Web-Projects/blog-static
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

Switch day 1 Coq post to use submodule'd code.

Browse Source
This commit is contained in:
Danila Fedorin 2021-01-01 18:46:35 -08:00
parent 06014eade9
commit 6f9a2ce092
1 changed files with 10 additions and 10 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

20
content/blog/00_aoc_coq.md
View File

@ -26,7 +26,7 @@ let's write a helper function that, given a number `x`, tries to find another nu
`y` such that `x + y = 2020`. In fact, rather than hardcoding the desired `y` such that `x + y = 2020`. In fact, rather than hardcoding the desired
sum to `2020`, let's just use another argument called `total`. The code is quite simple: sum to `2020`, let's just use another argument called `total`. The code is quite simple:
{{< codelines "Coq" "aoc-coq/day1.v" 7 14 >}} {{< codelines "Coq" "aoc-2020/day1.v" 11 18 >}}
Here, `is` is the list of numbers that we want to search. Here, `is` is the list of numbers that we want to search.
We proceed by case analysis: if the list is empty, we can't We proceed by case analysis: if the list is empty, we can't
@ -43,7 +43,7 @@ for our purposes when the argument being case analyzed is given first.
We can now use `find_matching` to define our `find_sum` function, which solves part 1. We can now use `find_matching` to define our `find_sum` function, which solves part 1.
Here's the code: Here's the code:
{{< codelines "Coq" "aoc-coq/day1.v" 16 24 >}} {{< codelines "Coq" "aoc-2020/day1.v" 20 28 >}}
For every `x` that we encounter in our input list `is`, we want to check if there's For every `x` that we encounter in our input list `is`, we want to check if there's
a matching number in the rest of the list. We only search the remainder of the list a matching number in the rest of the list. We only search the remainder of the list
@ -72,13 +72,13 @@ formally as follows:
And this is how we write it in Coq: And this is how we write it in Coq:
{{< codelines "Coq" "aoc-coq/day1.v" 26 27 >}} {{< codelines "Coq" "aoc-2020/day1.v" 30 31 >}}
The arrow, `->`, reads "implies". Other than that, I think this The arrow, `->`, reads "implies". Other than that, I think this
property reads pretty well. The proof, unfortunately, is a little bit more involved. property reads pretty well. The proof, unfortunately, is a little bit more involved.
Here are the first few lines: Here are the first few lines:
{{< codelines "Coq" "aoc-coq/day1.v" 28 31 >}} {{< codelines "Coq" "aoc-2020/day1.v" 32 35 >}}
We start with the `intros is` tactic, which is akin to saying We start with the `intros is` tactic, which is akin to saying
"consider a particular list of integers `is`". We do this without losing "consider a particular list of integers `is`". We do this without losing
@ -157,7 +157,7 @@ is zero. This means we're done with the base case!
The inductive case is the meat of this proof. Here's the corresponding part The inductive case is the meat of this proof. Here's the corresponding part
of the Coq source file: of the Coq source file:
{{< codelines "Coq" "aoc-coq/day1.v" 32 36 >}} {{< codelines "Coq" "aoc-2020/day1.v" 36 40 >}}
This time, the proof state is more complicated: This time, the proof state is more complicated:
@ -284,14 +284,14 @@ Coq proofs is to actually step through them in the IDE!
First on the list is `find_matching_skip`. Here's the type: First on the list is `find_matching_skip`. Here's the type:
{{< codelines "Coq" "aoc-coq/day1.v" 38 39 >}} {{< codelines "Coq" "aoc-2020/day1.v" 42 43 >}}
It reads: if we correctly find a number in a small list `is`, we can find that same number It reads: if we correctly find a number in a small list `is`, we can find that same number
even if another number is prepended to `is`. That makes sense: _adding_ a number to even if another number is prepended to `is`. That makes sense: _adding_ a number to
a list doesn't remove whatever we found in it! I used this lemma to prove another, a list doesn't remove whatever we found in it! I used this lemma to prove another,
`find_matching_works`: `find_matching_works`:
{{< codelines "Coq" "aoc-coq/day1.v" 49 50 >}} {{< codelines "Coq" "aoc-2020/day1.v" 53 54 >}}
This reads, if there _is_ an element `y` in `is` that adds up to `k` with `x`, then This reads, if there _is_ an element `y` in `is` that adds up to `k` with `x`, then
`find_matching` will find it. This is an important property. After all, if it didn't `find_matching` will find it. This is an important property. After all, if it didn't
@ -310,7 +310,7 @@ that all lists from this Advent of Code puzzle are guaranteed to have two number
add up to our goal, and that these numbers are not equal to each other. In Coq, add up to our goal, and that these numbers are not equal to each other. In Coq,
we state this as follows: we state this as follows:
{{< codelines "Coq" "aoc-coq/day1.v" 4 5 >}} {{< codelines "Coq" "aoc-2020/day1.v" 8 9 >}}
This defines a new property, `has_pair t is` (read "`is` has a pair of numbers that add to `t`"), This defines a new property, `has_pair t is` (read "`is` has a pair of numbers that add to `t`"),
which means: which means:
@ -323,7 +323,7 @@ which means:
When making claims about the correctness of our algorithm, we will assume that this When making claims about the correctness of our algorithm, we will assume that this
property holds. Finally, here's the theorem we want to prove: property holds. Finally, here's the theorem we want to prove:
{{< codelines "Coq" "aoc-coq/day1.v" 64 66 >}} {{< codelines "Coq" "aoc-2020/day1.v" 68 70 >}}
It reads, "for any total `k` and list `is`, if `is` has a pair of numbers that add to `k`, It reads, "for any total `k` and list `is`, if `is` has a pair of numbers that add to `k`,
then `find_sum` will return a pair of numbers `x` and `y` that add to `k`". then `find_sum` will return a pair of numbers `x` and `y` that add to `k`".
@ -335,7 +335,7 @@ we want to confirm that `find_sum` will find one of them. Finally, here is the p
I will not be able to go through it in detail in this post, but I did comment it to I will not be able to go through it in detail in this post, but I did comment it to
make it easier to read: make it easier to read:
{{< codelines "Coq" "aoc-coq/day1.v" 67 102 >}} {{< codelines "Coq" "aoc-2020/day1.v" 71 106 >}}
Coq seems happy with it, and so am I! The bug I mentioned earlier popped up on line 96. Coq seems happy with it, and so am I! The bug I mentioned earlier popped up on line 96.
I had accidentally made `find_sum` return `None` if it couldn't find a complement I had accidentally made `find_sum` return `None` if it couldn't find a complement