chore: fix some typos

Signed-off-by: acceptacross <csqcqs@gmail.com>

specification/src/lookup-argument.md

@@ -3,7 +3,7 @@
 The [Lookup Argument](https://eprint.iacr.org/2023/107.pdf) establishes that all elements of list $A = (a_0, \dots, a_\ell)$ also occur in list $B = (b_0, \dots, b_n)$.
 In this context, $A$ contains the values that are being looked up, while $B$ is the lookup table.[^1]
 Both lists $A$ and $B$ may contain duplicates.
-However, it is inefficient if $B$ does, and is therefor assumed not to.
+However, it is inefficient if $B$ does, and is therefore assumed not to.
 
 The example at the end of this section summarizes the necessary computations for the Lookup Argument.
 The rest of the section derives those computations.

tips/tip-0005/tip-0005.md

@@ -163,13 +163,13 @@ On the side of the server, two columns are needed to probe this identity in the
 
 On the client only *one* extension column is needed. Specifically, the running product `rpc` and formal derivative `fdc` are merged into a single column, the logarithmic derivative `ldc`.
 
-To update `ldc`, recall that the standard running product column `rpc` is defined to accumulate one *factor* in every row. Moreover, `ldc` is defined to contain the logarithmic derivative of `rpc` in every row, so we can use the eponymous property to populate it. Speficially, the would-have-been running product update rule `rpc* = rpc ⋅ (α - rlc*)` becomes `ldc* = ldc + 1/(α - rlc*)`, where the asterisk `*` indicates the respective element from the next row.
+To update `ldc`, recall that the standard running product column `rpc` is defined to accumulate one *factor* in every row. Moreover, `ldc` is defined to contain the logarithmic derivative of `rpc` in every row, so we can use the eponymous property to populate it. Specifically, the would-have-been running product update rule `rpc* = rpc ⋅ (α - rlc*)` becomes `ldc* = ldc + 1/(α - rlc*)`, where the asterisk `*` indicates the respective element from the next row.
 
 The update rules `sum* = sum + mul* / (α - rlc*)` and `ldc* = ldc + 1 / (α - rlc*)` can be converted to AIR constraints by multiplying left and right hand sides by `(α - rlc*)`.
 
 ### Cascade Construction
 
-The cascade construction arithmetizes a composite lookup gate in terms of multiple lookups into component gates followed by combining the looked-up outputs. For example, supose that a 16-bit wide map can be represented as the concatenation of two 8-bit wide lookups. Then this 16-bit wide map can be arithmetized with a cascade table as follows. The cascade table is the *server* authenticating 16-bit wide input-output pairs to the external client. Internally, every input or output element is represented as two limbs of 8 bits. To authenticate the 8-bit wide input-output pairs, the cascade table is the *client* of an 8-bit wide subset argument with an external server.
+The cascade construction arithmetizes a composite lookup gate in terms of multiple lookups into component gates followed by combining the looked-up outputs. For example, suppose that a 16-bit wide map can be represented as the concatenation of two 8-bit wide lookups. Then this 16-bit wide map can be arithmetized with a cascade table as follows. The cascade table is the *server* authenticating 16-bit wide input-output pairs to the external client. Internally, every input or output element is represented as two limbs of 8 bits. To authenticate the 8-bit wide input-output pairs, the cascade table is the *client* of an 8-bit wide subset argument with an external server.
 
 A cascade table consists of 5 base columns and 3 extension column. The extension columns are defined relative to challenges `a,b,c,d,β,γ`. The Latin letters denote weights used to compress columns, and the Greek letters denote indeterminates.
 
@@ -346,7 +346,7 @@ For Tip5 this means we have to add 3 columns per state element that undergoes th
 
 For Rescue-Prime and Rescue-Prime Optimized there are 16 columns that need to be expanded in this way.The total number of base columns is 16 + 16⋅3 = 64. The total number of columns is the same, 64, whether in technical terms of B-field equivalent terms.
 
-Note that for the parameters here, Rescue-Prime has 8 rounds and as a result the trace of one hash function invocation does not fit into 8 rows. To accomodate for this, one must either include columns for counting the round, or expand the trace into 16 rows and use periodic constraints or periodic interpolants.
+Note that for the parameters here, Rescue-Prime has 8 rounds and as a result the trace of one hash function invocation does not fit into 8 rows. To accommodate for this, one must either include columns for counting the round, or expand the trace into 16 rows and use periodic constraints or periodic interpolants.
 
 ## Performance