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14
docs/proofs/definitional.rst
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@ -141,17 +141,17 @@ 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:
like '2=2'. It is enough that both sides are *definitionally equal* like this:
.. code-block:: idris
onePlusOne : 1+1=2
onePlusOne = Refl
Both sides of this equation nomalise to 2 and so Refl matches and the
Both sides of this equation normalise to 2 and so Refl matches and the
proposition is proved.
We don't have to stick to terms, we can also use symbolic parameters so the
We don't have to stick to terms; we can also use symbolic parameters so the
following type checks:
.. code-block:: idris
@ -159,15 +159,15 @@ following type checks:
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
If a proposition/equality type is not definitionally equal but is still true
then it is *propositionally 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 definitionally equal,
in this example ``plusReducesL`` is *definitionally equal* but ``plusReducesR`` is
not (although it is *propositionaly equal*). The only difference between
not (although it is *propositionally equal*). The only difference between
them is the order of the operands.
.. code-block:: idris
@ -207,4 +207,4 @@ definitional equality is built in
Main> 1+1
2
In the following pages we discuss how to resolve propositionaly equalies.
In the following pages we discuss how to resolve propositional equalities.