diff --git a/src/simplifying/fol/ht.rs b/src/simplifying/fol/ht.rs
index 31207bcc..b1529db0 100644
--- a/src/simplifying/fol/ht.rs
+++ b/src/simplifying/fol/ht.rs
@@ -3,7 +3,10 @@ use crate::{
         apply::Apply as _,
         unbox::{fol::UnboxedFormula, Unbox as _},
     },
-    syntax_tree::fol::{AtomicFormula, BinaryConnective, Formula, Theory},
+    syntax_tree::fol::{
+        AtomicFormula, BinaryConnective, Comparison, Formula, GeneralTerm, Guard, IntegerTerm,
+        Quantification, Quantifier, Relation, Sort, SymbolicTerm, Theory, Variable,
+    },
 };
 
 pub fn simplify(theory: Theory) -> Theory {
@@ -14,13 +17,131 @@ pub fn simplify(theory: Theory) -> Theory {
 
 fn simplify_formula(formula: Formula) -> Formula {
     formula.apply_all(&mut vec![
+        Box::new(substitute_defined_variables),
+        Box::new(evaluate_comparisons_between_equal_terms),
         Box::new(remove_identities),
         Box::new(remove_annihilations),
         Box::new(remove_idempotences),
+        Box::new(remove_orphaned_variables),
+        Box::new(remove_empty_quantifications),
         Box::new(join_nested_quantifiers),
     ])
 }
 
+fn substitute_defined_variables(formula: Formula) -> Formula {
+    // Substitute defined variables in existential quantifications
+
+    fn find_definition(variable: &Variable, formula: &Formula) -> Option<GeneralTerm> {
+        match formula {
+            Formula::AtomicFormula(AtomicFormula::Comparison(comparison)) => comparison
+                .individuals()
+                .filter_map(|individual| match individual {
+                    (lhs, Relation::Equal, rhs) => Some((lhs, rhs)),
+                    _ => None,
+                })
+                .flat_map(|(lhs, rhs)| [(lhs, rhs), (rhs, lhs)])
+                .filter_map(|(x, term)| match (x, term, &variable.sort) {
+                    (GeneralTerm::Variable(name), _, Sort::General)
+                    | (
+                        GeneralTerm::IntegerTerm(IntegerTerm::Variable(name)),
+                        GeneralTerm::IntegerTerm(_),
+                        Sort::Integer,
+                    )
+                    | (
+                        GeneralTerm::SymbolicTerm(SymbolicTerm::Variable(name)),
+                        GeneralTerm::SymbolicTerm(_),
+                        Sort::Symbol,
+                    ) if variable.name == *name && !term.variables().contains(variable) => {
+                        Some(term)
+                    }
+                    _ => None,
+                })
+                .next()
+                .cloned(),
+
+            Formula::BinaryFormula {
+                connective: BinaryConnective::Conjunction,
+                lhs,
+                rhs,
+            } => find_definition(variable, lhs).or_else(|| find_definition(variable, rhs)),
+
+            _ => None,
+        }
+    }
+
+    match formula {
+        Formula::QuantifiedFormula {
+            quantification:
+                Quantification {
+                    quantifier: quantifier @ Quantifier::Exists,
+                    variables,
+                },
+            formula,
+        } => {
+            let mut formula = *formula;
+
+            for variable in variables.iter().rev() {
+                if let Some(definition) = find_definition(variable, &formula) {
+                    formula = formula.substitute(variable.clone(), definition);
+                }
+            }
+
+            Formula::quantify(formula, quantifier, variables)
+        }
+        x => x,
+    }
+}
+
+fn evaluate_comparisons_between_equal_terms(formula: Formula) -> Formula {
+    // Evaluate comparisons between structurally equal terms
+    // e.g. T  = T => #true
+    // e.g. T != T => #false
+    // e.g. T1 = T2 = T3 => T1 = T2 and T2 = T3 (side effect)
+
+    match formula {
+        Formula::AtomicFormula(AtomicFormula::Comparison(Comparison { term, guards })) => {
+            let mut formulas = vec![];
+
+            let mut lhs = term;
+            for Guard {
+                relation,
+                term: rhs,
+            } in guards
+            {
+                formulas.push(Formula::AtomicFormula(if lhs == rhs {
+                    match relation {
+                        // T  = T => #true
+                        // T >= T => #true
+                        // T <= T => #true
+                        Relation::Equal | Relation::GreaterEqual | Relation::LessEqual => {
+                            AtomicFormula::Truth
+                        }
+                        // T != T => #false
+                        // T >  T => #false
+                        // T <  T => #false
+                        Relation::NotEqual | Relation::Greater | Relation::Less => {
+                            AtomicFormula::Falsity
+                        }
+                    }
+                } else {
+                    AtomicFormula::Comparison(Comparison {
+                        term: lhs,
+                        guards: vec![Guard {
+                            relation,
+                            term: rhs.clone(),
+                        }],
+                    })
+                }));
+
+                lhs = rhs;
+            }
+
+            Formula::conjoin(formulas)
+        }
+        x => x,
+    }
+}
+
 fn remove_identities(formula: Formula) -> Formula {
     // Remove identities
     // e.g. F op E => F
@@ -112,6 +233,54 @@ fn remove_idempotences(formula: Formula) -> Formula {
     }
 }
 
+fn remove_orphaned_variables(formula: Formula) -> Formula {
+    // Remove orphaned variables in quantification
+    // e.g. q X Y F(X) => q X F(X)
+
+    match formula {
+        // forall X Y F(X) => forall X F(X)
+        // exists X Y F(X) => exists X F(X)
+        Formula::QuantifiedFormula {
+            quantification:
+                Quantification {
+                    quantifier,
+                    variables,
+                },
+            formula,
+        } => {
+            let free_variables = formula.free_variables();
+            let variables = variables
+                .into_iter()
+                .filter(|v| free_variables.contains(v))
+                .collect();
+
+            Formula::QuantifiedFormula {
+                quantification: Quantification {
+                    quantifier,
+                    variables,
+                },
+                formula,
+            }
+        }
+        x => x,
+    }
+}
+
+fn remove_empty_quantifications(formula: Formula) -> Formula {
+    // Remove empty quantifiers
+    // e.g. q F => F
+
+    match formula {
+        // forall F => F
+        // exists F => F
+        Formula::QuantifiedFormula {
+            quantification,
+            formula,
+        } if quantification.variables.is_empty() => *formula,
+        x => x,
+    }
+}
+
 pub(crate) fn join_nested_quantifiers(formula: Formula) -> Formula {
     // Remove nested quantifiers
     // e.g. q X ( q Y F(X,Y) ) => q X Y F(X,Y)
@@ -142,17 +311,74 @@ pub(crate) fn join_nested_quantifiers(formula: Formula) -> Formula {
 mod tests {
     use {
         super::{
-            join_nested_quantifiers, remove_annihilations, remove_idempotences, remove_identities,
-            simplify_formula,
+            evaluate_comparisons_between_equal_terms, join_nested_quantifiers,
+            remove_annihilations, remove_idempotences, remove_identities,
+            remove_orphaned_variables, simplify_formula,
+        },
+        crate::{
+            convenience::apply::Apply as _,
+            simplifying::fol::ht::{remove_empty_quantifications, substitute_defined_variables},
+            syntax_tree::fol::Formula,
         },
-        crate::{convenience::apply::Apply as _, syntax_tree::fol::Formula},
     };
 
+    #[test]
+    fn test_substitute_defined_variables() {
+        for (src, target) in [
+            (
+                "exists X$g (X$g = 1 and p(X$g))",
+                "exists X$g (1 = 1 and p(1))",
+            ),
+            (
+                "exists X$g (X$g = a and p(X$g))",
+                "exists X$g (a = a and p(a))",
+            ),
+            (
+                "exists X$i (X$i = 1 and p(X$i))",
+                "exists X$i (1 = 1 and p(1))",
+            ),
+            (
+                "exists X$i (X$i = a and p(X$i))",
+                "exists X$i (X$i = a and p(X$i))",
+            ),
+            (
+                "exists X$s (X$s = 1 and p(X$s))",
+                "exists X$s (X$s = 1 and p(X$s))",
+            ),
+            (
+                "exists X$s (X$s = a and p(X$s))",
+                "exists X$s (a = a and p(a))",
+            ),
+            (
+                "exists X$i (X$i = X$i + 1 and p(X$i))",
+                "exists X$i (X$i = X$i + 1 and p(X$i))",
+            ),
+            (
+                "exists X$i (X$i = 1 or p(X$i))",
+                "exists X$i (X$i = 1 or p(X$i))",
+            ),
+            (
+                "forall X$i (X$i = 1 and p(X$i))",
+                "forall X$i (X$i = 1 and p(X$i))",
+            ),
+        ] {
+            assert_eq!(
+                src.parse::<Formula>()
+                    .unwrap()
+                    .apply(&mut substitute_defined_variables),
+                target.parse().unwrap()
+            )
+        }
+    }
+
     #[test]
     fn test_simplify() {
         for (src, target) in [
             ("#true and #true and a", "a"),
             ("#true and (#true and a)", "a"),
+            ("forall X a", "a"),
+            ("X = X and a", "a"),
+            ("forall X (X = X)", "#true"),
         ] {
             assert_eq!(
                 simplify_formula(src.parse().unwrap()),
@@ -161,6 +387,36 @@ mod tests {
         }
     }
 
+    #[test]
+    fn test_evaluate_comparisons_between_equal_terms() {
+        for (src, target) in [
+            ("X = X", "#true"),
+            ("X = Y", "X = Y"),
+            ("X != X", "#false"),
+            ("X != Y", "X != Y"),
+            ("X > X", "#false"),
+            ("X > Y", "X > Y"),
+            ("X < X", "#false"),
+            ("X < Y", "X < Y"),
+            ("X >= X", "#true"),
+            ("X >= Y", "X >= Y"),
+            ("X <= X", "#true"),
+            ("X <= Y", "X <= Y"),
+            ("X$i + 1 = X$i + 1", "#true"),
+            ("X$i + 1 + 1 = X$i + 2", "X$i + 1 + 1 = X$i + 2"),
+            ("X = X = Y", "#true and X = Y"),
+            ("X != X = Y", "#false and X = Y"),
+            ("X = Y = Z", "X = Y and Y = Z"),
+        ] {
+            assert_eq!(
+                src.parse::<Formula>()
+                    .unwrap()
+                    .apply(&mut evaluate_comparisons_between_equal_terms),
+                target.parse().unwrap()
+            )
+        }
+    }
+
     #[test]
     fn test_remove_identities() {
         for (src, target) in [
@@ -207,6 +463,45 @@ mod tests {
         }
     }
 
+    #[test]
+    fn test_remove_orphaned_variables() {
+        for (src, target) in [
+            ("forall X Y Z (X = X)", "forall X (X = X)"),
+            ("exists X Y (X = Y)", "exists X Y (X = Y)"),
+            ("exists X Y Z (X = Y)", "exists X Y (X = Y)"),
+        ] {
+            assert_eq!(
+                src.parse::<Formula>()
+                    .unwrap()
+                    .apply(&mut remove_orphaned_variables),
+                target.parse().unwrap()
+            )
+        }
+    }
+
+    #[test]
+    fn test_remove_empty_quantifications() {
+        use crate::syntax_tree::fol::{Atom, AtomicFormula, Quantification, Quantifier};
+
+        let src = Formula::QuantifiedFormula {
+            quantification: Quantification {
+                quantifier: Quantifier::Forall,
+                variables: vec![],
+            },
+            formula: Box::new(Formula::AtomicFormula(AtomicFormula::Atom(Atom {
+                predicate_symbol: "a".into(),
+                terms: vec![],
+            }))),
+        };
+
+        let target = Formula::AtomicFormula(AtomicFormula::Atom(Atom {
+            predicate_symbol: "a".into(),
+            terms: vec![],
+        }));
+
+        assert_eq!(src.apply(&mut remove_empty_quantifications), target);
+    }
+
     #[test]
     fn test_join_nested_quantifiers() {
         for (src, target) in [
diff --git a/src/syntax_tree/fol.rs b/src/syntax_tree/fol.rs
index ddaef468..628ec619 100644
--- a/src/syntax_tree/fol.rs
+++ b/src/syntax_tree/fol.rs
@@ -143,6 +143,13 @@ impl SymbolicTerm {
             SymbolicTerm::Symbol(_) | SymbolicTerm::Variable(_) => IndexSet::new(),
         }
     }
+
+    pub fn substitute(self, var: Variable, term: SymbolicTerm) -> Self {
+        match self {
+            SymbolicTerm::Variable(s) if var.name == s && var.sort == Sort::Symbol => term,
+            _ => self,
+        }
+    }
 }
 
 #[derive(Clone, Debug, Eq, PartialEq, Hash)]
@@ -202,6 +209,12 @@ impl GeneralTerm {
                     "cannot substitute general term `{term}` for the integer variable `{var}`"
                 ),
             },
+            GeneralTerm::SymbolicTerm(t) if var.sort == Sort::Symbol => match term {
+                GeneralTerm::SymbolicTerm(term) => GeneralTerm::SymbolicTerm(t.substitute(var, term)),
+                _ => panic!(
+                    "cannot substitute general term `{term}` for the symbolic variable `{var}`"
+                ),
+            }
             t => t,
         }
     }