repeat lemmas (#25)

move lemmas from
[verified_go_stdlib](https://github.com/jcp19/verified_go_stdlib) to
`bytes`.

- the name of `LemmaRepeatCommutative` could be better, it is not
`Repeat` that is commutative but `++` (with the special arguments `b`
and `Repeat(b, n)`
- `verified_go_stdlib` has a lemma that ensures `Repeat(s, a)[: b *
len(s)] == Repeat(s, MinInt(a, b))`. Should we move this here as well?
It depends on a `MinInt` function that could be replaced by `math.Min`

bytes/repeat_lemmas.gobra

@@ -0,0 +1,71 @@
+package bytes
+
+//+gobra
+
+ghost
+ensures Repeat(b, 1) == b
+decreases
+func LemmaRepeatOnce(b seq[byte]) {
+	assert Repeat(b, 1) == b ++ Repeat(b, 0)
+}
+
+ghost
+requires n > 0
+ensures Repeat(b, n) == Repeat(b, n-1) ++ b
+decreases n
+func lemmaRepeatFlip(b seq[byte], n int) {
+	if n == 1 {
+		assert Repeat(b, n-1) == seq[byte]{}
+	} else {
+		assert Repeat(b, n) == b ++ Repeat(b, n-1)
+		lemmaRepeatFlip(b, n-1)
+		assert Repeat(b, n-1) == Repeat(b, n-2) ++ b
+	}
+}
+
+ghost
+requires n >= 0
+ensures  b ++ Repeat(b, n) == Repeat(b, n) ++ b
+decreases
+func LemmaRepeatCommutesWithConcat(b seq[byte], n int) {
+	assert b ++ Repeat(b, n) == Repeat(b, n+1)
+	lemmaRepeatFlip(b, n+1)
+}
+
+ghost
+requires n >= 0
+ensures  Repeat(b, 2 * n) == Repeat(b, n) ++ Repeat(b, n)
+decreases n
+func LemmaRepeatDoubled(b seq[byte], n int) {
+	if n != 0 {
+		assert Repeat(b, n) == b ++ Repeat(b, n - 1)
+		assert Repeat(b, n) ++ Repeat(b, n) == b ++ Repeat(b, n - 1) ++ b ++ Repeat(b, n - 1)
+		// order of repeating doesn't matter because it's always the same element that is repeated
+		assert b ++ Repeat(b, n - 1) ++ b ++ Repeat(b, n - 1) == b ++ (Repeat(b, n - 1) ++ b) ++ Repeat(b, n - 1)
+		LemmaRepeatCommutesWithConcat(b, n - 1)
+		assert b ++ (Repeat(b, n - 1) ++ b) ++ Repeat(b, n - 1) == b ++ (b ++ Repeat(b, n - 1)) ++ Repeat(b, n - 1)
+		assert b ++ Repeat(b, n - 1) ++ b ++ Repeat(b, n - 1) == b ++ b ++ Repeat(b, n - 1) ++ Repeat(b, n - 1)
+		LemmaRepeatDoubled(b, n - 1)
+		assert Repeat(b, n - 1) ++ Repeat(b, n - 1) == Repeat(b, 2 * (n - 1))
+		assert Repeat(b, 2 * n - 1) == b ++ Repeat(b, n - 1) ++ Repeat(b, n - 1)
+		assert Repeat(b, 2 * n) == b ++ b ++ Repeat(b, n - 1) ++ Repeat(b, n - 1)
+	} else {
+		assert Repeat(b, 2 * 0) == Repeat(b, 0) ++ Repeat(b, 0)
+	}
+}
+
+
+ghost
+requires n >= 0
+ensures  len(Repeat(s, n)) == len(s) * n
+decreases n
+func LemmaRepeatLength(s seq[byte], n int) {
+	switch {
+	case n == 0:
+		assert len(Repeat(s, n)) == 0
+	default:
+		assert Repeat(s, n) == s ++ Repeat(s, n-1)
+		LemmaRepeatLength(s, n-1)
+	}
+}
+