
foo _ _ | _ = False
data Parity : Nat -> Type where
Even : {n : _} -> Parity (n + n)
Odd : {n : _} ->Parity (S (n + n))
-- parity : (n : Nat) -> Parity n
-- parity Z = Even {n = Z}
-- parity (S Z) = Odd {n = Z}
-- parity (S (S k)) with (parity k)
-- parity (S (S (j + j))) | Even
-- = rewrite plusSuccRightSucc j j in Even {n = S j}
-- parity (S (S (S (j + j)))) | Odd
-- = rewrite plusSuccRightSucc j j in Odd {n = S j}
helpEven : (j : Nat) -> Parity (S j + S j) -> Parity (S (S (plus j j)))
helpEven j p = rewrite plusSuccRightSucc j j in p
helpOdd : (j : Nat) -> Parity (S (S (j + S j))) -> Parity (S (S (S (j + j))))
helpOdd j p = rewrite plusSuccRightSucc j j in p
parity : (n:Nat) -> Parity n
parity Z = Even {n=Z}
parity (S Z) = Odd {n=Z}
parity (S (S k)) with (parity k)
parity (S (S (j + j))) | Even = helpEven j (Even {n = S j})
parity (S (S (S (j + j)))) | Odd = helpOdd j (Odd {n = S j})
natToBin : Nat -> List Bool
natToBin Z = Nil
natToBin k with (parity k)
natToBin (j + j) | Even = False :: natToBin j
natToBin (S (j + j)) | Odd = True :: natToBin j

6
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@ -588,11 +588,11 @@ export
Show PartialReason where
show NotStrictlyPositive = "not strictly positive"
show (BadCall [n])
= "not terminating due to call to " ++ show n
= "possibly not terminating due to call to " ++ show n
show (BadCall ns)
= "not terminating due to calls to " ++ showSep ", " (map show ns)
= "possibly not terminating due to calls to " ++ showSep ", " (map show ns)
show (RecPath ns)
= "not terminating due to recursive path " ++ showSep " -> " (map show ns)
= "possibly not terminating due to recursive path " ++ showSep " -> " (map show ns)
public export
data Terminating

8
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@ -4,14 +4,14 @@ Main> Main.foo is total
Main> Main.ordElim is total
Main> Main.bar is total
Main> Main.swapR is total
Main> Main.loopy is not terminating due to recursive path Main.loopy
Main> Main.loopy is possibly not terminating due to recursive path Main.loopy
Main> Main.foom is total
Main> Main.pfoom is not terminating due to recursive path Main.pfoom -> Main.pfoom
Main> Main.pfoom is possibly not terminating due to recursive path Main.pfoom -> Main.pfoom
Main> Main.even is total
Main> Main.vtrans is not terminating due to recursive path Main.vtrans -> Main.vtrans
Main> Main.vtrans is possibly not terminating due to recursive path Main.vtrans -> Main.vtrans
Main> Main.GTree is total
Main> Main.size is total
Main> Main.qsortBad is not terminating due to recursive path Main.qsortBad -> Main.qsortBad
Main> Main.qsortBad is possibly not terminating due to recursive path Main.qsortBad -> Main.qsortBad
Main> Main.qsort is total
Main> Main.qsort' is total
Main> Main.mySorted is total

8
tests/idris2/total005/expected
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@ -1,9 +1,9 @@
1/1: Building Total (Total.idr)
Main> Main.Bad is not strictly positive
Main> Main.Bad1 is not strictly positive
Main> Main.Bad2 is not terminating due to call to Main.Bad1
Main> Main.foo is not terminating due to call to Main.MkBad'
Main> Main.foo2 is not terminating due to call to Main.MkBad'
Main> Main.Bad2 is possibly not terminating due to call to Main.Bad1
Main> Main.foo is possibly not terminating due to call to Main.MkBad'
Main> Main.foo2 is possibly not terminating due to call to Main.MkBad'
Main> Main.T is not strictly positive
Main> Main.T2 is not terminating due to call to Main.T
Main> Main.T2 is possibly not terminating due to call to Main.T
Main> Bye for now!

4
tests/idris2/total006/expected
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@ -1,8 +1,8 @@
1/1: Building Total (Total.idr)
Main> Main.count is total
Main> Main.badCount is not terminating due to recursive path Main.badCount -> Prelude.Functor implementation at Prelude.idr:873:1--877:1 -> Prelude.map
Main> Main.badCount is possibly not terminating due to recursive path Main.badCount -> Prelude.Functor implementation at Prelude.idr:873:1--877:1 -> Prelude.map
Main> Main.process is total
Main> Main.badProcess is not terminating due to recursive path Main.badProcess -> Main.badProcess -> Main.badProcess
Main> Main.badProcess is possibly not terminating due to recursive path Main.badProcess -> Main.badProcess -> Main.badProcess
Main> Main.doubleInt is total
Main> Main.main is total
Main> Bye for now!

2
tests/ttimp/total002/expected
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@ -5,5 +5,5 @@ Yaffle> Main.ack is total
Yaffle> Main.foo is total
Yaffle> Main.foo' is total
Yaffle> Main.foom is total
Yaffle> Main.pfoom is not terminating due to recursive path Main.pfoom -> Main.pfoom -> Main.pfoom
Yaffle> Main.pfoom is possibly not terminating due to recursive path Main.pfoom -> Main.pfoom -> Main.pfoom
Yaffle> Bye for now!

2
tests/ttimp/total003/expected
View File

@ -3,6 +3,6 @@ Written TTC
Yaffle> Main.Bad is not strictly positive
Yaffle> Main.MkBad is not strictly positive
Yaffle> Main.MkBad' is not strictly positive
Yaffle> Main.foo is not terminating due to call to Main.MkBad'
Yaffle> Main.foo is possibly not terminating due to call to Main.MkBad'
Yaffle> Main.T is not strictly positive
Yaffle> Bye for now!