Use auto instead of apply.

Apparently writing proof objects by hand is
easier than using tactics.

README.md

@@ -0,0 +1,55 @@
+# 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

day13.cr

@@ -0,0 +1,34 @@
+require "advent"
+INPU = input(2020, 13).lines
+INPUT = {INPU[0].to_i32, INPU[1].split(",")}
+
+def part1(input)
+  early, busses = input
+  busses.reject! &.==("x")
+  busses = busses.map &.to_i32
+  bbus = busses.min_by do |b|
+    (early / b).ceil * b
+  end
+  diff = bbus * (((early/bbus).ceil * bbus).to_i32 - early)
+end
+
+def part2(input)
+  _, busses = input
+  busses = busses.map_with_index do |x, i|
+    x.to_i32?.try { |n| { n, i } }
+  end
+  busses = busses.compact
+  n = 0_i64
+  iter = 1_i64
+  busses.each do |m, i|
+    while (n + i) % m != 0
+      n += iter
+    end
+    iter *= m
+  end
+  puts n
+  puts busses
+end
+
+puts part1(INPUT.clone)
+puts part2(INPUT.clone)

day14.cr

@@ -0,0 +1,60 @@
+require "advent"
+raw = input(2020, 14).lines
+input = [] of {UInt64, UInt64, Array(Int32), UInt64, UInt64}
+
+and_mask = (2_u64**63) + (2_u64**63-1)
+or_mask = 0_u64
+floats = [] of Int32
+
+raw.each do |line|
+  if data = line.match(/^mask = (.+)$/)
+    new_mask = data[1]
+    and_mask = (2_u64**63) + (2_u64**63-1)
+    or_mask = 0_u64
+    floats = [] of Int32
+    new_mask.reverse.chars.each_with_index do |c, i|
+      floats << i if c == 'X'
+      or_mask = or_mask | (1_u64 << i) if c == '1'
+      and_mask = and_mask & ~(1_u64 << i) if c == '0'
+    end
+  elsif data = line.match(/^mem\[(\d+)\] = (\d+)$/)
+    input << {and_mask, or_mask, floats, data[1].to_u64, data[2].to_u64}
+  else
+    raise "Invalid line"
+  end
+end
+
+def part1(input)
+  memory = {} of UInt64 => UInt64
+  input.each do |i|
+    and_mask, or_mask, floats, addr, v = i
+    memory[addr] = (v & and_mask) | or_mask
+  end
+  memory.values.sum
+end
+
+def generate_possible(indices, index, n, &block : UInt64 -> _)
+  if index == indices.size
+    yield n
+    return 
+  end
+  float_addr = indices[index]
+  no = n | (1_u64 << float_addr)
+  generate_possible(indices, index+1, no, &block)
+  nz = n & ~(1_u64 << float_addr)
+  generate_possible(indices, index+1, nz, &block)
+end
+
+def part2(input)
+  memory = {} of UInt64 => UInt64
+  input.each do |i|
+    _, or_mask, floats, addr, v = i
+    generate_possible(floats, 0, addr | or_mask) do |addr|
+      memory[addr] = v
+    end
+  end
+  memory.values.sum
+end
+
+puts part1(input)
+puts part2(input)

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

@@ -0,0 +1,47 @@
+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

@@ -5,6 +5,8 @@ Require Import Coq.Vectors.Fin.
 Require Import Coq.Program.Equality.
 Require Import Coq.Logic.Eqdep_dec.
 Require Import Coq.Arith.Peano_dec.
+Require Import Coq.Program.Wf.
+Require Import Lia.
 
 Module DayEight (Import M:Int).
   (* We need to coerce natural numbers into integers to add them. *)
@@ -27,6 +29,7 @@ Module DayEight (Import M:Int).
     | Ok : t -> run_result
     | Fail : set (fin n) -> run_result.
 
+  (* A single program state .*)
   Definition state n : Type := (fin (S n) * set (fin n) * t).
 
   (* An instruction is a pair of an opcode and an argument. *)
@@ -37,6 +40,18 @@ Module DayEight (Import M:Int).
      addresses, which are used for calculating jumps. *)
   Definition indices (n : nat) := VectorDef.t (fin n) n.
 
+  (* Change a jump to a nop, or a nop to a jump. *)
+  Definition swap (i : inst) : inst :=
+    match i with
+    | (add, t) => (add, t)
+    | (nop, t) => (jmp, t)
+    | (jmp, t) => (nop, t)
+    end.
+
+  Inductive swappable : inst -> Prop :=
+  | swap_nop : forall t, swappable (nop, t)
+  | swap_jmp : forall t, swappable (jmp, t).
+
   (* Compute the destination jump index, an integer. *)
   Definition jump_t {n} (pc : fin n) (off : t) : t :=
     M.add (nat_to_t (proj1_sig (to_nat pc))) off.
@@ -45,56 +60,23 @@ Module DayEight (Import M:Int).
      Not all inputs are valid, so this may fail. *)
   Definition valid_jump_t {n} (pc : fin n) (off : t) : option (fin (S n)) := @clamp (S n) (jump_t pc off).
 
+  (* Cast a fin n to a fin (S n). *)
   Fixpoint weaken_one {n} (f : fin n) : fin (S n) :=
     match f with
     | F1 => F1
     | FS f' => FS (weaken_one f')
     end.
 
-  (* One modification: we really want to use 'allowed' addresses,
+  (* Convert a nat to fin. *)
-     a set that shrinks as the program continues, rather than 'visited'
-     addresses, a set that increases as the program continues. *)
-  Inductive step_noswap {n} : input n -> state n -> state n -> Prop :=
-    | step_noswap_add : forall inp pc' v acc t,
-        nth inp pc' = (add, t) ->
-        set_mem Fin.eq_dec pc' v = true ->
-        step_noswap inp (weaken_one pc', v, acc) (FS pc', set_remove Fin.eq_dec pc' v, M.add acc t)
-    | step_noswap_nop : forall inp pc' v acc t,
-        nth inp pc' = (nop, t) ->
-        set_mem Fin.eq_dec pc' v = true ->
-        step_noswap inp (weaken_one pc', v, acc) (FS pc', set_remove Fin.eq_dec pc' v, acc)
-    | step_noswap_jmp : forall inp pc' pc'' v acc t,
-        nth inp pc' = (jmp, t) ->
-        set_mem Fin.eq_dec pc' v = true ->
-        valid_jump_t pc' t = Some pc'' ->
-        step_noswap inp (weaken_one pc', v, acc) (pc'', set_remove Fin.eq_dec pc' v, acc).
-
   Fixpoint nat_to_fin (n : nat) : fin (S n) :=
     match n with
     | O => F1
     | S n' => FS (nat_to_fin n')
     end.
 
-  Inductive run_noswap {n} : input n -> state n -> state n -> Prop :=
+  (* A finite natural is either its maximum value (aka nat_to_fin n),
-    | run_noswap_ok : forall inp v acc,
+     or it's not thatbig, which means it can be cast down to
-        run_noswap inp (nat_to_fin n, v, acc) (nat_to_fin n, v, acc)
+     a fin (pred n). *)
-    | run_noswap_fail : forall inp pc' v acc,
-        set_mem Fin.eq_dec pc' v = false ->
-        run_noswap inp (weaken_one pc', v, acc) (weaken_one pc', v, acc)
-    | run_noswap_trans : forall inp st st' st'',
-        step_noswap inp st st' -> run_noswap inp st' st'' -> run_noswap inp st st''.
-
-  Inductive valid_inst {n} : inst -> fin n -> Prop :=
-    | valid_inst_add : forall t f, valid_inst (add, t) f
-    | valid_inst_nop : forall t f f',
-        valid_jump_t f t = Some f' -> valid_inst (nop, t) f
-    | valid_inst_jmp : forall t f f',
-        valid_jump_t f t = Some f' -> valid_inst (jmp, t) f.
-
-  (* An input is valid if all its instructions are valid. *)
-  Definition valid_input {n} (inp : input n) : Prop := forall (pc : fin n),
-    valid_inst (nth inp pc) pc.
-
   Lemma fin_big_or_small : forall {n} (f : fin (S n)),
     (f = nat_to_fin n) \/ (exists (f' : fin n), f = weaken_one f').
   Proof.
@@ -115,303 +97,159 @@ Module DayEight (Import M:Int).
         reflexivity.
   Qed.
   
-  Lemma set_add_idempotent : forall {A:Type}
+  (* One modification: we really want to use 'allowed' addresses,
-    (Aeq_dec : forall x y : A, { x = y } + { x <> y })
+     a set that shrinks as the program continues, rather than 'visited'
-    (a : A) (s : set A), set_mem Aeq_dec a s = true -> set_add Aeq_dec a s = s.
+     addresses, a set that increases as the program continues. *)
-  Proof.
+  Inductive step_noswap {n} : inst -> (fin n * t) -> (fin (S n) * t) -> Prop :=
-    intros A Aeq_dec a s Hin.
+    | step_noswap_add : forall pc acc t,
-    induction s.
+        step_noswap (add, t) (pc, acc) (FS pc, M.add acc t)
-    - inversion Hin.
+    | step_noswap_nop : forall pc acc t,
-    - simpl. simpl in Hin.
+        step_noswap (nop, t) (pc, acc) (FS pc, acc)
-      destruct (Aeq_dec a a0).
+    | step_noswap_jmp : forall pc pc' acc t,
-      + reflexivity.
+        valid_jump_t pc t = Some pc' ->
-      + simpl. rewrite IHs; auto.
+        step_noswap (jmp, t) (pc, acc) (pc', acc).
-  Qed.
 
-  Theorem set_add_append : forall {A:Type}
+  Inductive done {n} : input n -> state n -> Prop :=
-    (Aeq_dec : forall x y : A, {x = y } + { x <> y })
+    | done_prog : forall inp v acc, done inp (nat_to_fin n, v, acc).
-    (a : A) (s : set A), set_mem Aeq_dec a s = false ->
-    set_add Aeq_dec a s = List.app s (List.cons a List.nil).
-  Proof.
-    induction s.
-    - reflexivity.
-    - intros Hnm. simpl in Hnm.
-      destruct (Aeq_dec a a0) eqn:Heq_dec.
-      + inversion Hnm.
-      + simpl. rewrite Heq_dec. rewrite IHs.
-        reflexivity. assumption.
-  Qed.
 
-  Lemma list_append_or_nil : forall {A:Type} (l : list A),
+  Inductive stuck {n} : input n -> state n -> Prop :=
-    l = List.nil \/ exists l' a, l = List.app l' (List.cons a List.nil).
+    | stuck_prog : forall inp pc' v acc,
-  Proof.
+        ~ set_In pc' v -> stuck inp (weaken_one pc', v, acc).
-    induction l.
-    - left. reflexivity.
-    - right. destruct IHl.
-      + exists List.nil. exists a.
-        rewrite H. reflexivity.
-      + destruct H as [l' [a' H]].
-        exists (List.cons a l'). exists a'.
-        rewrite H. reflexivity.
-  Qed.
 
-  Theorem list_append_induction : forall {A:Type}
+  Inductive run_noswap {n} : input n -> state n -> state n -> Prop :=
-    (P : list A -> Prop),
+    | run_noswap_ok : forall inp st, done inp st -> run_noswap inp st st
-    P List.nil -> (forall (a : A) (l : list A), P l -> P (List.app l (List.cons a (List.nil)))) ->
+    | run_noswap_fail : forall inp st, stuck inp st -> run_noswap inp st st
-    forall l, P l.
+    | run_noswap_trans : forall inp pc pc' v acc acc' st',
-  Proof. Admitted.
+        set_In pc v ->
+        step_noswap (nth inp pc) (pc, acc) (pc', acc') ->
+        run_noswap inp (pc', set_remove Fin.eq_dec pc v, acc') st' ->
+        run_noswap inp (weaken_one pc, v, acc) st'.
 
+  Inductive run_swap {n} : input n -> state n -> state n -> Prop :=
+    | run_swap_normal : forall inp pc pc' v acc acc' st',
+        set_In pc v ->
+        ~ swappable (nth inp pc) ->
+        step_noswap (nth inp pc) (pc, acc) (pc', acc') ->
+        run_swap inp (pc', set_remove Fin.eq_dec pc v, acc') st' ->
+        run_swap inp (weaken_one pc, v, acc) st'
+    | run_swap_swapped_ok : forall inp pc pc' v acc acc' st',
+        set_In pc v ->
+        swappable (nth inp pc) ->
+        step_noswap (swap (nth inp pc)) (pc, acc) (pc', acc') ->
+        run_noswap inp (pc', set_remove Fin.eq_dec pc v, acc') st' ->
+        done inp st' ->
+        run_swap inp (weaken_one pc, v, acc) st'
+    | run_swap_swapped_next : forall inp pc pc'w pc'n v acc acc'w acc'n st'w st'n,
+        set_In pc v ->
+        swappable (nth inp pc) ->
+        step_noswap (swap (nth inp pc)) (pc, acc) (pc'w, acc'w) ->
+        run_noswap inp (pc'w, set_remove Fin.eq_dec pc v, acc'w) st'w ->
+        stuck inp st'w ->
+        step_noswap (nth inp pc) (pc, acc) (pc'n, acc'n) ->
+        run_swap inp (pc'n, set_remove Fin.eq_dec pc v, acc'n) st'n ->
+        run_swap inp (weaken_one pc, v, acc) st'n.
 
-  Theorem set_induction : forall {A:Type}
+  Inductive valid_inst {n} : inst -> fin n -> Prop :=
-    (Aeq_dec : forall x y : A, { x = y } + {x <> y })
+    | valid_inst_add : forall t f, valid_inst (add, t) f
-    (P : set A -> Prop),
+    | valid_inst_nop : forall t f f',
-    P (@empty_set A) -> (forall (a : A) (s' : set A), P s' -> P (set_add Aeq_dec a s')) ->
+        valid_jump_t f t = Some f' -> valid_inst (nop, t) f
-    forall (s : set A), P s.
+    | valid_inst_jmp : forall t f f',
-  Proof. Admitted.
+        valid_jump_t f t = Some f' -> valid_inst (jmp, t) f.
 
-  Lemma weaken_one_inj : forall n (f1 f2 : fin n),
+  (* An input is valid if all its instructions are valid. *)
-    (weaken_one f1 = weaken_one f2 -> f1 = f2).
+  Definition valid_input {n} (inp : input n) : Prop := forall (pc : fin n),
-  Proof. 
+    valid_inst (nth inp pc) pc.
-    remember (fun {n} (a b : fin n) => weaken_one a = weaken_one b -> a = b) as P.
-    (* Base case for rect2 *)
-    assert (forall n, @P (S n) F1 F1).
-    {rewrite HeqP. intros n Heq. reflexivity. }
-    (* 'Impossible' cases for rect2. *)
-    assert (forall {n} (f : fin n), P (S n) F1 (FS f)).
-    {rewrite HeqP. intros n f Heq. simpl in Heq. inversion Heq. }
-    assert (forall {n} (f : fin n), P (S n) (FS f) F1).
-    {rewrite HeqP. intros n f Heq. simpl in Heq. inversion Heq. }
-    (* Recursive case for rect2. *)
-    assert (forall {n} (f g : fin n), P n f g -> P (S n) (FS f) (FS g)).
-    {rewrite HeqP. intros n f g IH Heq.
-      simpl in Heq. injection Heq as Heq'.
-      apply inj_pair2_eq_dec in Heq'.
-      - rewrite IH. reflexivity. assumption.
-      - apply eq_nat_dec. }
 
-    (* Actually apply recursion. *)
+  Section ValidInput.
-    (* This can't be _the_ way to do this. *)
+    Variable n : nat.
-    intros n.
+    Variable inp : input n.
-    specialize (@Fin.rect2 P H H0 H1 H2 n) as Hind.
+    Hypothesis Hv : valid_input inp.
-    rewrite HeqP in Hind. apply Hind.
-  Qed.
 
-  Lemma weaken_neq_to_fin : forall {n} (f : fin (S n)),
+    Theorem valid_input_can_step : forall pc acc, exists pc' acc',
-    nat_to_fin (S n) <> weaken_one f.
+      step_noswap (nth inp pc) (pc, acc) (pc', acc').
-  Proof.
+    Proof.
-    apply Fin.rectS; intros n Heq.
+      intros pc acc.
-    - inversion Heq.
+      destruct (nth inp pc) eqn:Hop.
-    - intros IH. simpl. intros Heq'.
+      destruct o.
-      injection Heq' as Hinj. apply inj_pair2_eq_dec in Hinj.
+      - exists (FS pc). exists (M.add acc t0). apply step_noswap_add. 
-      + simpl in IH. apply IH. apply Hinj.
+      - exists (FS pc). exists acc. eapply step_noswap_nop.
-      + apply eq_nat_dec.
+      - specialize (Hv pc). rewrite Hop in Hv. inversion Hv; subst.
-  Qed.
+        exists f'. exists acc. eapply step_noswap_jmp. apply H0.
+    Qed.
 
-  Lemma add_pc_safe_step : forall {n} (inp : input n) (pc : fin (S n)) i is acc st',
+    (* A program is either done, stuck (at an invalid/visited address), or can step. *)
-    step_noswap inp (pc, is, acc) st' ->
+    Theorem valid_input_progress : forall pc v acc,
-    exists st'', step_noswap inp (pc, (set_add Fin.eq_dec i is), acc) st''.
+      (pc = nat_to_fin n /\ done inp (pc, v, acc)) \/
-  Proof.
+      (exists pcs, pc = weaken_one pcs /\
-    intros n inp pc' i is acc st' Hstep.
+        ((~ set_In pcs v /\ stuck inp (pc, v, acc)) \/
-    inversion Hstep.
+         (exists pc' acc', set_In pcs v /\
-    - eexists. apply step_noswap_add. apply H4.
+         step_noswap (nth inp pcs) (pcs, acc) (pc', acc')))).
-      apply set_mem_correct2. apply set_add_intro1.
+    Proof.
-      apply set_mem_correct1 with Fin.eq_dec. assumption.
+      intros pc v acc.
-    - eexists. eapply step_noswap_nop. apply H4.
+      (* Have we reached the end? *)
-      apply set_mem_correct2. apply set_add_intro1.
+      destruct (fin_big_or_small pc).
-      apply set_mem_correct1 with Fin.eq_dec. assumption.
+      (* We're at the end, so we're done. *)
-    - eexists. eapply step_noswap_jmp. apply H3.
+      left. rewrite H. split. reflexivity. apply done_prog.
-      apply set_mem_correct2. apply set_add_intro1.
+      (* We're not at the end. *)
-      apply set_mem_correct1 with Fin.eq_dec. assumption.
+      right. destruct H as [pcs H]. exists pcs. rewrite H. split. reflexivity.
-      apply H6.
+      (* We're not at the end. Is the PC valid? *)
-  Qed.
+      destruct (set_In_dec Fin.eq_dec pcs v).
+      - (* It is. *)
+        right.
+        destruct (valid_input_can_step pcs acc) as [pc' [acc' Hstep]].
+        exists pc'; exists acc'; auto.
+      - (* It is not. *)
+        left. split; auto. apply stuck_prog; auto.
+    Qed.
 
-  Lemma remove_pc_safe_run : forall {n} (inp : input n) i pc v acc st',
+    Theorem list_length_induction {X : Type} (P : list X -> Prop) :
-    run_noswap inp (pc, set_add Fin.eq_dec i v, acc) st' ->
+      (forall l, (forall l', length l' < length l -> P l') -> P l) ->
-    exists st'', run_noswap inp (pc, v, acc) st''.
+      forall l, P l.
-  Proof.
+    Proof.
-    intros n inp i pc v acc st' Hr.
+      intros Hrec.
-    dependent induction Hr.
+      assert (forall (l l' : list X), length l' <= length l -> P l').
-    - eexists. eapply run_noswap_ok.
+      { induction l; intros l' Hlen; apply Hrec; intros l'0 Hlen0.
-    - eexists. eapply run_noswap_fail.
+        - simpl in Hlen. lia.
-      apply set_mem_complete1 in H.
+        - apply IHl. simpl in Hlen. lia. }
-      apply set_mem_complete2.
+      intros l. apply H with l. lia.
-      intros Hin. apply H. apply set_add_intro. right. apply Hin.
+    Qed.
-    - inversion H; subst; destruct (set_mem Fin.eq_dec pc' v) eqn:Hm.
-  Admitted.
 
-  Lemma add_pc_safe_run : forall {n} (inp : input n) i pc v acc st',
+    Theorem set_remove_length : forall (f : fin n) (s : set (fin n)),
-    run_noswap inp (pc, v, acc) st' ->
+      set_In f s ->
-    exists st'', run_noswap inp (pc, (set_add Fin.eq_dec i v), acc) st''.
+      length (set_remove Fin.eq_dec f s) < length s.
-  Proof.
+    Proof.
-    intros n inp i pc v acc st' Hr.
+      intros f s Hin.
-    destruct (set_mem Fin.eq_dec i v) eqn:Hm.
+      induction s.
-    (* If i is already in the set, nothing changes. *)
+      - inversion Hin.
-    rewrite set_add_idempotent.
+      - simpl. destruct (Fin.eq_dec f a) eqn:Heq.
-    exists st'. assumption. assumption.
+        + unfold lt. apply le_n. (* Why couldn't lia get this one? *)
-    (* Otherwise, the behavior might have changed.. *)
+        + inversion Hin; subst. exfalso. apply n0. auto.
-    destruct (fin_big_or_small pc).
+          apply IHs in H. simpl. lia.
-    - (* If we're done, we're done no matter what. *)
+    Qed.
-      eexists. rewrite H. eapply run_noswap_ok.
-    - (* The PC points somewhere inside. We tried (and maybe failed)
-         to execute and instruction. The challenging part
-         is that adding i may change the outcome from 'fail' to 'ok' *)
-      destruct H as [pc' Heq].
-      generalize dependent st'.
-      induction v using (@set_induction (fin n) Fin.eq_dec);
-      intros st' Hr.
-      + (* Our set of valid states is nearly empty. One step,
-           and it runs dry. *)
-        simpl. destruct (Fin.eq_dec pc' i) eqn:Heq_dec.
-        * (* The PC is the one allowed state. *)
-          remember (nth inp pc') as h. destruct h. destruct o.
-          { (* Addition. *)
-            destruct (fin_big_or_small (FS pc')).
-            - (* The additional step puts as at the end. *)
-              eexists. eapply run_noswap_trans.
-              + rewrite Heq. apply step_noswap_add.
-                symmetry. apply Heqh. simpl. rewrite Heq_dec. reflexivity.
-              + rewrite H. apply run_noswap_ok.
-            - (* The additional step puts us somewhere else. *)
-              destruct H as [f' H].
-              eexists. eapply run_noswap_trans.
-              + rewrite Heq. apply step_noswap_add.
-                symmetry. apply Heqh. simpl. rewrite Heq_dec. reflexivity.
-              + rewrite H. apply run_noswap_fail.
-                simpl. rewrite Heq_dec. reflexivity. }
-          { (* No-op *) admit. }
-          { (* Jump*) admit. }
-        * (* The PC is not. We're done. *)
-          eexists. rewrite Heq. eapply run_noswap_fail.
-          simpl. rewrite Heq_dec. reflexivity. 
-      + destruct (set_mem Fin.eq_dec a v) eqn:Hm'.
-        * unfold fin. rewrite (set_add_idempotent Fin.eq_dec a).
 
-            
+    Theorem valid_input_terminates : forall (pc : fin (S n)) (v : set (fin n)) (acc : t),
-          { apply step_noswap_nop.
+      (exists pc', run_noswap inp (pc, v, acc) pc').
-            - symmetry. apply Heqh.
+    Proof.
-            - simpl. rewrite Heq_dec. reflexivity. }
+      intros pc v. generalize dependent pc.
-
+      induction v using list_length_induction.
-      (*
+      intros pc acc.
-      dependent induction Hr; subst.
+      destruct (valid_input_progress pc l acc) as [[_ Hd]|[pc' [Hw [[_ Hst]|[pc'' [acc'' [Hin Hst]]]]]]].
-      + (* We can't be in the OK state, since we already covered
+      - (* We're done. *)
-           that earlier. *)
+        eexists. apply run_noswap_ok. assumption.
-        destruct n. inversion pc'.
+      - (* We're stuck. *)
-        apply weaken_neq_to_fin in Heq as [].
+        eexists. apply run_noswap_fail. assumption.
-      + apply weaken_one_inj in Heq as Hs. subst.
+      - (* We can make a step. This will remove our current PC from the valid list, *)
-        destruct (Fin.eq_dec pc'0 i) eqn:Heq_dec.
+        edestruct (H (set_remove Fin.eq_dec pc' l)).
-        * admit.
+        (* Since the PC must be in the list, removing it makes the list smaller. *)
-        * eexists. eapply run_noswap_fail.
+        apply (set_remove_length _ _ Hin).
-          assert (~set_In pc'0 v).
+        (* Without the current PC, our valid set shrinks.
-          { apply (set_mem_complete1 Fin.eq_dec). assumption. }
+           Since this is the inductive step, we have assumed
-          assert (~set_In pc'0 (List.cons i List.nil)).
+           that programs with smaller sets of valid PCs always
-          { simpl. intros [Heq'|[]]. apply n0. auto. }
+           terminate. Thus, after we make the step, we're done. *)
-          assert (~set_In pc'0 (set_union Fin.eq_dec v (List.cons i List.nil))).
+        exists x. subst. eapply run_noswap_trans.
-          { intros Hin. apply set_union_iff in Hin as [Hf|Hf].
+        + auto.
-            - apply H0. apply Hf.
+        + apply Hst.
-            - apply H1. apply Hf. }
+        + apply H0.
-          simpl in H2. apply set_mem_complete2. assumption.
+    Qed.
-      + apply (add_pc_safe_step _ _ i) in H as [st''' Hr'].
+  End ValidInput.
-        eexists. eapply run_noswap_trans.
-        apply Hr'. destruct st' as [[pc'' v''] acc''].
-         specialize (IHHr i pc'' v'' acc'').*)
-
-
-    (*    intros n inp i pc v.
-    generalize dependent i.
-    generalize dependent pc.
-    induction v; intros pc i acc st Hr.
-    - inversion Hr; subst.
-      + eexists. apply run_noswap_ok.
-      + destruct (Fin.eq_dec pc' i) eqn:Heq_dec.
-        * admit.
-        * eexists. apply run_noswap_fail.
-          simpl. rewrite Heq_dec. reflexivity.
-      + inversion H; subst; simpl in H7; inversion H7.
-    - inversion Hr; subst.
-      + eexists. apply run_noswap_ok.
-      + destruct (Fin.eq_dec pc' i) eqn:Heq_dec.
-        * admit.
-        * eexists. apply run_noswap_fail.
-          simpl. rewrite Heq_dec. simpl in H4.
-          apply H4.
-      + 
-       destruct (nth inp pc') as [op t]. *)
-
-  Admitted.
-   
-  Theorem valid_input_terminates  : forall n (inp : input n) st,
-    valid_input inp -> exists st', run_noswap inp st st'.
-  Proof.
-    intros n inp st.
-    destruct st as [[pc is] acc].
-    generalize dependent inp.
-    generalize dependent pc.
-    generalize dependent acc.
-    induction is using (@set_induction (fin n) Fin.eq_dec); intros acc pc inp Hv;
-    (* The PC may point past the end of the
-       array, or it may not. *)
-    destruct (fin_big_or_small pc);
-    (* No matter what, if it's past the end
-       of the array, we're done, *)
-    try (eexists (pc, _, acc); rewrite H; apply run_noswap_ok).
-    - (* It's not past the end of the array,
-         and the 'allowed' list is empty.
-         Evaluation fails. *)
-      destruct H as [f' Heq].
-      exists (pc, Datatypes.nil, acc).
-      rewrite Heq. apply run_noswap_fail. reflexivity.
-    - (* We're not past the end of the array. However,
-         adding a new valid index still guarantees
-         evaluation terminates. *)
-      specialize (IHis acc pc inp Hv) as [st' Hr].
-      apply add_pc_safe_run with st'. assumption.
-  Qed.
-
-  (*
-      (* It's not past the end of the array,
-         and we're in the inductive case on is. *)
-      destruct H as [pc' Heq].
-      destruct (Fin.eq_dec pc' a) eqn:Heq_dec.
-      + (* This PC is allowed. *)
-        (* That must mean we have a non-empty list. *)
-        remember (nth inp pc') as h. destruct h as [op t].
-        (* Unfortunately, we can't do eexists at the top
-           level, since that will mean the final state
-           has to be the same for every op. *)
-        destruct op.
-        (* Addition. *)
-        { destruct (IHis (M.add acc t) (FS pc') inp Hv) as [st' Htrans].
-          eexists. eapply run_noswap_trans.
-          rewrite Heq. apply step_noswap_add.
-          - symmetry. apply Heqh.
-          - simpl. rewrite Heq_dec. reflexivity.
-          - simpl. rewrite Heq_dec. apply Htrans. }
-        (* No-ops *)
-        { destruct (IHis acc (FS pc') inp Hv) as [st' Htrans].
-          eexists. eapply run_noswap_trans.
-          rewrite Heq. apply step_noswap_nop with t.
-          - symmetry. apply Heqh.
-          - simpl. rewrite Heq_dec. reflexivity. 
-          - simpl. rewrite Heq_dec. apply Htrans. }
-        (* Jump. *) 
-        { (* A little more interesting. We need to know that the jump is valid. *)
-          assert (Hv' : valid_inst (jmp, t) pc').
-          { specialize (Hv pc'). rewrite <- Heqh in Hv. assumption. }
-          inversion Hv'.
-          (* Now, proceed as usual. *)
-          destruct (IHis acc f' inp Hv) as [st' Htrans].
-          eexists. eapply run_noswap_trans.
-          rewrite Heq. apply step_noswap_jmp with t.
-          - symmetry. apply Heqh.
-          - simpl. rewrite Heq_dec. reflexivity. 
-          - apply H0.
-          - simpl. rewrite Heq_dec. apply Htrans. }
-      + (* The top PC is not allowed. *)
-        specialize (IHis acc pc inp Hv) as [st' Hr].
-     apply add_pc_safe_run with st'. assumption. *)
-  Qed.
-
-
-    (* Stoppped here. *)
-  Admitted.
 End DayEight.