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211 lines
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211 lines
6.6 KiB
ReStructuredText
Before we discuss the details of theorem proving in Idris, we will describe
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some fundamental concepts:
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- Propositions and judgments
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- Boolean and constructive logic
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- Curry-Howard correspondence
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- Definitional and propositional equalities
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- Axiomatic and constructive approaches
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Propositions and Judgments
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==========================
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Propositions are the subject of our proofs. Before the proof, we can't
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formally say if they are true or not. If the proof is successful then the
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result is a 'judgment'. For instance, if the ``proposition`` is,
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+-------+
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| 1+1=2 |
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+-------+
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When we prove it, the ``judgment`` is,
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+------------+
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| 1+1=2 true |
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+------------+
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Or if the ``proposition`` is,
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+-------+
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| 1+1=3 |
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+-------+
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we can't prove it is true, but it is still a valid proposition and perhaps we
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can prove it is false so the ``judgment`` is,
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+-------------+
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| 1+1=3 false |
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+-------------+
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This may seem a bit pedantic but it is important to be careful: in mathematics
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not every proposition is true or false. For instance, a proposition may be
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unproven or even unprovable.
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So the logic here is different from the logic that comes from boolean algebra.
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In that case what is not true is false and what is not false is true. The logic
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we are using here does not have this law, the "Law of Excluded Middle", so we
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cannot use it.
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A false proposition is taken to be a contradiction and if we have a
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contradiction then we can prove anything, so we need to avoid this. Some
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languages, used in proof assistants, prevent contradictions.
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The logic we are using is called constructive (or sometimes intuitional)
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because we are constructing a 'database' of judgments.
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Curry-Howard correspondence
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---------------------------
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So how do we relate these proofs to Idris programs? It turns out that there is
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a correspondence between constructive logic and type theory. They have the same
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structure and we can switch back and forth between the two notations.
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The way that this works is that a proposition is a type so...
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.. code-block:: idris
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Main> 1 + 1 = 2
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2 = 2
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Main> :t 1 + 1 = 2
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(fromInteger 1 + fromInteger 1) === fromInteger 2 : Type
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...is a proposition and it is also a type. The
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following will also produce an equality type:
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.. code-block:: idris
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Main> 1 + 1 = 3
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2 = 3
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Both of these are valid propositions so both are valid equality types. But how
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do we represent a true judgment? That is, how do we denote 1+1=2 is true but not
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1+1=3? A type that is true is inhabited, that is, it can be constructed. An
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equality type has only one constructor 'Refl' so a proof of 1+1=2 is
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.. code-block:: idris
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onePlusOne : 1+1=2
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onePlusOne = Refl
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Now that we can represent propositions as types other aspects of
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propositional logic can also be translated to types as follows:
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+----------+-------------------+--------------------------+
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| | propositions | example of possible type |
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+----------+-------------------+--------------------------+
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| A | x=y | |
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+----------+-------------------+--------------------------+
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| B | y=z | |
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+----------+-------------------+--------------------------+
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| and | A /\\ B | Pair(x=y,y=z) |
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+----------+-------------------+--------------------------+
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| or | A \\/ B | Either(x=y,y=z) |
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+----------+-------------------+--------------------------+
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| implies | A -> B | (x=y) -> (y=z) |
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+----------+-------------------+--------------------------+
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| for all | y=z | |
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+----------+-------------------+--------------------------+
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| exists | y=z | |
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+----------+-------------------+--------------------------+
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And (conjunction)
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-----------------
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We can have a type which corresponds to conjunction:
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.. code-block:: idris
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AndIntro : a -> b -> A a b
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There is a built in type called 'Pair'.
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Or (disjunction)
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----------------
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We can have a type which corresponds to disjunction:
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.. code-block:: idris
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data Or : Type -> Type -> Type where
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OrIntroLeft : a -> A a b
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OrIntroRight : b -> A a b
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There is a built in type called 'Either'.
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Definitional and Propositional Equalities
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-----------------------------------------
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We have seen that we can 'prove' a type by finding a way to construct a term.
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In the case of equality types there is only one constructor which is ``Refl``.
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We have also seen that each side of the equation does not have to be identical
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like '2=2'. It is enough that both sides are *definitionally equal* like this:
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.. code-block:: idris
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onePlusOne : 1+1=2
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onePlusOne = Refl
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Both sides of this equation normalise to 2 and so Refl matches and the
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proposition is proved.
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We don't have to stick to terms; we can also use symbolic parameters so the
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following type checks:
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.. code-block:: idris
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varIdentity : m = m
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varIdentity = Refl
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If a proposition/equality type is not definitionally equal but is still true
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then it is *propositionally equal*. In this case we may still be able to prove
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it but some steps in the proof may require us to add something into the terms
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or at least to take some sideways steps to get to a proof.
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Especially when working with equalities containing variable terms (inside
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functions) it can be hard to know which equality types are definitionally equal,
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in this example ``plusReducesL`` is *definitionally equal* but ``plusReducesR`` is
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not (although it is *propositionally equal*). The only difference between
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them is the order of the operands.
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.. code-block:: idris
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plusReducesL : (n:Nat) -> plus Z n = n
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plusReducesL n = Refl
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plusReducesR : (n:Nat) -> plus n Z = n
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plusReducesR n = Refl
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Checking ``plusReducesR`` gives the following error:
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.. code-block:: idris
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Proofs.idr:21:18--23:1:While processing right hand side of Main.plusReducesR at Proofs.idr:21:1--23:1:
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Can't solve constraint between:
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plus n Z
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and
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n
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So why is ``Refl`` able to prove some equality types but not others?
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The first answer is that ``plus`` is defined by recursion on its first
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argument. So, when the first argument is ``Z``, it reduces, but not when the
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second argument is ``Z``.
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If an equality type can be proved/constructed by using ``Refl`` alone it is known
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as a *definitional equality*. In order to be definitionally equal both sides
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of the equation must normalise to the same value.
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So when we type ``1+1`` in Idris it is immediately reduced to 2 because
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definitional equality is built in
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.. code-block:: idris
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Main> 1+1
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2
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In the following pages we discuss how to resolve propositional equalities.
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