Fix article usage in documentation and code comments

README.md

@@ -33,7 +33,7 @@ verification logic can be delayed to query and decision phase.
 - This implementation has low-degree test built-in, and can handle RS-IOP.
 - Multiple oracles with same evaluation domain share a merkle tree and be submitted in one round, which greatly reduces
   verification overhead and number of constraints.
-- Each leaf of an low-degree oracle is a coset instead of an individual field element, which significantly reduces
+- Each leaf of a low-degree oracle is a coset instead of an individual field element, which significantly reduces
   merkle tree overhead on FRI query.
 
 ## Build Guide

ark-bcs/src/bcs/constraints/verifier.rs

@@ -237,7 +237,7 @@ where
 
     /// Verify without LDT.
     ///
-    /// ** Warning **: If verifier tries to get an low-degree oracle, no LDT
+    /// ** Warning **: If verifier tries to get a low-degree oracle, no LDT
     /// will be automatically performed.
     pub fn verify_with_dummy_ldt<V, S>(
         cs: ConstraintSystemRef<CF>,

ark-bcs/src/bcs/verifier.rs

@@ -221,7 +221,7 @@ where
 
     /// Verify without LDT.
     ///
-    /// ** Warning **: If verifier tries to get an low-degree oracle, no LDT
+    /// ** Warning **: If verifier tries to get a low-degree oracle, no LDT
     /// will be automatically performed.
     pub fn verify_with_dummy_ldt<V, S>(
         sponge: S,

ark-bcs/src/iop/message.rs

@@ -18,7 +18,7 @@ use super::{
 pub struct MsgRoundRef {
     pub(crate) index: usize,
     pub(crate) trace: TraceInfo,
-    /// Whether this round refers to an virtual oracle.
+    /// Whether this round refers to a virtual oracle.
     pub(crate) is_virtual: bool,
 }
 

ark-bcs/src/iop/mod.rs

@@ -16,7 +16,7 @@ pub mod verifier;
 /// Prover parameter used by IOP Prover. Any IOP prover parameter is a superset
 /// of IOP verifier parameter.
 pub trait ProverParam: Clone + Debug {
-    /// Verifier state should be a improper subset of `self`.
+    /// Verifier state should be an improper subset of `self`.
     type VerifierParameter: VerifierParam;
     /// Derive verifier parameter from prover parameter.
     fn to_verifier_param(&self) -> Self::VerifierParameter;

ark-bcs/src/ldt/constraints/rl_ldt.rs

@@ -241,7 +241,7 @@ impl<F: PrimeField + Absorb> LDTWithGadget<F> for LinearCombinationLDT<F> {
 
 /// return evaluation of x^{degree_to_raise} at specific location (with coset
 /// structure) For now, we assume the offset of codeword domain is constant.
-/// TODO: in the future, the offset can also be an variable.
+/// TODO: in the future, the offset can also be a variable.
 fn degree_raise_poly_query<F: PrimeField>(
     domain: Radix2CosetDomain<F>,
     degree_to_raise: u64,

ark-bcs/src/ldt/mod.rs

@@ -21,7 +21,7 @@ pub mod constraints;
 /// LDT that runs FRI on a random linear combination.
 pub mod rl_ldt;
 
-/// Trait for LDT, which is an public coin IOP.
+/// Trait for LDT, which is a public coin IOP.
 pub trait LDT<F: PrimeField + Absorb> {
     /// Parameters of `Self`
     type LDTParameters: Clone;