diff --git a/ast/src/analyzed/display.rs b/ast/src/analyzed/display.rs
index aaf943ce7..e4d812253 100644
--- a/ast/src/analyzed/display.rs
+++ b/ast/src/analyzed/display.rs
@@ -71,6 +71,7 @@ impl<T: Display> Display for FunctionValueDefinition<T> {
                 write!(f, " = {}", items.iter().map(|i| i.to_string()).join(" + "))
             }
             FunctionValueDefinition::Query(e) => write!(f, "(i) query {e}"),
+            FunctionValueDefinition::Expression(e) => write!(f, " = {e}"),
         }
     }
 }
diff --git a/ast/src/analyzed/mod.rs b/ast/src/analyzed/mod.rs
index c7565505e..30c27bc2f 100644
--- a/ast/src/analyzed/mod.rs
+++ b/ast/src/analyzed/mod.rs
@@ -61,6 +61,12 @@ impl<T> Analyzed<T> {
         self.definitions_in_source_order(PolynomialType::Committed)
     }
 
+    pub fn intermediate_polys_in_source_order(
+        &self,
+    ) -> Vec<&(Polynomial, Option<FunctionValueDefinition<T>>)> {
+        self.definitions_in_source_order(PolynomialType::Intermediate)
+    }
+
     pub fn definitions_in_source_order(
         &self,
         poly_type: PolynomialType,
@@ -96,12 +102,12 @@ impl<T> Analyzed<T> {
     /// so that they are contiguous again.
     /// There must not be any reference to the removed polynomials left.
     pub fn remove_polynomials(&mut self, to_remove: &BTreeSet<PolyID>) {
-        // TODO intermediate polys
         let replacements: BTreeMap<PolyID, PolyID> = [
             // We have to do it separately because we need to re-start the counter
             // for each kind.
             self.committed_polys_in_source_order(),
             self.constant_polys_in_source_order(),
+            self.intermediate_polys_in_source_order(),
         ]
         .map(|polys| {
             polys
@@ -239,6 +245,7 @@ pub enum FunctionValueDefinition<T> {
     Mapping(Expression<T>),
     Array(Vec<RepeatedArray<T>>),
     Query(Expression<T>),
+    Expression(Expression<T>),
 }
 
 /// An array of elements that might be repeated.
diff --git a/ast/src/analyzed/util.rs b/ast/src/analyzed/util.rs
index 9636a82aa..bb7730f2c 100644
--- a/ast/src/analyzed/util.rs
+++ b/ast/src/analyzed/util.rs
@@ -22,6 +22,7 @@ where
                 .iter_mut()
                 .flat_map(|e| e.pattern.iter_mut())
                 .try_for_each(|e| previsit_expression_mut(e, f)),
+            Some(FunctionValueDefinition::Expression(e)) => previsit_expression_mut(e, f),
             None => ControlFlow::Continue(()),
         })?;
 
@@ -66,6 +67,7 @@ where
                 .iter_mut()
                 .flat_map(|e| e.pattern.iter_mut())
                 .try_for_each(|e| postvisit_expression_mut(e, f)),
+            Some(FunctionValueDefinition::Expression(e)) => postvisit_expression_mut(e, f),
             None => ControlFlow::Continue(()),
         })?;
 
diff --git a/ast/src/parsed/display.rs b/ast/src/parsed/display.rs
index 920488520..074cdb932 100644
--- a/ast/src/parsed/display.rs
+++ b/ast/src/parsed/display.rs
@@ -370,6 +370,9 @@ impl<T: Display> Display for FunctionDefinition<T> {
             FunctionDefinition::Query(params, value) => {
                 write!(f, "({}) query {value}", params.join(", "),)
             }
+            FunctionDefinition::Expression(e) => {
+                write!(f, " = {e}")
+            }
         }
     }
 }
diff --git a/ast/src/parsed/mod.rs b/ast/src/parsed/mod.rs
index 8a5b5f205..d886c90ad 100644
--- a/ast/src/parsed/mod.rs
+++ b/ast/src/parsed/mod.rs
@@ -289,7 +289,6 @@ pub struct FunctionCall<T> {
 }
 
 /// The definition of a function (excluding its name):
-/// Either a param-value mapping or an array expression.
 #[derive(Debug, PartialEq, Eq, PartialOrd, Ord, Clone)]
 pub enum FunctionDefinition<T> {
     /// Parameter-value-mapping.
@@ -298,6 +297,8 @@ pub enum FunctionDefinition<T> {
     Array(ArrayExpression<T>),
     /// Prover query.
     Query(Vec<String>, Expression<T>),
+    /// Expression, for intermediate polynomials
+    Expression(Expression<T>),
 }
 
 #[derive(Debug, PartialEq, Eq, PartialOrd, Ord, Clone)]
@@ -451,6 +452,7 @@ where
                 postvisit_expression_mut(e, f)
             }
             FunctionDefinition::Array(ae) => postvisit_expression_in_array_expression_mut(ae, f),
+            FunctionDefinition::Expression(e) => postvisit_expression_mut(e, f),
         },
         PilStatement::PolynomialCommitDeclaration(_, _, None)
         | PilStatement::Include(_, _)
diff --git a/backend/src/pilcom_cli/json_exporter/mod.rs b/backend/src/pilcom_cli/json_exporter/mod.rs
index e04329f87..fc8eb1cac 100644
--- a/backend/src/pilcom_cli/json_exporter/mod.rs
+++ b/backend/src/pilcom_cli/json_exporter/mod.rs
@@ -34,7 +34,7 @@ pub fn export<T: FieldElement>(analyzed: &Analyzed<T>) -> JsonValue {
             StatementIdentifier::Definition(name) => {
                 if let (poly, Some(value)) = &analyzed.definitions[name] {
                     if poly.poly_type == PolynomialType::Intermediate {
-                        if let FunctionValueDefinition::Mapping(value) = value {
+                        if let FunctionValueDefinition::Expression(value) = value {
                             let expression_id = exporter.extract_expression(value, 1);
                             assert_eq!(
                                 expression_id,
diff --git a/compiler/src/lib.rs b/compiler/src/lib.rs
index 57140fbd9..346a40d18 100644
--- a/compiler/src/lib.rs
+++ b/compiler/src/lib.rs
@@ -163,6 +163,7 @@ pub fn compile_asm_string<T: FieldElement>(
         );
         return Ok((pil_file_path, None));
     }
+
     fs::write(pil_file_path.clone(), format!("{pil}")).unwrap();
 
     let pil_file_name = pil_file_path.file_name().unwrap();
diff --git a/compiler/tests/asm.rs b/compiler/tests/asm.rs
index 079d8e44b..cb54d5681 100644
--- a/compiler/tests/asm.rs
+++ b/compiler/tests/asm.rs
@@ -162,6 +162,22 @@ fn full_pil_constant() {
     gen_halo2_proof(f, Default::default());
 }
 
+#[test]
+#[should_panic = "assertion failed: poly.is_fixed() || poly.is_witness()"]
+fn intermediate() {
+    let f = "intermediate.asm";
+    verify_asm::<GoldilocksField>(f, Default::default());
+    gen_halo2_proof(f, Default::default());
+}
+
+#[test]
+#[should_panic = "assertion failed: poly.is_fixed() || poly.is_witness()"]
+fn intermediate_nested() {
+    let f = "intermediate_nested.asm";
+    verify_asm::<GoldilocksField>(f, Default::default());
+    gen_halo2_proof(f, Default::default());
+}
+
 #[test]
 fn book() {
     for f in fs::read_dir("../test_data/asm/book/").unwrap() {
diff --git a/executor/src/constant_evaluator/mod.rs b/executor/src/constant_evaluator/mod.rs
index afda27ea8..e140cce2c 100644
--- a/executor/src/constant_evaluator/mod.rs
+++ b/executor/src/constant_evaluator/mod.rs
@@ -74,6 +74,9 @@ fn generate_values<T: FieldElement>(
             values
         }
         FunctionValueDefinition::Query(_) => panic!("Query used for fixed column."),
+        FunctionValueDefinition::Expression(_) => {
+            panic!("Expression used for fixed column, only expected for intermediate polynomials")
+        }
     }
 }
 
diff --git a/pil_analyzer/src/pil_analyzer.rs b/pil_analyzer/src/pil_analyzer.rs
index fd7e21379..22fa80e8c 100644
--- a/pil_analyzer/src/pil_analyzer.rs
+++ b/pil_analyzer/src/pil_analyzer.rs
@@ -147,7 +147,7 @@ impl<T: FieldElement> PILContext<T> {
                     name,
                     None,
                     PolynomialType::Intermediate,
-                    Some(FunctionDefinition::Mapping(vec![], value)),
+                    Some(FunctionDefinition::Expression(value)),
                 );
             }
             PilStatement::PublicDeclaration(start, name, polynomial, index) => {
@@ -321,12 +321,14 @@ impl<T: FieldElement> PILContext<T> {
         let name = poly.absolute_name.clone();
 
         let value = value.map(|v| match v {
+            FunctionDefinition::Expression(expr) => {
+                assert!(!have_array_size);
+                assert!(poly.poly_type == PolynomialType::Intermediate);
+                FunctionValueDefinition::Expression(self.process_expression(expr))
+            }
             FunctionDefinition::Mapping(params, expr) => {
                 assert!(!have_array_size);
-                assert!(
-                    poly.poly_type == PolynomialType::Constant
-                        || poly.poly_type == PolynomialType::Intermediate
-                );
+                assert!(poly.poly_type == PolynomialType::Constant);
                 FunctionValueDefinition::Mapping(self.process_function(params, expr))
             }
             FunctionDefinition::Query(params, expr) => {
@@ -660,4 +662,20 @@ namespace T(65536);
         }
         assert_eq!(input, formatted);
     }
+
+    #[test]
+    fn intermediate() {
+        let input = r#"namespace N(65536);
+    col witness x;
+    col intermediate = x;
+    intermediate = intermediate;
+"#;
+        let expected = r#"namespace N(65536);
+    col witness x;
+    col intermediate = N.x;
+    N.intermediate = N.intermediate;
+"#;
+        let formatted = process_pil_file_contents::<GoldilocksField>(input).to_string();
+        assert_eq!(formatted, expected);
+    }
 }
diff --git a/pilopt/src/lib.rs b/pilopt/src/lib.rs
index 3c86e1e3c..7f6617681 100644
--- a/pilopt/src/lib.rs
+++ b/pilopt/src/lib.rs
@@ -96,6 +96,7 @@ fn constant_value<T: FieldElement>(function: &FunctionValueDefinition<T>) -> Opt
             }
         }
         FunctionValueDefinition::Query(_) => None,
+        FunctionValueDefinition::Expression(_) => None,
     }
 }
 
@@ -411,6 +412,22 @@ mod test {
     N.Z = ((1 + N.A) * 2);
     N.A = (1 + N.A);
     N.Z = (1 + N.A);
+"#;
+        let optimized = optimize(process_pil_file_contents::<GoldilocksField>(input)).to_string();
+        assert_eq!(optimized, expectation);
+    }
+
+    #[test]
+    fn intermediate() {
+        let input = r#"namespace N(65536);
+        col witness x;
+        col intermediate = x;
+        intermediate = intermediate;
+    "#;
+        let expectation = r#"namespace N(65536);
+    col witness x;
+    col intermediate = N.x;
+    N.intermediate = N.intermediate;
 "#;
         let optimized = optimize(process_pil_file_contents::<GoldilocksField>(input)).to_string();
         assert_eq!(optimized, expectation);
diff --git a/test_data/asm/intermediate.asm b/test_data/asm/intermediate.asm
new file mode 100644
index 000000000..979e3f0f6
--- /dev/null
+++ b/test_data/asm/intermediate.asm
@@ -0,0 +1,9 @@
+machine Intermediate(latch, operation_id) {
+    constraints {
+        col fixed latch = [1]*;
+        col fixed operation_id = [0]*;
+        col witness x;
+        col intermediate = x;
+        intermediate = intermediate;
+    }
+}
\ No newline at end of file
diff --git a/test_data/asm/intermediate_nested.asm b/test_data/asm/intermediate_nested.asm
new file mode 100644
index 000000000..54326c671
--- /dev/null
+++ b/test_data/asm/intermediate_nested.asm
@@ -0,0 +1,11 @@
+machine Intermediate(latch, operation_id) {
+	constraints {
+		col fixed latch = [1]*;
+		col fixed operation_id = [0]*;
+		col witness x;
+		col intermediate = x;
+		col int2 = intermediate;
+		col int3 = int2 + intermediate;
+		int3 = 2 * x;
+	}
+}
\ No newline at end of file