Fix documentation

This commit is contained in:
Julien Cretin
2020-09-30 18:37:33 +02:00
parent d0ad46b868
commit f9f428dcbf

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@@ -214,7 +214,7 @@
//! We want to show that the next `N - 1` compactions won't move the tail past the //! We want to show that the next `N - 1` compactions won't move the tail past the
//! last page of their window, with `I` the initial window: //! last page of their window, with `I` the initial window:
//! //!
//! ``` //! ```text
//! forall 1 <= i <= N - 1, t_{I + i} <= (I + i + N - 1) * Q //! forall 1 <= i <= N - 1, t_{I + i} <= (I + i + N - 1) * Q
//! ``` //! ```
//! //!
@@ -224,19 +224,19 @@
//! first page of the window with the last entry possibly overlapping on the next //! first page of the window with the last entry possibly overlapping on the next
//! page. //! page.
//! //!
//! ``` //! ```text
//! forall j, t_{j + 1} = t_j + |h_{j + 1}| - |h_j| + 1 //! forall j, t_{j + 1} = t_j + |h_{j + 1}| - |h_j| + 1
//! ``` //! ```
//! //!
//! By induction, we have: //! By induction, we have:
//! //!
//! ``` //! ```text
//! t_{I + i} <= t_I + |h_{I + i}| - |h_I| + i //! t_{I + i} <= t_I + |h_{I + i}| - |h_I| + i
//! ``` //! ```
//! //!
//! We have the following properties: //! We have the following properties:
//! //!
//! ``` //! ```text
//! t_I <= h_I + V //! t_I <= h_I + V
//! |h_{I + i}| - |h_I| <= h_{I + i} - h_I //! |h_{I + i}| - |h_I| <= h_{I + i} - h_I
//! h_{I + i} <= (I + i) * Q + M //! h_{I + i} <= (I + i) * Q + M
@@ -244,7 +244,7 @@
//! //!
//! Replacing into our previous equality, we can conclude: //! Replacing into our previous equality, we can conclude:
//! //!
//! ``` //! ```text
//! t_{I + i} = t_I + |h_{I + i}| - |h_I| + i //! t_{I + i} = t_I + |h_{I + i}| - |h_I| + i
//! <= h_I + V + (I + i) * Q + M - h_I + i //! <= h_I + V + (I + i) * Q + M - h_I + i
//! = (N - 1) * (Q - 1) - M + (I + i) * Q + M + i //! = (N - 1) * (Q - 1) - M + (I + i) * Q + M + i
@@ -256,7 +256,7 @@
//! We also want to show that after `N - 1` compactions, the remaining capacity is //! We also want to show that after `N - 1` compactions, the remaining capacity is
//! available without compaction. //! available without compaction.
//! //!
//! ``` //! ```text
//! V - (t_{I + N - 1} - h_{I + N - 1}) >= // The available words in the window. //! V - (t_{I + N - 1} - h_{I + N - 1}) >= // The available words in the window.
//! C - (|t_{I + N - 1}| - |h_{I + N - 1}|) // The remaining capacity. //! C - (|t_{I + N - 1}| - |h_{I + N - 1}|) // The remaining capacity.
//! + 1 // Reserved for Clear. //! + 1 // Reserved for Clear.
@@ -264,14 +264,14 @@
//! //!
//! We can replace the definition of `C` and simplify: //! We can replace the definition of `C` and simplify:
//! //!
//! ``` //! ```text
//! V - (t_{I + N - 1} - h_{I + N - 1}) >= V - N - (|t_{I + N - 1}| - |h_{I + N - 1}|) + 1 //! V - (t_{I + N - 1} - h_{I + N - 1}) >= V - N - (|t_{I + N - 1}| - |h_{I + N - 1}|) + 1
//! iff t_{I + N - 1} - h_{I + N - 1} <= |t_{I + N - 1}| - |h_{I + N - 1}| + N - 1 //! iff t_{I + N - 1} - h_{I + N - 1} <= |t_{I + N - 1}| - |h_{I + N - 1}| + N - 1
//! ``` //! ```
//! //!
//! We have the following properties: //! We have the following properties:
//! //!
//! ``` //! ```text
//! t_{I + N - 1} = t_I + |h_{I + N - 1}| - |h_I| + N - 1 //! t_{I + N - 1} = t_I + |h_{I + N - 1}| - |h_I| + N - 1
//! |t_{I + N - 1}| - |h_{I + N - 1}| = |t_I| - |h_I| // Compaction preserves capacity. //! |t_{I + N - 1}| - |h_{I + N - 1}| = |t_I| - |h_I| // Compaction preserves capacity.
//! |h_{I + N - 1}| - |t_I| <= h_{I + N - 1} - t_I //! |h_{I + N - 1}| - |t_I| <= h_{I + N - 1} - t_I
@@ -279,7 +279,7 @@
//! //!
//! From which we conclude: //! From which we conclude:
//! //!
//! ``` //! ```text
//! t_{I + N - 1} - h_{I + N - 1} <= |t_{I + N - 1}| - |h_{I + N - 1}| + N - 1 //! t_{I + N - 1} - h_{I + N - 1} <= |t_{I + N - 1}| - |h_{I + N - 1}| + N - 1
//! iff t_I + |h_{I + N - 1}| - |h_I| + N - 1 - h_{I + N - 1} <= |t_I| - |h_I| + N - 1 //! iff t_I + |h_{I + N - 1}| - |h_I| + N - 1 - h_{I + N - 1} <= |t_I| - |h_I| + N - 1
//! iff t_I + |h_{I + N - 1}| - h_{I + N - 1} <= |t_I| //! iff t_I + |h_{I + N - 1}| - h_{I + N - 1} <= |t_I|