1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
// This file is part of Substrate.

// Copyright (C) 2019-2020 Parity Technologies (UK) Ltd.
// SPDX-License-Identifier: Apache-2.0

// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
// 	http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.

//! Infinite precision unsigned integer for substrate runtime.

use num_traits::Zero;
use sp_std::{cmp::Ordering, ops, prelude::*, vec, cell::RefCell, convert::TryFrom};

// A sensible value for this would be half of the dword size of the host machine. Since the
// runtime is compiled to 32bit webassembly, using 32 and 64 for single and double respectively
// should yield the most performance.
/// Representation of a single limb.
pub type Single = u32;
/// Representation of two limbs.
pub type Double = u64;
/// Difference in the number of bits of [`Single`] and [`Double`].
const SHIFT: usize = 32;
/// short form of _Base_. Analogous to the value 10 in base-10 decimal numbers.
const B: Double = Single::max_value() as Double + 1;

/// Splits a [`Double`] limb number into a tuple of two [`Single`] limb numbers.
pub fn split(a: Double) -> (Single, Single) {
	let al = a as Single;
	let ah = (a >> SHIFT) as Single;
	(ah, al)
}

/// Assumed as a given primitive.
///
/// Multiplication of two singles, which at most yields 1 double.
pub fn mul_single(a: Single, b: Single) -> Double {
	let a: Double = a.into();
	let b: Double = b.into();
	a * b
}

/// Assumed as a given primitive.
///
/// Addition of two singles, which at most takes a single limb of result and a carry,
/// returned as a tuple respectively.
pub fn add_single(a: Single, b: Single) -> (Single, Single) {
	let a: Double = a.into();
	let b: Double = b.into();
	let q = a + b;
	let (carry, r) = split(q);
	(r, carry)
}

/// Assumed as a given primitive.
///
/// Division of double by a single limb. Always returns a double limb of quotient and a single
/// limb of remainder.
fn div_single(a: Double, b: Single) -> (Double, Single) {
	let b: Double = b.into();
	let q = a / b;
	let r = a % b;
	// both conversions are trivially safe.
	(q, r as Single)
}

/// Simple wrapper around an infinitely large integer, represented as limbs of [`Single`].
#[derive(Clone, Default)]
pub struct BigUint {
	/// digits (limbs) of this number (sorted as msb -> lsd).
	pub(crate) digits: Vec<Single>,
}

impl BigUint {
	/// Create a new instance with `size` limbs. This prevents any number with zero limbs to be
	/// created.
	///
	/// The behavior of the type is undefined with zero limbs.
	pub fn with_capacity(size: usize) -> Self {
		Self { digits: vec![0; size.max(1)] }
	}

	/// Raw constructor from custom limbs. If `limbs` is empty, `Zero::zero()` implementation is
	/// used.
	pub fn from_limbs(limbs: &[Single]) -> Self {
		if !limbs.is_empty() {
			Self { digits: limbs.to_vec() }
		} else {
			Zero::zero()
		}
	}

	/// Number of limbs.
	pub fn len(&self) -> usize { self.digits.len() }

	/// A naive getter for limb at `index`. Note that the order is lsb -> msb.
	///
	/// #### Panics
	///
	/// This panics if index is out of range.
	pub fn get(&self, index: usize) -> Single {
		self.digits[self.len() - 1 - index]
	}

	/// A naive getter for limb at `index`. Note that the order is lsb -> msb.
	pub fn checked_get(&self, index: usize) -> Option<Single> {
		let i = self.len().checked_sub(1)?;
		let j = i.checked_sub(index)?;
		self.digits.get(j).cloned()
	}

	/// A naive setter for limb at `index`. Note that the order is lsb -> msb.
	///
	/// #### Panics
	///
	/// This panics if index is out of range.
	pub fn set(&mut self, index: usize, value: Single) {
		let len = self.digits.len();
		self.digits[len - 1 - index] = value;
	}

	/// returns the least significant limb of the number.
	///
	/// #### Panics
	///
	/// While the constructor of the type prevents this, this can panic if `self` has no digits.
	pub fn lsb(&self) -> Single {
		self.digits[self.len() - 1]
	}

	/// returns the most significant limb of the number.
	///
	/// #### Panics
	///
	/// While the constructor of the type prevents this, this can panic if `self` has no digits.
	pub fn msb(&self) -> Single {
		self.digits[0]
	}

	/// Strips zeros from the left side (the most significant limbs) of `self`, if any.
	pub fn lstrip(&mut self) {
		// by definition, a big-int number should never have leading zero limbs. This function
		// has the ability to cause this. There is nothing to do if the number already has 1
		// limb only. call it a day and return.
		if self.len().is_zero() { return; }
		let index = self.digits.iter().position(|&elem| elem != 0).unwrap_or(self.len() - 1);

		if index > 0 {
			self.digits = self.digits[index..].to_vec()
		}
	}

	/// Zero-pad `self` from left to reach `size` limbs. Will not make any difference if `self`
	/// is already bigger than `size` limbs.
	pub fn lpad(&mut self, size: usize) {
		let n = self.len();
		if n >= size { return; }
		let pad = size - n;
		let mut new_digits = (0..pad).map(|_| 0).collect::<Vec<Single>>();
		new_digits.extend(self.digits.iter());
		self.digits = new_digits;
	}

	/// Adds `self` with `other`. self and other do not have to have any particular size. Given
	/// that the `n = max{size(self), size(other)}`, it will produce a number with `n + 1`
	/// limbs.
	///
	/// This function does not strip the output and returns the original allocated `n + 1`
	/// limbs. The caller may strip the output if desired.
	///
	/// Taken from "The Art of Computer Programming" by D.E. Knuth, vol 2, chapter 4.
	pub fn add(self, other: &Self) -> Self {
		let n = self.len().max(other.len());
		let mut k: Double = 0;
		let mut w = Self::with_capacity(n + 1);

		for j in 0..n {
			let u = Double::from(self.checked_get(j).unwrap_or(0));
			let v = Double::from(other.checked_get(j).unwrap_or(0));
			let s = u + v + k;
			w.set(j, (s % B) as Single);
			k = s / B;
		}
		// k is always 0 or 1.
		w.set(n, k as Single);
		w
	}

	/// Subtracts `other` from `self`. self and other do not have to have any particular size.
	/// Given that the `n = max{size(self), size(other)}`, it will produce a number of size `n`.
	///
	/// If `other` is bigger than `self`, `Err(B - borrow)` is returned.
	///
	/// Taken from "The Art of Computer Programming" by D.E. Knuth, vol 2, chapter 4.
	pub fn sub(self, other: &Self) -> Result<Self, Self> {
		let n = self.len().max(other.len());
		let mut k = 0;
		let mut w = Self::with_capacity(n);
		for j in 0..n {
			let s = {
				let u = Double::from(self.checked_get(j).unwrap_or(0));
				let v = Double::from(other.checked_get(j).unwrap_or(0));
				let mut needs_borrow = false;
				let mut t = 0;

				if let Some(v) = u.checked_sub(v) {
					if let Some(v2) = v.checked_sub(k) {
						t = v2 % B;
						k = 0;
					} else {
						needs_borrow = true;
					}
				} else {
					needs_borrow = true;
				}
				if needs_borrow {
					t = u + B - v - k;
					k = 1;
				}
				t
			};
			// PROOF: t either comes from `v2 % B`, or from `u + B - v - k`. The former is
			// trivial. The latter will not overflow this branch will only happen if the sum of
			// `u - v - k` part has been negative, hence `u + B - v - k < b`.
			w.set(j, s as Single);
		}

		if k.is_zero() {
			Ok(w)
		} else {
			Err(w)
		}
	}

	/// Multiplies n-limb number `self` with m-limb number `other`.
	///
	/// The resulting number will always have `n + m` limbs.
	///
	/// This function does not strip the output and returns the original allocated `n + m`
	/// limbs. The caller may strip the output if desired.
	///
	/// Taken from "The Art of Computer Programming" by D.E. Knuth, vol 2, chapter 4.
	pub fn mul(self, other: &Self) -> Self {
		let n = self.len();
		let m = other.len();
		let mut w = Self::with_capacity(m + n);

		for j in 0..n {
			if self.get(j) == 0 {
				// Note: `with_capacity` allocates with 0. Explicitly set j + m to zero if
				// otherwise.
				continue;
			}

			let mut k = 0;
			for i in 0..m {
				// PROOF: (B−1) × (B−1) + (B−1) + (B−1) = B^2 −1 < B^2. addition is safe.
				let t =
					mul_single(self.get(j), other.get(i))
						+ Double::from(w.get(i + j))
						+ Double::from(k);
				w.set(i + j, (t % B) as Single);
				// PROOF: (B^2 - 1) / B < B. conversion is safe.
				k = (t / B) as Single;
			}
			w.set(j + m, k);
		}
		w
	}

	/// Divides `self` by a single limb `other`. This can be used in cases where the original
	/// division cannot work due to the divisor (`other`) being just one limb.
	///
	/// Invariant: `other` cannot be zero.
	pub fn div_unit(self, mut other: Single) -> Self {
		other = other.max(1);
		let n = self.len();
		let mut out = Self::with_capacity(n);
		let mut r: Single = 0;
		// PROOF: (B-1) * B + (B-1) still fits in double
		let with_r = |x: Double, r: Single| { Double::from(r) * B + x };
		for d in (0..n).rev() {
			let (q, rr) = div_single(with_r(self.get(d).into(), r), other) ;
			out.set(d, q as Single);
			r = rr;
		}
		out
	}

	/// Divides an `n + m` limb self by a `n` limb `other`. The result is a `m + 1` limb
	/// quotient and a `n` limb remainder, if enabled by passing `true` in `rem` argument, both
	/// in the form of an option's `Ok`.
	///
	/// - requires `other` to be stripped and have no leading zeros.
	/// - requires `self` to be stripped and have no leading zeros.
	/// - requires `other` to have at least two limbs.
	/// - requires `self` to have a greater length compared to `other`.
	///
	/// All arguments are examined without being stripped for the above conditions. If any of
	/// the above fails, `None` is returned.`
	///
	/// Taken from "The Art of Computer Programming" by D.E. Knuth, vol 2, chapter 4.
	pub fn div(self, other: &Self, rem: bool) -> Option<(Self, Self)> {
		if other.len() <= 1
			|| other.msb() == 0
			|| self.msb() == 0
			|| self.len() <= other.len()
		{
			return None
		}
		let n = other.len();
		let m = self.len() - n;

		let mut q = Self::with_capacity(m + 1);
		let mut r = Self::with_capacity(n);

		// PROOF: 0 <= normalizer_bits < SHIFT 0 <= normalizer < B. all conversions are
		// safe.
		let normalizer_bits = other.msb().leading_zeros() as Single;
		let normalizer = (2 as Single).pow(normalizer_bits as u32) as Single;

		// step D1.
		let mut self_norm = self.mul(&Self::from(normalizer));
		let mut other_norm = other.clone().mul(&Self::from(normalizer));

		// defensive only; the mul implementation should always create this.
		self_norm.lpad(n + m + 1);
		other_norm.lstrip();

		// step D2.
		for j in (0..=m).rev() {
			// step D3.0 Find an estimate of q[j], named qhat.
			let (qhat, rhat) = {
				// PROOF: this always fits into `Double`. In the context of Single = u8, and
				// Double = u16, think of 255 * 256 + 255 which is just u16::max_value().
				let dividend =
					Double::from(self_norm.get(j + n))
						* B
						+ Double::from(self_norm.get(j + n - 1));
				let divisor = other_norm.get(n - 1);
				div_single(dividend, divisor)
			};

			// D3.1 test qhat
			// replace qhat and rhat with RefCells. This helps share state with the closure
			let qhat = RefCell::new(qhat);
			let rhat = RefCell::new(Double::from(rhat));

			let test = || {
				// decrease qhat if it is bigger than the base (B)
				let qhat_local = *qhat.borrow();
				let rhat_local = *rhat.borrow();
				let predicate_1 = qhat_local >= B;
				let predicate_2 = {
					let lhs = qhat_local * Double::from(other_norm.get(n - 2));
					let rhs = B * rhat_local + Double::from(self_norm.get(j + n - 2));
					lhs > rhs
				};
				if predicate_1 || predicate_2 {
					*qhat.borrow_mut() -= 1;
					*rhat.borrow_mut() += Double::from(other_norm.get(n - 1));
					true
				} else {
					false
				}
			};

			test();
			while (*rhat.borrow() as Double) < B {
				if !test() { break; }
			}

			let qhat = qhat.into_inner();
			// we don't need rhat anymore. just let it go out of scope when it does.

			// step D4
			let lhs = Self { digits: (j..=j+n).rev().map(|d| self_norm.get(d)).collect() };
			let rhs = other_norm.clone().mul(&Self::from(qhat));

			let maybe_sub = lhs.sub(&rhs);
			let mut negative = false;
			let sub = match maybe_sub {
				Ok(t) => t,
				Err(t) => { negative = true; t }
			};
			(j..=j+n).for_each(|d| { self_norm.set(d, sub.get(d - j)); });

			// step D5
			// PROOF: the `test()` specifically decreases qhat until it is below `B`. conversion
			// is safe.
			q.set(j, qhat as Single);

			// step D6: add back if negative happened.
			if negative {
				q.set(j, q.get(j) - 1);
				let u = Self { digits: (j..=j+n).rev().map(|d| self_norm.get(d)).collect() };
				let r = other_norm.clone().add(&u);
				(j..=j+n).rev().for_each(|d| { self_norm.set(d, r.get(d - j)); })
			}
		}

		// if requested, calculate remainder.
		if rem {
			// undo the normalization.
			if normalizer_bits > 0 {
				let s = SHIFT as u32;
				let nb = normalizer_bits;
				for d in 0..n-1 {
					let v = self_norm.get(d) >> nb
						| self_norm.get(d + 1).overflowing_shl(s - nb).0;
					r.set(d, v);
				}
				r.set(n - 1, self_norm.get(n - 1) >> normalizer_bits);
			} else {
				r = self_norm;
			}
		}

		Some((q, r))
	}
}

impl sp_std::fmt::Debug for BigUint {
	#[cfg(feature = "std")]
	fn fmt(&self, f: &mut sp_std::fmt::Formatter<'_>) -> sp_std::fmt::Result {
		write!(
			f,
			"BigUint {{ {:?} ({:?})}}",
			self.digits,
			u128::try_from(self.clone()).unwrap_or(0),
		)
	}

	#[cfg(not(feature = "std"))]
	fn fmt(&self, _: &mut sp_std::fmt::Formatter<'_>) -> sp_std::fmt::Result {
		Ok(())
	}

}

impl PartialEq for BigUint {
	fn eq(&self, other: &Self) -> bool {
		self.cmp(other) == Ordering::Equal
	}
}

impl Eq for BigUint {}

impl Ord for BigUint {
	fn cmp(&self, other: &Self) -> Ordering {
		let lhs_first = self.digits.iter().position(|&e| e != 0);
		let rhs_first = other.digits.iter().position(|&e| e != 0);

		match (lhs_first, rhs_first) {
			// edge cases that should not happen. This basically means that one or both were
			// zero.
			(None, None) => Ordering::Equal,
			(Some(_), None) => Ordering::Greater,
			(None, Some(_)) => Ordering::Less,
			(Some(lhs_idx), Some(rhs_idx)) => {
				let lhs = &self.digits[lhs_idx..];
				let rhs = &other.digits[rhs_idx..];
				let len_cmp = lhs.len().cmp(&rhs.len());
				match len_cmp {
					Ordering::Equal => lhs.cmp(rhs),
					_ => len_cmp,
				}
			}
		}
	}
}

impl PartialOrd for BigUint {
	fn partial_cmp(&self, other: &Self) -> Option<Ordering> {
		Some(self.cmp(other))
	}
}

impl ops::Add for BigUint {
	type Output = Self;
	fn add(self, rhs: Self) -> Self::Output {
		self.add(&rhs)
	}
}

impl ops::Sub for BigUint {
	type Output = Self;
	fn sub(self, rhs: Self) -> Self::Output {
		self.sub(&rhs).unwrap_or_else(|e| e)
	}
}

impl ops::Mul for BigUint {
	type Output = Self;
	fn mul(self, rhs: Self) -> Self::Output {
		self.mul(&rhs)
	}
}

impl Zero for BigUint {
	fn zero() -> Self {
		Self { digits: vec![Zero::zero()] }
	}

	fn is_zero(&self) -> bool {
		self.digits.iter().all(|d| d.is_zero())
	}
}

macro_rules! impl_try_from_number_for {
	($([$type:ty, $len:expr]),+) => {
		$(
			impl TryFrom<BigUint> for $type {
				type Error = &'static str;
				fn try_from(mut value: BigUint) -> Result<$type, Self::Error> {
					value.lstrip();
					let error_message = concat!("cannot fit a number into ", stringify!($type));
					if value.len() * SHIFT > $len {
						Err(error_message)
					} else {
						let mut acc: $type = Zero::zero();
						for (i, d) in value.digits.iter().rev().cloned().enumerate() {
							let d: $type = d.into();
							acc += d << (SHIFT * i);
						}
						Ok(acc)
					}
				}
			}
		)*
	};
}
// can only be implemented for sizes bigger than two limb.
impl_try_from_number_for!([u128, 128], [u64, 64]);

macro_rules! impl_from_for_smaller_than_word {
	($($type:ty),+) => {
		$(impl From<$type> for BigUint {
			fn from(a: $type) -> Self {
				Self { digits: vec! [a.into()] }
			}
		})*
	}
}
impl_from_for_smaller_than_word!(u8, u16, Single);

impl From<Double> for BigUint {
	fn from(a: Double) -> Self {
		let (ah, al) = split(a);
		Self { digits: vec![ah, al] }
	}
}

#[cfg(test)]
pub mod tests {
	use super::*;

	fn with_limbs(n: usize) -> BigUint {
		BigUint { digits: vec![1; n] }
	}

	#[test]
	fn split_works() {
		let a = SHIFT / 2;
		let b = SHIFT * 3 / 2;
		let num: Double = 1 << a | 1 << b;
		// example when `Single = u8`
		// assert_eq!(num, 0b_0001_0000_0001_0000)
		assert_eq!(split(num), (1 << a, 1 << a));
	}

	#[test]
	fn strip_works() {
		let mut a = BigUint::from_limbs(&[0, 1, 0]);
		a.lstrip();
		assert_eq!(a.digits, vec![1, 0]);

		let mut a = BigUint::from_limbs(&[0, 0, 1]);
		a.lstrip();
		assert_eq!(a.digits, vec![1]);

		let mut a = BigUint::from_limbs(&[0, 0]);
		a.lstrip();
		assert_eq!(a.digits, vec![0]);

		let mut a = BigUint::from_limbs(&[0, 0, 0]);
		a.lstrip();
		assert_eq!(a.digits, vec![0]);
	}

	#[test]
	fn lpad_works() {
		let mut a = BigUint::from_limbs(&[0, 1, 0]);
		a.lpad(2);
		assert_eq!(a.digits, vec![0, 1, 0]);

		let mut a = BigUint::from_limbs(&[0, 1, 0]);
		a.lpad(3);
		assert_eq!(a.digits, vec![0, 1, 0]);

		let mut a = BigUint::from_limbs(&[0, 1, 0]);
		a.lpad(4);
		assert_eq!(a.digits, vec![0, 0, 1, 0]);
	}

	#[test]
	fn equality_works() {
		assert_eq!(
			BigUint { digits: vec![1, 2, 3] } == BigUint { digits: vec![1, 2, 3] },
			true,
		);
		assert_eq!(
			BigUint { digits: vec![3, 2, 3] } == BigUint { digits: vec![1, 2, 3] },
			false,
		);
		assert_eq!(
			BigUint { digits: vec![0, 1, 2, 3] } == BigUint { digits: vec![1, 2, 3] },
			true,
		);
	}

	#[test]
	fn ordering_works() {
		assert!(BigUint { digits: vec![0] } < BigUint { digits: vec![1] });
		assert!(BigUint { digits: vec![0] } == BigUint { digits: vec![0] });
		assert!(BigUint { digits: vec![] } == BigUint { digits: vec![0] });
		assert!(BigUint { digits: vec![] } == BigUint { digits: vec![] });
		assert!(BigUint { digits: vec![] } < BigUint { digits: vec![1] });

		assert!(BigUint { digits: vec![1, 2, 3] } == BigUint { digits: vec![1, 2, 3] });
		assert!(BigUint { digits: vec![0, 1, 2, 3] } == BigUint { digits: vec![1, 2, 3] });

		assert!(BigUint { digits: vec![1, 2, 4] } > BigUint { digits: vec![1, 2, 3] });
		assert!(BigUint { digits: vec![0, 1, 2, 4] } > BigUint { digits: vec![1, 2, 3] });
		assert!(BigUint { digits: vec![1, 2, 1, 0] } > BigUint { digits: vec![1, 2, 3] });

		assert!(BigUint { digits: vec![0, 1, 2, 1] } < BigUint { digits: vec![1, 2, 3] });
	}

	#[test]
	fn can_try_build_numbers_from_types() {
		use sp_std::convert::TryFrom;
		assert_eq!(u64::try_from(with_limbs(1)).unwrap(), 1);
		assert_eq!(u64::try_from(with_limbs(2)).unwrap(), u32::max_value() as u64 + 2);
		assert_eq!(
			u64::try_from(with_limbs(3)).unwrap_err(),
			"cannot fit a number into u64",
		);
		assert_eq!(
			u128::try_from(with_limbs(3)).unwrap(),
			u32::max_value() as u128 + u64::max_value() as u128 + 3
		);
	}

	#[test]
	fn zero_works() {
		assert_eq!(BigUint::zero(), BigUint { digits: vec![0] });
		assert_eq!(BigUint { digits: vec![0, 1, 0] }.is_zero(), false);
		assert_eq!(BigUint { digits: vec![0, 0, 0] }.is_zero(), true);

		let a = BigUint::zero();
		let b = BigUint::zero();
		let c = a * b;
		assert_eq!(c.digits, vec![0, 0]);
	}

	#[test]
	fn sub_negative_works() {
		assert_eq!(
			BigUint::from(10 as Single).sub(&BigUint::from(5 as Single)).unwrap(),
			BigUint::from(5 as Single)
		);
		assert_eq!(
			BigUint::from(10 as Single).sub(&BigUint::from(10 as Single)).unwrap(),
			BigUint::from(0 as Single)
		);
		assert_eq!(
			BigUint::from(10 as Single).sub(&BigUint::from(13 as Single)).unwrap_err(),
			BigUint::from((B - 3) as Single),
		);
	}

	#[test]
	fn mul_always_appends_one_digit() {
		let a = BigUint::from(10 as Single);
		let b = BigUint::from(4 as Single);
		assert_eq!(a.len(), 1);
		assert_eq!(b.len(), 1);

		let n = a.mul(&b);

		assert_eq!(n.len(), 2);
		assert_eq!(n.digits, vec![0, 40]);
	}

	#[test]
	fn div_conditions_work() {
		let a = BigUint { digits: vec![2] };
		let b = BigUint { digits: vec![1, 2] };
		let c = BigUint { digits: vec![1, 1, 2] };
		let d = BigUint { digits: vec![0, 2] };
		let e = BigUint { digits: vec![0, 1, 1, 2] };

		assert!(a.clone().div(&b, true).is_none());
		assert!(c.clone().div(&a, true).is_none());
		assert!(c.clone().div(&d, true).is_none());
		assert!(e.clone().div(&a, true).is_none());

		assert!(c.clone().div(&b, true).is_some());
	}

	#[test]
	fn div_unit_works() {
		let a = BigUint { digits: vec![100] };
		let b = BigUint { digits: vec![1, 100] };

		assert_eq!(a.clone().div_unit(1), a);
		assert_eq!(a.clone().div_unit(0), a);
		assert_eq!(a.clone().div_unit(2), BigUint::from(50 as Single));
		assert_eq!(a.clone().div_unit(7), BigUint::from(14 as Single));

		assert_eq!(b.clone().div_unit(1), b);
		assert_eq!(b.clone().div_unit(0), b);
		assert_eq!(b.clone().div_unit(2), BigUint::from(((B + 100) / 2) as Single));
		assert_eq!(b.clone().div_unit(7), BigUint::from(((B + 100) / 7) as Single));

	}
}