From e8cc77a7d5918327d2fa353761d1f39b905a63e6 Mon Sep 17 00:00:00 2001
From: James Ray <16969914+jamesray1@users.noreply.github.com>
Date: Fri, 26 Jan 2018 12:50:48 +1100
Subject: [PATCH 1/3] Replace all _(\{[A-Z]+\b)([^\}]) with _{mathbf$1}$2...

...using Libre Open Office with match case and regular expressions ticked as options.
---
 Paper.tex | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/Paper.tex b/Paper.tex
index c834c995..54766bda 100644
--- a/Paper.tex
+++ b/Paper.tex
@@ -1075,7 +1075,7 @@ \subsection{Ommer Validation}
 
 The validation of ommer headers means nothing more than verifying that each ommer header is both a valid header and satisfies the relation of $N$th-generation ommer to the present block where $N \leq 6$. The maximum of ommer headers is two. Formally:
 \begin{equation}
-\lVert B_\mathbf{U} \rVert \leqslant 2 \bigwedge_{U \in B_\mathbf{U}} V(U) \; \wedge \; k(U, P(B_H)_H, 6)
+\lVert B_\mathbf{U} \rVert \leqslant 2 \bigwedge_{mathbf{U} \in B_\mathbf{U}} V(U) \; \wedge \; k(U, P(B_H)_H, 6)
 \end{equation}
 
 where $k$ denotes the ``is-kin'' property:
@@ -1108,7 +1108,7 @@ \subsection{Reward Application}
 \\ \nonumber
 \Omega(B, \boldsymbol{\sigma}) & \equiv & \boldsymbol{\sigma}': \boldsymbol{\sigma}' = \boldsymbol{\sigma} \quad \text{except:} \\
 \qquad\boldsymbol{\sigma}'[{B_H}_c]_b & = & \boldsymbol{\sigma}[{B_H}_c]_b + (1 + \frac{\lVert B_\mathbf{U}\rVert}{32})R_b \\
-\qquad\forall_{U \in B_\mathbf{U}}: \\ \nonumber
+\qquad\forall_{mathbf{U} \in B_\mathbf{U}}: \\ \nonumber
 \boldsymbol{\sigma}'[U_c] & = & \begin{cases}
 \varnothing &\text{if}\ \boldsymbol{\sigma}[U_c] = \varnothing\ \wedge\ R = 0 \\
 \mathbf{a}' &\text{otherwise}
@@ -1383,7 +1383,7 @@ \section{Modified Merkle Patricia Tree}\label{app:trie}
 
 When considering such a sequence, we use the common numeric subscript notation to refer to a tuple's key or value, thus:
 \begin{equation}
-\forall_{I \in \mathfrak{I}} I \equiv (I_0, I_1)
+\forall_{mathbf{I} \in \mathfrak{I}} I \equiv (I_0, I_1)
 \end{equation}
 
 Any series of bytes may also trivially be viewed as a series of nibbles, given an endian-specific notation; here we assume big-endian. Thus:
@@ -1421,7 +1421,7 @@ \section{Modified Merkle Patricia Tree}\label{app:trie}
 \begin{equation}
 c(\mathfrak{I}, i) \equiv \begin{cases}
 \texttt{\small RLP}\Big( \big(\texttt{\small HP}(I_0[i .. (\lVert I_0\rVert - 1)], true), I_1 \big) \Big) & \text{if} \quad \lVert \mathfrak{I} \rVert = 1 \quad \text{where} \; \exists I: I \in \mathfrak{I} \\
-\texttt{\small RLP}\Big( \big(\texttt{\small HP}(I_0[i .. (j - 1)], false), n(\mathfrak{I}, j) \big) \Big) & \text{if} \quad i \ne j \quad \text{where} \; j = \arg \max_x : \exists \mathbf{l}: \lVert \mathbf{l} \rVert = x : \forall_{I \in \mathfrak{I}}: I_0[0 .. (x - 1)] = \mathbf{l} \\
+\texttt{\small RLP}\Big( \big(\texttt{\small HP}(I_0[i .. (j - 1)], false), n(\mathfrak{I}, j) \big) \Big) & \text{if} \quad i \ne j \quad \text{where} \; j = \arg \max_x : \exists \mathbf{l}: \lVert \mathbf{l} \rVert = x : \forall_{mathbf{I} \in \mathfrak{I}}: I_0[0 .. (x - 1)] = \mathbf{l} \\
 \texttt{\small RLP}\Big( (u(0), u(1), ..., u(15), v) \Big) & \text{otherwise} \quad \text{where} \begin{array}[t]{rcl}
 u(j) & \equiv & n(\{ I : I \in \mathfrak{I} \wedge I_0[i] = j \}, i + 1) \\
 v & = & \begin{cases}

From 5b60d755c1c08e7fb5b9f3dfdc869ec3320d3ffb Mon Sep 17 00:00:00 2001
From: James Ray <16969914+jamesray1@users.noreply.github.com>
Date: Tue, 30 Jan 2018 12:19:28 +1100
Subject: [PATCH 2/3] Add a missing forward slash

---
 Paper.tex | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/Paper.tex b/Paper.tex
index 54766bda..4b7d658d 100644
--- a/Paper.tex
+++ b/Paper.tex
@@ -1075,7 +1075,7 @@ \subsection{Ommer Validation}
 
 The validation of ommer headers means nothing more than verifying that each ommer header is both a valid header and satisfies the relation of $N$th-generation ommer to the present block where $N \leq 6$. The maximum of ommer headers is two. Formally:
 \begin{equation}
-\lVert B_\mathbf{U} \rVert \leqslant 2 \bigwedge_{mathbf{U} \in B_\mathbf{U}} V(U) \; \wedge \; k(U, P(B_H)_H, 6)
+\lVert B_\mathbf{U} \rVert \leqslant 2 \bigwedge_{\mathbf{U} \in B_\mathbf{U}} V(U) \; \wedge \; k(U, P(B_H)_H, 6)
 \end{equation}
 
 where $k$ denotes the ``is-kin'' property:

From 34b9991f4d4692a6da379208f6b7bbd12f65413c Mon Sep 17 00:00:00 2001
From: James Ray <16969914+jamesray1@users.noreply.github.com>
Date: Tue, 30 Jan 2018 12:24:04 +1100
Subject: [PATCH 3/3] Add missing forward slashes before mathbfs

---
 Paper.tex | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/Paper.tex b/Paper.tex
index 4b7d658d..6ce11a88 100644
--- a/Paper.tex
+++ b/Paper.tex
@@ -1108,7 +1108,7 @@ \subsection{Reward Application}
 \\ \nonumber
 \Omega(B, \boldsymbol{\sigma}) & \equiv & \boldsymbol{\sigma}': \boldsymbol{\sigma}' = \boldsymbol{\sigma} \quad \text{except:} \\
 \qquad\boldsymbol{\sigma}'[{B_H}_c]_b & = & \boldsymbol{\sigma}[{B_H}_c]_b + (1 + \frac{\lVert B_\mathbf{U}\rVert}{32})R_b \\
-\qquad\forall_{mathbf{U} \in B_\mathbf{U}}: \\ \nonumber
+\qquad\forall_{\mathbf{U} \in B_{\mathbf{U}}}: \\ \nonumber
 \boldsymbol{\sigma}'[U_c] & = & \begin{cases}
 \varnothing &\text{if}\ \boldsymbol{\sigma}[U_c] = \varnothing\ \wedge\ R = 0 \\
 \mathbf{a}' &\text{otherwise}
@@ -1383,7 +1383,7 @@ \section{Modified Merkle Patricia Tree}\label{app:trie}
 
 When considering such a sequence, we use the common numeric subscript notation to refer to a tuple's key or value, thus:
 \begin{equation}
-\forall_{mathbf{I} \in \mathfrak{I}} I \equiv (I_0, I_1)
+\forall_{\mathbf{I} \in \mathfrak{I}} I \equiv (I_0, I_1)
 \end{equation}
 
 Any series of bytes may also trivially be viewed as a series of nibbles, given an endian-specific notation; here we assume big-endian. Thus:
@@ -1421,7 +1421,7 @@ \section{Modified Merkle Patricia Tree}\label{app:trie}
 \begin{equation}
 c(\mathfrak{I}, i) \equiv \begin{cases}
 \texttt{\small RLP}\Big( \big(\texttt{\small HP}(I_0[i .. (\lVert I_0\rVert - 1)], true), I_1 \big) \Big) & \text{if} \quad \lVert \mathfrak{I} \rVert = 1 \quad \text{where} \; \exists I: I \in \mathfrak{I} \\
-\texttt{\small RLP}\Big( \big(\texttt{\small HP}(I_0[i .. (j - 1)], false), n(\mathfrak{I}, j) \big) \Big) & \text{if} \quad i \ne j \quad \text{where} \; j = \arg \max_x : \exists \mathbf{l}: \lVert \mathbf{l} \rVert = x : \forall_{mathbf{I} \in \mathfrak{I}}: I_0[0 .. (x - 1)] = \mathbf{l} \\
+\texttt{\small RLP}\Big( \big(\texttt{\small HP}(I_0[i .. (j - 1)], false), n(\mathfrak{I}, j) \big) \Big) & \text{if} \quad i \ne j \quad \text{where} \; j = \arg \max_x : \exists \mathbf{l}: \lVert \mathbf{l} \rVert = x : \forall_{\mathbf{I} \in \mathfrak{I}}: I_0[0 .. (x - 1)] = \mathbf{l} \\
 \texttt{\small RLP}\Big( (u(0), u(1), ..., u(15), v) \Big) & \text{otherwise} \quad \text{where} \begin{array}[t]{rcl}
 u(j) & \equiv & n(\{ I : I \in \mathfrak{I} \wedge I_0[i] = j \}, i + 1) \\
 v & = & \begin{cases}