diff --git a/book/src/SUMMARY.md b/book/src/SUMMARY.md
index e52c1d0..5f11e6f 100644
--- a/book/src/SUMMARY.md
+++ b/book/src/SUMMARY.md
@@ -2,7 +2,7 @@
 
 [halo2](README.md)
 - [Concepts](concepts.md)
-  - [Background](concepts/background.md)
+  - [Proof systems](concepts/proofs.md)
   - [UltraPLONK Arithmetization](concepts/arithmetization.md)
   - [Cores](concepts/cores.md)
   - [Chips](concepts/chips.md)
diff --git a/book/src/concepts/background.md b/book/src/concepts/proofs.md
similarity index 89%
rename from book/src/concepts/background.md
rename to book/src/concepts/proofs.md
index 296aedc..de7ec1f 100644
--- a/book/src/concepts/background.md
+++ b/book/src/concepts/proofs.md
@@ -1,4 +1,4 @@
-# Background
+# Proof systems
 
 The aim of any ***proof system*** is to be able to prove ***instances*** of ***statements***.
 
@@ -10,6 +10,10 @@ Normally the statement will also have ***private inputs***. The private inputs a
 intermediate values that make an instance of a statement hold, are collectively called a
 ***witness*** for that instance.
 
+> The intermediate values depend on how we express the statement. We assume that we can
+> compute them efficiently from the private and public inputs (if that were not the case
+> then we would consider them part of the private inputs).
+
 A ***Non-interactive Argument of Knowledge*** allows a ***prover*** to create a ***proof***
 for a given instance and witness. The proof is data that can be used to convince a
 ***verifier*** that the creator of the proof knew a witness for which the statement holds on