chore: typos

src/folding-and-sonobe.md

@@ -2,7 +2,7 @@
 
 ## Folding schemes overview
 
-Folding schemes efficitently achieve incrementally verifiable computation (IVC), where the prover recursively proves the correct execution of the incremental computations.
+Folding schemes efficiently achieve incrementally verifiable computation (IVC), where the prover recursively proves the correct execution of the incremental computations.
 Once the IVC iterations are completed, the IVC proof is compressed into the Decider proof, a zkSNARK proof which proves that applying $n$ times the $F$ function (the circuit being folded) to the initial state ($z_0$) results in the final state ($z_n$).
 
 <p align="center">

src/usage/frontend-circom.md

@@ -1,30 +1,30 @@
 # Circom frontend
- > **Note**: Circom frontend will be significantly slower than the Arkworks frontend.
+ > **Note**: Circom frontend will be significantly slower than the Arkworks frontend. We explain below how to implement a custom `step_native` function with your circom circuits to speed things up!
  
 Experimental frontend using [arkworks/circom-compat](https://github.com/arkworks-rs/circom-compat).
 
 We can define the circuit to be folded in Circom. The only interface that we need to fit in is:
 
 ```c
-template FCircuit() {
+template FCircuit(ivc_state_len, aux_inputs_len) {
-    signal input ivc_input[1]; // IVC state
+    signal input ivc_input[ivc_state_len]; // IVC state
-    signal input external_inputs[2]; // not state
+    signal input external_inputs[aux_inputs_len]; // not state, 
 
-    signal output ivc_output[1]; // next IVC state
+    signal output ivc_output[ivc_state_len]; // next IVC state
     
     // [...]
 }
 component main {public [ivc_input]} = Example();
 ```
 
-The `ivc_input` is the array that defines the initial state, and the `ivc_output` is the array that defines the output state after the step.
+The `ivc_input` is the array that defines the initial state, and the `ivc_output` is the array that defines the output state after the step. Both need to be of the same size. The `external_inputs` array expects auxiliary input values.
 
 So for example, the following circuit does the traditional example at each step, which proves knowledge of $x$ such that $y==x^3 + x + e_0 + e_1$ for a known $y$ ($e_i$ are the `external_inputs[i]`):
 
 ```c
 pragma circom 2.0.3;
 
-template Example () {
+template CubicCircuit() {
     signal input ivc_input[1]; // IVC state
     signal input external_inputs[2]; // not state