fix recap.md

docs/src/plonk/recap.md

@@ -9,7 +9,7 @@ We use the following notation.
 
 The symbol $\mathbb{F}$ denotes a finite field. It is fixed all along. The symbol $\omega$ denotes a primitive root of unity in $\mathbb{F}$.
 
-All polynomials have coefficients in $\mathbb{F}$ and the variable is usually denoted by $X$. We denote polynomials by single letters like $p, a, b, z$. We only denote them as $z(X)$ when we want to emphasize the fact that it is a polynomial in $X$, or we need that to explicitly define a polynomial from another one. For example when composing a polynomial $z$ with the polynomial $\omega X$, the result being denoted by $z' := z(\omega X)$. The symbol $'$ is **not** used to denote derivatives.
+All polynomials have coefficients in $\mathbb{F}$ and the variable is usually denoted by $X$. We denote polynomials by single letters like $p, a, b, z$. We only denote them as $z(X)$ when we want to emphasize the fact that it is a polynomial in $X$, or we need to explicitly define a polynomial from another one. For example when composing a polynomial $z$ with the polynomial $\omega X$, the result being denoted by $z' := z(\omega X)$. The symbol $'$ is **not** used to denote derivatives.
 
 When interpolating at a domain $H=\{h_0, \dots, h_n\} \subset \mathbb{F}$, the symbols $L_i$ denote the Lagrange basis. That is $L_i$ is the polynomial such that $L_i(h_j) = 0$ for all $j\neq i$, and that $L_i(h_i) = 1$.
 
@@ -38,7 +38,7 @@ The idea here is that the verifier holds some value $x$, say $x=3$. He gives it
 
 In the context of PLONK, both the inputs and outputs of the program are considered _public inputs_. This may sound odd, but it is because these are the inputs to the verification algorithm. This is the algorithm that takes, in this case, the tuple $(3, 8, \pi)$ and outputs _Accept_ if the toy program was executed with input $x=3$, some private value $e$ not revealed to the verifier, and out came $8$. Otherwise it outputs _Reject_.
 
-PLONK can be used to delegate program executions to untrusted parties, but it can also be used as a proof of knowledge. Our program could be used by a prover to demostrate that she knows the multiplicative inverse of some value $x$ in the finite field without revealing it. She would do it by sending the verifier the tuple $(x, 0, \pi)$, where $\pi$ is the proof of the execution of our toy program.
+PLONK can be used to delegate program executions to untrusted parties, but it can also be used as a proof of knowledge. Our program could be used by a prover to demonstrate that she knows the multiplicative inverse of some value $x$ in the finite field without revealing it. She would do it by sending the verifier the tuple $(x, 0, \pi)$, where $\pi$ is the proof of the execution of our toy program.
 
 In our toy example this is pointless because inverting field elements is easily performed by any verifier. But change our program to the following and you get proofs of knowledge of the preimage of SHA256 digests.
 
@@ -444,4 +444,4 @@ $$
 
 This last step is not obvious. You can check the paper to see the proof. Anyway, this is the equation you'll recognize below in the description of the protocol.
 
-Randomness is a delicate matter and an important part of the protocol is where it comes from, who chooses it and when they choose it. Check out the protocol to see how it works.
+Randomness is a delicate matter and an important part of the protocol is where it comes from, who chooses it and when they choose it. Check out the protocol to see how it works.