\hyperlink{H_cancel_n}{\hcancel{n}}, etc. + grammar

Paper.tex

@@ -473,7 +473,7 @@ \subsubsection{Block Header Validity}
 \newcommand{\expdiffsymb}{\ensuremath{\epsilon}}
 \newcommand{\diffadjustment}{x}
 
-\hypertarget{block_difficulty_H__d}{}The canonical difficulty of a block of header $H$ is defined as $D(H)$:
+\hypertarget{block_difficulty_H__d}{}\linkdest{H__d}The canonical difficulty of a block of header $H$ is defined as $D(H)$:
 \begin{equation}
 D(H) \equiv \begin{dcases}
 \mindifficulty & \text{if} \quad H_{\mathrm{i}} = 0\\
@@ -523,7 +523,7 @@ \subsubsection{Block Header Validity}
 \end{equation}
 with $(n, m) = \mathtt{PoW}(H_{\hcancel{n}}, H_{\mathrm{n}}, \mathbf{d})$.
 
-\hypertarget{block_header_without_nonce_and_mixmash_h__cancel_n}{}Where $H_{\hcancel{n}}$ is the new block's header $H$, but \textit{without} the nonce and mix-hash components, $\mathbf{d}$ being the current DAG, a large data set needed to compute the mix-hash, and $\mathtt{PoW}$ is the proof-of-work function (see section \ref{ch:pow}): this evaluates to an array with the first item being the mix-hash, to proof that a correct DAG has been used, and the second item being a pseudo-random number cryptographically dependent on $H$ and $\mathbf{d}$. Given an approximately uniform distribution in the range $[0, 2^{64})$, the expected time to find a solution is proportional to the difficulty, $H_{\mathrm{d}}$.
+\hypertarget{block_header_without_nonce_and_mixmash_h__cancel_n}{}\linkdest{H_cancel_n}Where $H_{\hcancel{n}}$ is the new block's header $H$, but \textit{without} the nonce and mix-hash components, $\mathbf{d}$ being the current DAG, a large data set needed to compute the mix-hash, and $\mathtt{PoW}$ is the proof-of-work function (see section \ref{ch:pow}): this evaluates to an array with the first item being the mix-hash, to proof that a correct DAG has been used, and the second item being a pseudo-random number cryptographically dependent on $H$ and $\mathbf{d}$. Given an approximately uniform distribution in the range $[0, 2^{64})$, the expected time to find a solution is proportional to the difficulty, $H_{\mathrm{d}}$.
 
 This is the foundation of the security of the blockchain and is the fundamental reason why a malicious node cannot propagate newly created blocks that would otherwise overwrite (``rewrite'') history. Because the nonce must satisfy this requirement, and because its satisfaction depends on the contents of the block and in turn its composed transactions, creating new, valid, blocks is difficult and, over time, requires approximately the total compute power of the trustworthy portion of the mining peers.
 
@@ -1135,17 +1135,17 @@ \subsection{State \& Nonce Validation}\label{sec:statenoncevalidation}
 \begin{equation}
 \Gamma(B) \equiv \begin{cases}
 \boldsymbol{\sigma}_0 & \text{if} \quad P(B_{H}) = \varnothing \\
-\boldsymbol{\sigma}_{\mathrm{i}}: \mathtt{\small TRIE}(L_{S}(\boldsymbol{\sigma}_{\mathrm{i}})) = {P(B_{H})_{H}}_{\mathrm{r}} & \text{otherwise}
+\boldsymbol{\sigma}_{\mathrm{i}}: \mathtt{\small \hyperlink{trie}{TRIE}}(L_{S}(\boldsymbol{\sigma}_{\mathrm{i}})) = {P(B_{H})_{H}}_{\mathrm{r}} & \text{otherwise}
 \end{cases}
 \end{equation}
 
 Here, $\mathtt{\small TRIE}(L_{S}(\boldsymbol{\sigma}_{\mathrm{i}}))$ means the hash of the root node of a trie of state $\boldsymbol{\sigma}_{\mathrm{i}}$; it is assumed that implementations will store this in the state database, trivial and efficient since the trie is by nature an immutable data structure.
 
-\hypertarget{Phi}{}And finally define $\Phi$, the block transition function, which maps an incomplete block $B$ to a complete block $B'$:
+\hypertarget{Phi}{}And finally we define $\Phi$, the block transition function, which maps an incomplete block $B$ to a complete block $B'$:
 \begin{eqnarray}
 \Phi(B) & \equiv & B': \quad B' = B^* \quad \text{except:} \\
-B'_{\mathrm{n}} & = & n: \quad x \leqslant \frac{2^{256}}{H_{\mathrm{d}}} \\
-B'_{\mathrm{m}} & = & m \quad \text{with } (x, m) = \mathtt{PoW}(B^*_{\hcancel{n}}, n, \mathbf{d}) \\
+B'_{\mathrm{n}} & = & n: \quad x \leqslant \frac{2^{256}}{\hyperlink{H__d}{H_{\mathrm{d}}}} \\
+B'_{\mathrm{m}} & = & m \quad \text{with } (x, m) = \mathtt{PoW}(B^*_{\hyperlink{H_cancel_n}{\hcancel{n}}}, n, \mathbf{d}) \\
 B^* & \equiv & B \quad \text{except:} \quad B^*_{\mathrm{r}} = r(\Pi(\Gamma(B), B))
 \end{eqnarray}
 With $\mathbf{d}$ being a dataset as specified in appendix \ref{app:ethash}.