Problem:
RMQ and RAQ

Write a program which manipulates a sequence $A$ = {$a_0, a_1, ..., a_{n-1}$} with the following operations:

$add(s, t, x)$ : add $x$ to $a_s, a_{s+1}, ..., a_t$.

$find(s, t)$ : report the minimum value in $a_s, a_{s+1}, ..., a_t$.

Note that the initial values of $a_i ( i = 0, 1, ..., n-1 )$ are 0.

Input

$n$ $q$
$query_1$
$query_2$
:
$query_q$

In the first line, $n$ (the number of elements in $A$) and $q$ (the number of queries) are given. Then, $i$th query $query_i$ is given in the following format:

0 $s$ $t$ $x$

or

1 $s$ $t$

The first digit represents the type of the query. '0' denotes $add(s, t, x)$ and '1' denotes $find(s, t)$.

Output

For each $find$ query, print the minimum value.

Constraints

$1 ≤ n ≤ 100000$

$1 ≤ q ≤ 100000$

$0 ≤ s ≤ t < n$

$-1000 ≤ x ≤ 1000$

Sample Input 1

6 7
0 1 3 1
0 2 4 -2
1 0 5
1 0 1
0 3 5 3
1 3 4
1 0 5

Sample Output 1

-2
0
1
-1

Solution in Ruby (judge verdict: Accepted)

class LazySegmentTreeBase
  def initialize(arr, l = 0, r = arr.size - 1)
    @l, @r = l, r
    @lazy = nil
    if l == r
      @agg = arr[l]
    else
      c = (l + r) / 2
      @left_child = self.class.new(arr, l, c)
      @right_child = self.class.new(arr, c + 1, r)
      @agg = aggregation_of(@left_child.agg, @right_child.agg)
    end
  end

  attr_reader :agg, :l, :r

  def update(l, r, x)
    if l == @l && r == @r
      defer(x)
    else
      force
      if r <= @left_child.r
        @left_child.update(l, r, x)
      elsif @right_child.l <= l
        @right_child.update(l, r, x)
      else
        @left_child.update(l, @left_child.r, x)
        @right_child.update(@right_child.l, r, x)
      end
      @left_child.force
      @right_child.force
      @agg = aggregation_of(@left_child.agg, @right_child.agg)
    end
  end

  def force
    return if !@lazy
    x = @lazy
    @lazy = nil
    whole_update(x)
    @left_child.update(@left_child.l, @left_child.r, x) if @left_child
    @right_child.update(@right_child.l, @right_child.r, x) if @right_child
  end

  def query(l, r)
    force
    return @agg if l == @l && r == @r
    if r <= @left_child.r
      return @left_child.query(l, r)
    elsif @right_child.l <= l
      return @right_child.query(l, r)
    else
      agg_left = @left_child.query(l, @left_child.r)
      agg_right = @right_child.query(@right_child.l, r)
      return aggregation_of(agg_left, agg_right)
    end
  end
end

class LazySegmentTree < LazySegmentTreeBase
  def initialize(arr, l = 0, r = arr.size - 1)
    super(arr, l, r)
  end

  def defer(x)
    if @lazy
      @lazy += x
    else
      @lazy = x
    end
  end

  def whole_update(x)
    @agg += x
  end

  def aggregation_of(x, y)
    [x, y].min
  end
end

n, q = gets.split.map(&:to_i)
seg = LazySegmentTree.new(Array.new(n, 0))
q.times do
  c, x, y, z = gets.split.map(&:to_i)
  if c == 0
    seg.update(x, y, z)
  else
    puts seg.query(x, y)
  end
end