Use auto instead of apply.

README.md

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+# Advent of Code 2020 Solutions
+Here's to my first 50 star year since 2017! The goal
+was to make it to top 100 at least once this year,
+and it finally came true on day 22 (and again in day 25
+part 2, but that hardly counts). Honestly,
+I am not at all happy with myself, though.
+
+## Kinds of Solutions
+I "raced" in [Crystal](https://crystal-lang.org), but also tried my hand at
+formal verification in [Coq](https://coq.inria.fr/), and tried out
+an APL dialect called [J](https://jsoftware.com/) for fun.
+I didn't always clean my race day solutions, particularly the ones to the
+"hard" days. I will write about my Coq solutions on [my site](https://danilafe.com).
+
+## Rankings
+Here's the (rather embarassing) table with my times.
+```
+      -------Part 1--------   -------Part 2--------
+Day       Time  Rank  Score       Time  Rank  Score
+ 25   00:07:18   105      0   00:07:24    94      7
+ 24   00:25:27  1496      0   00:43:46  1165      0
+ 23   00:27:30   887      0   01:35:00  1057      0
+ 22   00:03:44    42     59   01:16:52  2242      0
+ 21   00:21:57   612      0   00:26:05   439      0
+ 20   00:27:52   465      0   03:00:50   845      0
+ 19   00:14:56   123      0   00:48:50   439      0
+ 18   00:26:17  1209      0   00:28:01   574      0
+ 17   00:17:23   400      0   00:19:14   281      0
+ 16   00:11:50   627      0   01:03:40  2040      0
+ 15   00:18:11  1772      0   00:30:38  1924      0
+ 14   00:18:05  1220      0   00:39:57  1173      0
+ 13   00:04:59   198      0   01:05:11  1606      0
+ 12   00:06:19   206      0   00:14:17   245      0
+ 11   00:13:23   414      0   00:19:11   261      0
+ 10   00:05:04   293      0   00:31:04  1493      0
+  9   00:08:55  1721      0   00:26:50  3540      0
+  8   00:04:23   275      0   00:09:56   256      0
+  7   00:17:46   832      0   00:24:22   615      0
+  6   00:03:31   385      0   00:09:38   999      0
+  5   00:08:21   839      0   00:27:23  3404      0
+  4   00:07:40   786      0   00:20:26   440      0
+  3   00:02:46   119      0   00:07:40   448      0
+  2   00:03:48   341      0   00:06:01   272      0
+  1   00:07:52  1072      0   00:09:00   748      0
+  ```
+
+## Crystal for Competitive Programming
+I really enjoyed writing Crystal for the Advent of Code, but there were a few reasons
+why it wasn't perfect for the task.
+
+* When you're not banking on brute force speed, you lose some time to the compiler.
+* Numbers are _always_ 32-bit by default, and require constant `_i64` suffixes everywhere
+in the code when they're involved.
+* Type annotations (however necessary they are) for hashes and arrays make refactoring
+a little bit slower.

day1.ijs

@@ -0,0 +1,3 @@
+c =. > 0 ". each cutopen 1!:1 < jpath '~/projects/AoC/2020/year2020day1.txt.cache'
+>./,(c*/c)*2020=c+/c
+>./,(c*/c*/c)*2020=c+/c+/c

day15.cr

@@ -0,0 +1,24 @@
+require "advent"
+INPUT = input(2020, 15).split(",").map(&.to_i32) 
+
+def run(input, times)
+  ls = {} of Int32 => Int32
+  temp = 0
+  (times-1).times do |i|
+    n = input[i]? || temp
+    temp = i - (ls[n]? || i)
+    ls[n] = i
+  end
+  return temp
+end
+
+def part1(input)
+  run(input, 2020)
+end
+
+def part2(input)
+  run(input, 30000000)
+end
+
+puts part1(INPUT.clone)
+puts part2(INPUT.clone)

day16.cr

@@ -0,0 +1,65 @@
+require "advent"
+
+def parse_range(str)
+  data = str.match(/([a-z ]+): (\d+)-(\d+) or (\d+)-(\d+)/).not_nil!
+  return {data[1], (data[2].to_i32..data[3].to_i32), (data[4].to_i32.. data[5].to_i32)}
+end
+
+fields, your, nearby = input(2020, 16).split("\n\n")
+fields = fields.lines.map { |l| parse_range(l) }
+your = your.lines[1].split(",").map(&.to_i32)
+nearby = nearby.lines[1..].map(&.split(",").map(&.to_i32))
+
+def part1(input)
+  fields, your, nearby = input
+  all_ranges = [] of Range(Int32, Int32)
+  fields.each do |a|
+    all_ranges << a[1] << a[2]
+  end
+  nearby.select! do |nb|
+    nb.all? do |n|
+      all_ranges.any? &.includes?(n)
+    end
+  end
+  nearby << your
+  field_map = {} of String => Set(Int32)
+  fields.each do |f|
+    field_map[f[0]] = Set(Int32).new
+    nearby[0].size.times do |i|
+      next if field_map.values.includes? i
+      all_match = nearby.all? do |nb|
+        f[1].includes?(nb[i]) || f[2].includes?(nb[i])
+      end
+      if all_match
+        field_map[f[0]] << i
+      end
+    end
+  end
+  sum = 1_u64
+  numbers = (0...nearby[0].size).to_a
+  solved = {} of String => Int32
+  while solved.size != fields.size
+    cleared = [] of {String, Int32}
+    field_map.each do |k, v|
+      next unless v.size == 1
+      cleared << {k, v.to_a[0]}
+    end
+    cleared.each do |f,n|
+      solved[f] = n
+      field_map.each do |k, v|
+        v.delete n
+      end
+    end
+  end
+  fields.each do |f|
+    next unless f[0].starts_with? "departure"
+    sum *= your[solved[f[0]]]
+  end
+  sum
+end
+
+def part2(input)
+end
+
+puts part1({fields, your, nearby})
+puts part2({fields, your, nearby})

day17.cr

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+require "advent"
+require "benchmark"
+
+INPUT = input(2020, 17).lines.map(&.chars) 
+
+def solve(input, dim)
+  step = input.clone
+  cubes = Set(Array(Int32)).new
+  new_cubes = Set(Array(Int32)).new
+  input.each_with_index do |row, y|
+    row.each_with_index do |c, x|
+      cubes << [x,y].concat([0] * (dim-2)) if c == '#'
+    end
+  end
+
+  8.times do |i|
+    print '.'
+    neighbor_count = Hash(Array(Int32), Int32).new(0)
+    Array.product([[-1,0,1]] * dim).each do |diff|
+      next if diff.all? &.==(0)
+      cubes.each do |c|
+        neighbor_count[c.zip_with(diff) { |a,b| a+b }] += 1
+      end
+    end
+
+    new_cubes.clear
+    neighbor_count.each do |n, i|
+      new_cubes << n if i == 3 || (cubes.includes?(n) && i == 2)
+    end
+    new_cubes, cubes = cubes, new_cubes
+  end
+  cubes.size
+end
+
+def part1(input)
+  solve(input, 3)
+end
+
+def part2(input)
+  solve(input, 4)
+end
+
+(3..).each do |i|
+  print "Dim #{i} "
+  bm = Benchmark.measure { puts " #{solve(INPUT, i)}" }
+  puts bm.real * 1000
+end

day18.cr

@@ -0,0 +1,54 @@
+require "advent"
+INPUT = input(2020, 18).lines
+
+class Array(T)
+  def push_op(op)
+    r = pop
+    l = pop
+    self << ((op == '*') ? (l*r) : (l+r))
+  end
+
+  def has_op?
+    !empty? && last != '('
+  end
+end
+
+def translate(toks, prec)
+  output = [] of Int64
+  stack = [] of Char
+  toks.each do |tok|
+    case tok
+    when .number? then output << tok.to_i64
+    when '(' then stack << '('
+    when ')'
+      while stack.has_op?
+        output.push_op(stack.pop)
+      end
+      stack.pop
+    else
+      while stack.has_op? && prec[stack.last] < prec[tok]
+        output.push_op(stack.pop)
+      end
+      stack << tok
+    end
+  end
+  while stack.has_op?
+    output.push_op(stack.pop)
+  end
+  output.last
+end
+
+def part1(input)
+  input.sum do |line|
+    translate(line.chars.reject &.==(' '), {'*' => 0, '+' => 0})
+  end
+end
+
+def part2(input)
+  input.sum do |line|
+    translate(line.chars.reject &.==(' '), {'*' => 1, '+' => 0})
+  end
+end
+
+puts part1(INPUT.clone)
+puts part2(INPUT.clone)

day19.cr

@@ -0,0 +1,68 @@
+require "advent"
+
+rlines, _, strings = input(2020, 19).partition("\n\n")
+strings = strings.lines
+rules = {} of String => String
+rlines.lines.map do |l|
+  rule, _, text = l.partition(": ")
+  rules[rule] = text
+end
+
+alias Matcher = Proc(String,Int32,Array(Int32))
+
+def char(c)
+  ->(str : String, i : Int32) { str[i]? == c ? [i+1] : [] of Int32 }
+end
+
+def any(ps)
+  ->(str : String, i : Int32) { ps.flat_map { |p| p.call(str, i) } }
+end
+
+def seq(ps)
+  ->(str : String, i : Int32) {
+    base = [i]
+    ps.each do |p|
+      base = base.flat_map { |i| p.call(str, i) }
+    end
+    base
+  }
+end
+
+def to_regex(rules, rule)
+end
+
+def build_rule(rules, built, rule) : Matcher
+  if exists = built[rule]?
+    return exists 
+  end
+
+  body = rules[rule]
+  return built[rule] = char body[1] if body.matches? /"."/
+  
+  branches = [] of Matcher
+  top = any(branches)
+  built[rule] = top
+  branches.concat(body.split(" | ").map { |b| seq(b.split(" ").map { |subrule| build_rule(rules, built, subrule) }) })
+  top
+end
+
+def part1(input)
+  rules, lines = input
+  matcher = build_rule(rules, {} of String => Matcher, "0")
+  lines.count do |l|
+    matcher.call(l, 0).includes? l.size
+  end
+end
+
+def part2(input)
+  rules, lines = input
+  rules["8"] = "42 | 42 8"
+  rules["11"] = "42 31 | 42 11 31"
+  matcher = build_rule(rules, {} of String => Matcher, "0")
+  lines.count do |l|
+    matcher.call(l, 0).includes? l.size
+  end
+end
+
+puts part1({rules,strings})
+puts part2({rules,strings})

day2.ijs

@@ -0,0 +1,6 @@
+r =: >"1 cut each cutopen 1!:1 < jpath '~/projects/AoC/2020/year2020day2.txt.cache'
+rs =: >0".each'-'cut"1>(0}"1 r)
+cs =: 0{"1>1{"1 r
+ss =: 2{"1 r
++/1>**/|:rs-+/"1 cs=>ss
++/1=+/|:(rs-1){"1 cs=>ss

day20.cr

@@ -0,0 +1,166 @@
+require "advent"
+tiles = input(2020, 20).split "\n\n"
+tiles.pop
+thash = {} of Int32 => Array(Array(Char))
+tiles = tiles.map do |t|
+  tl = t.lines
+  tid = tl[0].match(/Tile (\d+):/).not_nil![1].to_i32
+  tls = tl[1..]
+  thash[tid] = tls.map &.chars
+end
+
+class Array(T)
+  def matches?(other)
+    zip_with(other) { |l,r| l == r }.all?
+  end
+
+  def rotate
+    reverse.transpose
+  end
+end
+
+def check(side_a, side_b)
+  return :normal if side_a.matches?(side_b)
+  return :flip if side_a.matches?(side_b.reverse)
+  return nil
+end
+
+def check_all(side_a, other, other_t)
+  check(side_a, other.first) || check(side_a, other.last) || check(side_a, other_t.first) || check(side_a, other_t.last)
+end
+
+MONSTER = [ {0, 1}, {1, 0}, {4, 0}, {5, 1}, {6, 1}, {7, 0}, {10, 0}, {11,1}, {12, 1}, {13, 0}, {16, 0},
+            {17, 1}, {18, 1}, {18, 2}, {19, 1} ]
+
+def stitch(m, thash, corner)
+  image = Array(Array(Char)).new(12*8) do |y|
+    Array(Char).new(12*8) do |x|
+      '.'
+    end
+  end
+
+  tile = thash[corner]
+  tile_id = corner
+  tile.reverse! if m[corner].has_key? :top
+  tile.each &.reverse! if m[corner].has_key? :left
+
+  12.times do |row|
+    tile = tile.not_nil!
+
+    row_tile = tile
+    row_id = tile_id
+
+    12.times do |col|
+      row_tile = row_tile.not_nil!
+
+      (0..7).each do |y|
+        (0..7).each do |x|
+          image[col*8+y][row*8+x] = row_tile[y+1][x+1]
+        end
+      end
+
+      matches = nil
+      thash.each do |other_id, other_tile|
+        next if matches
+        next if other_id == row_id
+        4.times do
+          if row_tile.last.matches? other_tile.first
+            row_id = other_id
+            matches = other_tile
+          elsif row_tile.last.matches? other_tile.first.reverse
+            row_id = other_id
+            matches = other_tile.map &.reverse
+          end
+          other_tile = other_tile.rotate
+        end
+      end
+
+      row_tile = matches
+    end
+
+    rot = tile.rotate
+    matches = nil
+    thash.each do |other_id, other_tile|
+      next if matches
+      next if other_id == tile_id
+
+      4.times do
+        if rot.last.matches? other_tile.first
+          tile_id = other_id
+          matches = other_tile.rotate.rotate.rotate
+        elsif rot.last.matches? other_tile.first.reverse
+          tile_id = other_id
+          matches = other_tile.map(&.reverse).rotate.rotate.rotate
+        end
+        other_tile = other_tile.rotate
+      end
+    end
+    tile = matches
+  end
+  image
+end
+
+def find_dragons(stitch)
+  dragons = [] of {Int32,Int32}
+  stitch.each_with_index do |row, y|
+    row.each_with_index do |c, x|
+      is_dragon =  MONSTER.all? do |dx, dy|
+        (stitch[y+dy]?.try &.[x+dx]?) == '#'
+      end
+      dragons << {x,y} if is_dragon
+    end
+  end
+  dragons
+end
+
+def find_all_dragons(stitch)
+  dragons = [] of {Int32,Int32}
+  4.times do
+    dragons.concat find_dragons(stitch)
+    dragons.concat find_dragons(stitch.reverse)
+    stitch = stitch.rotate
+  end
+  dragons
+end
+
+def match(thash, tile)
+  matches = {} of Symbol => {Int32, Symbol}
+  cs = thash[tile]
+  tcs = cs.transpose
+
+  thash.each do |t, ocs|
+    next if t == tile
+    tocs = ocs.transpose
+    top = check_all(cs.first, ocs, tocs)
+    bottom = check_all(cs.last, ocs, tocs)
+    left = check_all(tcs.first, ocs, tocs)
+    right = check_all(tcs.last, ocs, tocs)
+
+    matches[:top]    = {t, top} if top
+    matches[:left]   = {t, left} if left
+    matches[:bottom] = {t, bottom} if bottom
+    matches[:right]  = {t, right} if right
+  end
+  matches
+end
+
+def part1(input)
+  corners = input.select do |t, i|
+    match(input, t).size == 2
+  end
+  corners.keys.map(&.to_i64).product
+end
+
+def part2(input)
+  matches = {} of Int32 => Hash(Symbol, {Int32, Symbol})
+  corners = input.select do |t, i|
+    matches[t] = match(input, t)
+    match(input, t).size == 2
+  end
+  stitched = stitch(matches, input, corners.first[0])
+  dragons = find_all_dragons(stitched)
+  stitched.sum(&.count(&.==('#'))) - (dragons.size * MONSTER.size)
+end
+
+puts part1(thash.clone)
+puts part2(thash.clone)

day21.cr

@@ -0,0 +1,31 @@
+require "advent"
+input = input(2020, 21).lines.map do |line|
+  data = line.match(/^([a-z ]+) \(contains (.+)\)$/).not_nil!
+  {data[1].split(" ").to_set, data[2].split(", ").to_set}
+end
+
+allergens = input.flat_map(&.last.to_a).to_set
+ingredients = input.flat_map(&.first.to_a).to_set
+
+allergen_sets = {} of String => Set(String)
+allergens.each do |a|
+  input.each do |ings, als|
+    next unless als.includes? a
+    allergen_sets[a] ||= ings
+    allergen_sets[a] &= ings
+  end
+end
+
+safe = ingredients.reject { |i| allergen_sets.any? &.last.includes?(i) }
+puts "Part 1: #{input.sum &.first.count { |i| safe.includes? i }}"
+
+known_allergens = {} of String => String
+while allergen_sets.size > known_allergens.size
+  allergen_sets.each do |a, s|
+    next unless s.size == 1
+    new_known = s.first
+    known_allergens[a] = new_known
+    allergen_sets.each &.last.delete(new_known)
+  end
+end
+puts "Part 2: #{known_allergens.to_a.sort_by(&.first).map(&.last).join(",")}"

day22.cr

@@ -0,0 +1,66 @@
+require "advent"
+first, second = input(2020, 22).split("\n\n")
+first = first.lines[1..].map &.to_i32
+second = second.lines[1..].map &.to_i32
+
+class Array(T)
+  def pop_left
+    delete_at(0)
+  end
+
+  def score
+    total = 0
+    s = size
+    each_with_index do |n, i|
+      total += n * (s - i)
+    end
+    total
+  end
+end
+
+def no_rec(deck1, deck2)
+  while !deck1.empty? && !deck2.empty?
+    f, s = deck1.delete_at(0), deck2.delete_at(0)
+    if f > s
+      deck1 << f << s
+    else
+      deck2 << s << f
+    end
+  end
+  return !deck1.empty?
+end
+
+def rec(deck1, deck2)
+  seen = Set({Int32, Int32}).new
+  while !deck1.empty? && !deck2.empty?
+    key = {deck1.score, deck2.score}
+    return true if seen.includes?(key)
+    seen << key
+
+    f = deck1.delete_at(0)
+    s = deck2.delete_at(0)
+
+    p1wins = (f <= deck1.size) && (s <= deck2.size) ? rec(deck1[0..f-1], deck2[0..s-1]) : f > s
+    if p1wins
+      deck1 << f << s
+    else
+      deck2 << s << f
+    end
+  end
+  return !deck1.empty?
+end
+
+def run(first, second, proc)
+  ((proc.call(first, second)) ? first : second).score
+end
+
+def part1(input)
+  run(input[0], input[1], ->no_rec(Array(Int32), Array(Int32)))
+end
+
+def part2(input)
+  run(input[0], input[1], ->rec(Array(Int32), Array(Int32)))
+end
+
+puts part1({first, second}.clone)
+puts part2({first, second}.clone)

day23.cr

@@ -0,0 +1,117 @@
+require "advent"
+
+input = input(2020, 23).lines[0].chars.map &.to_i32
+
+class Array(T)
+  def get(i)
+    self[i % size]
+  end
+
+  def del(i)
+    delete_at(i % size)
+  end
+end
+
+class Node
+  property next : Node
+  property int : Int32
+
+  def initialize(@int, @next)
+  end
+
+  def insert(other : Node, n = 1)
+    set_next = other
+    (n-1).times { set_next = set_next.@next }
+    set_next.next = @next
+    @next = other
+  end
+
+  def remove(n = 1)
+    curr = self
+    to_return = self.next
+    n.times { curr = curr.@next }
+    @next = curr.@next
+    to_return
+  end
+
+  def find(n)
+    start = self
+    while start.int != n
+      start = start.next
+    end
+    return start
+  end
+  
+  def to_s(n)
+    return "" if n < 0
+    return "#{@int} -> #{@next.to_s(n-1)}"
+  end
+
+  def includes?(n, count)
+    return false if count <= 0
+    return @int == n || @next.includes?(n, count-1)
+  end
+end
+
+class Cups
+  getter size : Int32
+  getter head : Node
+
+  def initialize(list : Array(Int32))
+    @cache = {} of Int32 => Node
+    h = list.delete_at(0)
+    temp = uninitialized Node
+    @cache[h] = @head = Node.new(h, temp)
+    @size = list.size
+    @local_min = list.min.as(Int32)
+    @local_max = list.max.as(Int32)
+    @head.next = @head
+    curr = @head
+    list.each do |n|
+      @cache[n] = new = Node.new(n, curr.next)
+      curr.insert(new)
+      curr = new
+    end
+  end
+  
+  def to_s
+    @head.to_s(@size)
+  end
+
+  def step
+    first = @head
+    after = first.remove(3)
+    m = first.int - 1
+    while after.includes?(m, 3) || m < @local_min
+      m = (m < @local_min) ? @local_max : (m - 1)
+    end
+    @cache[m].insert(after, 3)
+    @head = @head.next
+  end
+end
+
+def play(input, count)
+  cups = Cups.new input
+  count.times { |n| cups.step }
+  cups
+end
+
+def part1(input)
+  cups = play(input.clone, 100)
+  cups.head.find(1).next.to_s(cups.size - 1).gsub(" -> ", "")
+end
+
+def part2(input)
+  list = input.clone
+  max = input.max
+  (1000000 - input.size).times do
+    max += 1
+    list << max
+  end
+  cups = play(list, 10000000)
+  one = cups.head.find(1)
+  one.next.int.to_i64 * one.next.next.int
+end
+
+puts "Part 1: #{part1(input)}"
+puts "Part 2: #{part2(input)}"

day24.cr

@@ -0,0 +1,78 @@
+require "advent"
+
+INPUT = input(2020, 24).lines.map &.tiles.map { |t| MATCHES[t] }
+MATCHES = { "se" => {0.5, -1.0}, "sw" => {-0.5, -1.0}, "ne" => {0.5, 1.0}, "nw" => {-0.5, 1.0}, "e" => {1.0, 0.0}, "w" => {-1.0, 0.0} }
+
+class String
+  def tiles
+    i = 0
+    tiles = [] of String
+    while i < size
+      if self[i] == 's' || self[i] == 'n'
+        tiles << self[i..i+1] 
+        i += 2
+      else
+        tiles << self[i..i]
+        i += 1
+      end
+    end
+    tiles
+  end
+end
+
+class Array(T)
+  def pos
+    curr = {0.0, 0.0}
+    each do |c|
+      curr = curr.add c
+    end
+    curr
+  end
+end
+
+def part1(input)
+  counts = Hash({Float64,Float64}, Int32).new(0)
+  input.each do |i|
+    counts[i.pos] += 1
+  end
+  counts.count &.[1].%(2).==(1)
+end
+
+struct Tuple(*T)
+  def neighbors
+    MATCHES.values.map &.add(self)
+  end
+end
+
+def part2(input)
+  state = Hash({Float64,Float64}, Bool).new(false)
+  counts = Hash({Float64,Float64}, Int32).new(0)
+
+  input.each do |i|
+    state[i.pos] ^= true
+  end
+
+  100.times do |i|
+    counts.clear
+    state.each do |t, f|
+      next unless f
+      t.neighbors.each do |n|
+        counts[n] += 1
+      end
+    end
+
+    new_state = state.clone.clear
+    state.each do |k, v|
+      next unless v
+      new_state[k] = !(counts[k] == 0 || (counts[k] > 2))
+    end
+    counts.each do |k, v|
+      new_state[k] = true if (!state[k]) && v == 2
+    end
+    state = new_state
+  end
+  state.count &.[1]
+end
+
+puts part1(INPUT.clone)
+puts part2(INPUT.clone)

day25.cr

@@ -0,0 +1,28 @@
+require "advent"
+INPUT = input(2020, 25).lines.map(&.to_i64) 
+
+def transform(s)
+  i = 1_i64
+  c = 0
+  loop do
+    c += 1
+    i = (i * s) % 20201227
+    yield i, c
+  end
+end
+
+def find_size(s, goal)
+  transform(s) do |n, c|
+    return c if n == goal
+  end
+end
+
+def part1(input)
+  goal = find_size(7, input[0])
+  puts goal
+  transform(input[1]) do |n, c|
+    return n if c == goal
+  end
+end
+
+puts part1(INPUT.clone)

day3.ijs

@@ -0,0 +1,5 @@
+r =: > cutopen 1!:1 < jpath '~/projects/AoC/2020/year2020day3.txt.cache'
+rn =: 4 : '+/''#''=((#|:y)|x*i.#y){"0 1 y'
+rnx =: 3 : '(1{y) rn 0{"2((0{-y)]\r)'
+rnx 1 3
+*/ rnx"1 (_2 [\ 1 1 1 3 1 5 1 7 2 1)

day5.ijs

@@ -0,0 +1,4 @@
+bs2n =: 3 : '+/ "1(|.*/\(1,(_1+#y)$2)) *"(1) 0 "."0 y'
+ns =: bs2n "1 'F0B1L0R1' charsub > cutopen 1!:1 < jpath '~/projects/AoC/2020/year2020day5.txt.cache'
+>./ns
+(<./ns)+I.-.((<./ns)+i.#ns)e.ns

day8.v

@@ -202,35 +202,54 @@ Module DayEight (Import M:Int).
         left. split; auto. apply stuck_prog; auto.
     Qed.
 
-    (* A valid input always terminates, either by getting to the end of the program,
-       or by looping and thus getting stuck. *)
-    Program Fixpoint valid_input_terminates (pc : fin (S n)) (v : set (fin n)) (acc : t) (Hnd : List.NoDup v)
-      { measure (length v) }:
-      (exists pc', run_noswap inp (pc, v, acc) pc') :=
-      match valid_input_progress pc v acc with
-      | or_introl (conj Heq Hdone) => _
-      | or_intror (ex_intro _ pcs (conj Hw w)) =>
-          match w with
-          | or_introl (conj Hnin Hstuck) => _
-          | or_intror (ex_intro _ pc' (ex_intro _ acc' (conj Hin Hst))) =>
-              match valid_input_terminates pc' (set_remove Fin.eq_dec pcs v) acc' (set_remove_nodup Fin.eq_dec pcs Hnd) with
-              | ex_intro _ pc'' Hrun => _
-              end
-          end
-      end.
-    Obligation 1. eexists. apply run_noswap_ok. assumption. Qed.
-    Obligation 2. eexists. apply run_noswap_fail. assumption. Qed.
-    Obligation 3. 
-      clear Heq_anonymous. clear valid_input_terminates. clear Hst.
-      induction v.
-      - inversion Hin.
-      - destruct (Fin.eq_dec pcs a) eqn:Heq_dec.
-        + simpl. rewrite Heq_dec. lia.
-        + inversion Hnd; subst.
-          inversion Hin. subst. exfalso. apply n0. auto.
-          specialize (IHv H2 H).
-          simpl. rewrite Heq_dec. simpl. lia.
+    Theorem list_length_induction {X : Type} (P : list X -> Prop) :
+      (forall l, (forall l', length l' < length l -> P l') -> P l) ->
+      forall l, P l.
+    Proof.
+      intros Hrec.
+      assert (forall (l l' : list X), length l' <= length l -> P l').
+      { induction l; intros l' Hlen; apply Hrec; intros l'0 Hlen0.
+        - simpl in Hlen. lia.
+        - apply IHl. simpl in Hlen. lia. }
+      intros l. apply H with l. lia.
+    Qed.
+
+    Theorem set_remove_length : forall (f : fin n) (s : set (fin n)),
+      set_In f s ->
+      length (set_remove Fin.eq_dec f s) < length s.
+    Proof.
+      intros f s Hin.
+      induction s.
+      - inversion Hin.
+      - simpl. destruct (Fin.eq_dec f a) eqn:Heq.
+        + unfold lt. apply le_n. (* Why couldn't lia get this one? *)
+        + inversion Hin; subst. exfalso. apply n0. auto.
+          apply IHs in H. simpl. lia.
+    Qed.
+
+    Theorem valid_input_terminates : forall (pc : fin (S n)) (v : set (fin n)) (acc : t),
+      (exists pc', run_noswap inp (pc, v, acc) pc').
+    Proof.
+      intros pc v. generalize dependent pc.
+      induction v using list_length_induction.
+      intros pc acc.
+      destruct (valid_input_progress pc l acc) as [[_ Hd]|[pc' [Hw [[_ Hst]|[pc'' [acc'' [Hin Hst]]]]]]].
+      - (* We're done. *)
+        eexists. apply run_noswap_ok. assumption.
+      - (* We're stuck. *)
+        eexists. apply run_noswap_fail. assumption.
+      - (* We can make a step. This will remove our current PC from the valid list, *)
+        edestruct (H (set_remove Fin.eq_dec pc' l)).
+        (* Since the PC must be in the list, removing it makes the list smaller. *)
+        apply (set_remove_length _ _ Hin).
+        (* Without the current PC, our valid set shrinks.
+           Since this is the inductive step, we have assumed
+           that programs with smaller sets of valid PCs always
+           terminate. Thus, after we make the step, we're done. *)
+        exists x. subst. eapply run_noswap_trans.
+        + auto.
+        + apply Hst.
+        + apply H0.
     Qed.
-    Obligation 4. eexists. eapply run_noswap_trans; auto. apply Hst. apply Hrun. Qed.
   End ValidInput.
 End DayEight.