How can I describe this property of divisibility in Idris? - idris

Question

I want to prove that if b (Integer) divides a (Integer), then b also divides a * c (where c is an Integer). First, I need to reformulate the problem into a computer understandable problem, here was an attempt:
-- If a is divisible by b, then there exists an integer such that a = b * n
divisibleBy : (a, b : Integer) ->
(n : Integer **
(a = b * n))
-- If b | a then b | ac.
alsoDividesMultiples : (a, b, c : Integer) ->
(divisibleBy a b) ->
(divisibleBy (a * c) b)
However, I am getting TypeUnification failure. I am not really sure what is wrong.
|
7 | alsoDividesMultiples : (a, b, c : Integer) ->
| ^
When checking type of Numbris.Divisibility.alsoDividesMultiples:
Type mismatch between
(n : Integer ** a = b * n) (Type of divisibleBy a b)
and
Type (Expected type)
Specifically:
Type mismatch between
(n : Integer ** a = prim__mulBigInt b n)
and
TypeUnification failure
In context:
a : Integer
b : Integer
c : Integer
{a_509} : Integer
{b_510} : Integer

Answer 1

In Idris, propositions are represented by types, while proofs of propositions are represented by elements of those types. The basic issue here is that you have defined divisibleBy as a function which returns an element (i.e. a proof) rather than a type (proposition). Thus, as you've defined it here, divisbleBy actually purports to be a proof that all integers are divisible by all other integers, which is clearly not true! I think what you're actually looking for is something like this.
DivisibleBy : Integer -> Integer -> Type
DivisibleBy a b = (n : Integer ** a = b * n)
alsoDividesMultiples : (a, b, c : Integer) ->
DivisibleBy a b ->
DivisibleBy (a * c) b