diff --git a/R.py b/R.py
index 4ce4b04..1b0fa39 100644
--- a/R.py
+++ b/R.py
@@ -12,7 +12,8 @@ from logic import (
 )
 from model import Model, ModelFunction, ModelValue, satisfiable
 from generate_model import generate_model
-# from vsp import has_vsp
+from vsp import has_vsp
+from smt import smt_is_loaded
 
 
 # ===================================================
@@ -56,12 +57,17 @@ disjunction_rules = {
     Rule({Conjunction(x, Disjunction(y, z)),}, Disjunction(Conjunction(x, y), Conjunction(x, z)))
 }
 
+falsification_rules = {
+    # At least one value is non-designated
+    Rule(set(), x)
+}
+
 
 logic_rules = implication_rules | negation_rules | conjunction_rules | disjunction_rules
 
 operations = {Negation, Conjunction, Disjunction, Implication}
 
-R_logic = Logic(operations, logic_rules, "R")
+R_logic = Logic(operations, logic_rules, falsification_rules, "R")
 
 # ===============================
 
@@ -69,36 +75,36 @@ R_logic = Logic(operations, logic_rules, "R")
 Example 2-Element Model of R
 """
 
-a0 = ModelValue("a0")
-a1 = ModelValue("a1")
+a0 = ModelValue("0")
+a1 = ModelValue("1")
 
 carrier_set = {a0, a1}
 
 mnegation = ModelFunction(1, {
     a0: a1,
     a1: a0
-})
+}, "¬")
 
 mimplication = ModelFunction(2, {
     (a0, a0): a1,
     (a0, a1): a1,
     (a1, a0): a0,
     (a1, a1): a1
-})
+}, "→")
 
 mconjunction = ModelFunction(2, {
     (a0, a0): a0,
     (a0, a1): a0,
     (a1, a0): a0,
     (a1, a1): a1
-})
+}, "∧")
 
 mdisjunction = ModelFunction(2, {
     (a0, a0): a0,
     (a0, a1): a1,
     (a1, a0): a1,
     (a1, a1): a1
-})
+}, "∨")
 
 
 designated_values = {a1}
@@ -117,11 +123,18 @@ interpretation = {
 
 print(R_model_2)
 
+print(f"Does {R_model_2.name} satisfy the logic R?", satisfiable(R_logic, R_model_2, interpretation))
+
+if smt_is_loaded():
+    print(has_vsp(R_model_2, mimplication, True, True))
+else:
+    print("Z3 not setup, skipping VSP check...")
+
 
 # =================================
 
 """
-Generate models of R of a specified size
+Generate models of R of a specified size using the slow approach
 """
 
 print("*" * 30)
@@ -130,14 +143,20 @@ model_size = 2
 print("Generating models of Logic", R_logic.name, "of size", model_size)
 solutions = generate_model(R_logic, model_size, print_model=False)
 
-print(f"Found {len(solutions)} satisfiable models")
+if smt_is_loaded():
+    num_satisfies_vsp = 0
+    for model, interpretation in solutions:
+        negation_defined = Negation in interpretation
+        conj_disj_defined = Conjunction in interpretation and Disjunction in interpretation
+        if has_vsp(model, interpretation[Implication], negation_defined, conj_disj_defined).has_vsp:
+            num_satisfies_vsp += 1
+
+    print(f"Found {len(solutions)} satisfiable models of size {model_size}, {num_satisfies_vsp} of which satisfy VSP")
 
-# for model, interpretation in solutions:
-#     print(has_vsp(model, interpretation))
 
 print("*" * 30)
 
-######
+# =================================
 
 """
 Showing the smallest model for R that has the
@@ -146,12 +165,12 @@ variable sharing property.
 This model has 6 elements.
 """
 
-a0 = ModelValue("a0")
-a1 = ModelValue("a1")
-a2 = ModelValue("a2")
-a3 = ModelValue("a3")
-a4 = ModelValue("a4")
-a5 = ModelValue("a5")
+a0 = ModelValue("0")
+a1 = ModelValue("1")
+a2 = ModelValue("2")
+a3 = ModelValue("3")
+a4 = ModelValue("4")
+a5 = ModelValue("5")
 
 carrier_set = { a0, a1, a2, a3, a4, a5 }
 designated_values = {a1, a2, a3, a4, a5 }
@@ -312,4 +331,26 @@ interpretation = {
 
 print(R_model_6)
 print(f"Model {R_model_6.name} satisfies logic {R_logic.name}?", satisfiable(R_logic, R_model_6, interpretation))
-# print(has_vsp(R_model_6, interpretation))
+if smt_is_loaded():
+    print(has_vsp(R_model_6, mimplication, True, True))
+else:
+    print("Z3 not loaded, skipping VSP check...")
+
+"""
+Generate models of R of a specified size using the SMT approach
+"""
+
+from vsp import logic_has_vsp
+
+size = 7
+print(f"Searching for a model of size {size} which witness VSP...")
+if smt_is_loaded():
+    solution = logic_has_vsp(R_logic, size)
+    if solution is None:
+        print(f"No models found of size {size} which witness VSP")
+    else:
+        model, vsp_result = solution
+        print(vsp_result)
+        print(model)
+else:
+    print("Z3 not setup, skipping...")
\ No newline at end of file
diff --git a/generate_model.py b/generate_model.py
index 5a46efc..ceb9681 100644
--- a/generate_model.py
+++ b/generate_model.py
@@ -67,7 +67,7 @@ def only_rules_with(rules: Set[Rule], operation: Operation) -> List[Rule]:
 
 def possible_interpretations(
         logic: Logic, carrier_set: Set[ModelValue],
-        designated_values: Set[ModelValue]):
+        designated_values: Set[ModelValue], debug: bool):
     """
     Consider every possible interpretation of operations
     within the specified logic given the carrier set of
@@ -100,7 +100,7 @@ def possible_interpretations(
             passed_functions = candidate_functions
         if len(passed_functions) == 0:
             raise Exception("No interpretation satisfies the axioms for the operation " + str(operation))
-        else:
+        elif debug:
             print(
                   f"Operation {operation.symbol} has {len(passed_functions)} candidate functions"
             )
@@ -120,7 +120,7 @@ def possible_interpretations(
 
 def generate_model(
         logic: Logic, number_elements: int, num_solutions: int = -1,
-        print_model=False) -> List[Tuple[Model, Interpretation]]:
+        print_model=False, debug=False) -> List[Tuple[Model, Interpretation]]:
     """
     Generate the specified number of models that
     satisfy a logic of a certain size.
@@ -136,9 +136,10 @@ def generate_model(
 
     for designated_values in possible_designated_values:
         designated_values = set(designated_values)
-        print("Considering models for designated values", set_to_str(designated_values))
+        if debug:
+            print("Considering models for designated values", set_to_str(designated_values))
 
-        possible_interps = possible_interpretations(logic, carrier_set, designated_values)
+        possible_interps = possible_interpretations(logic, carrier_set, designated_values, debug)
         for interpretation in possible_interps:
             is_valid = True
             model = Model(carrier_set, set(interpretation.values()), designated_values)
diff --git a/logic.py b/logic.py
index 44c5c6b..1cb383e 100644
--- a/logic.py
+++ b/logic.py
@@ -85,7 +85,7 @@ class Logic:
             name: Optional[str] = None):
         self.operations = operations
         self.rules = rules
-        self.falsifies = falsifies
+        self.falsifies = falsifies if falsifies is not None else set()
         self.name = str(abs(hash((
             frozenset(operations),
             frozenset(rules)
diff --git a/smt.py b/smt.py
index 3946975..4394713 100644
--- a/smt.py
+++ b/smt.py
@@ -1,18 +1,26 @@
 from itertools import product
-from typing import Dict, Generator, Optional, Tuple
+from typing import Dict, Generator, Optional, Set, Tuple, TYPE_CHECKING
 
 from logic import Logic, Operation, Rule, PropositionalVariable, Term, OpTerm, get_prop_vars_from_rule
 from model import Model, ModelValue, ModelFunction
 
-from z3 import (
-    And, BoolSort, Context, EnumSort, Function, Implies, Or, sat, Solver, z3
-)
+SMT_LOADED = True
+try:
+    from z3 import (
+        And, BoolSort, Context, EnumSort, Function, Implies, Or, sat, Solver, z3
+    )
+except ImportError:
+    SMT_LOADED = False
+
+def smt_is_loaded() -> bool:
+    global SMT_LOADED
+    return SMT_LOADED
 
 def term_to_smt(
     t: Term,
-    op_mapping: Dict[Operation, z3.FuncDeclRef],
-    var_mapping: Dict[PropositionalVariable, z3.DatatypeRef]
-) -> z3.DatatypeRef:
+    op_mapping: Dict[Operation, "z3.FuncDeclRef"],
+    var_mapping: Dict[PropositionalVariable, "z3.DatatypeRef"]
+) -> "z3.DatatypeRef":
     """Convert a logic term to its SMT representation."""
     if isinstance(t, PropositionalVariable):
         return var_mapping[t]
@@ -26,10 +34,10 @@ def term_to_smt(
 
 def logic_rule_to_smt_constraints(
     rule: Rule,
-    IsDesignated: z3.FuncDeclRef,
+    IsDesignated: "z3.FuncDeclRef",
     smt_carrier_set,
-    op_mapping: Dict[Operation, z3.FuncDeclRef]
-) -> Generator[z3.BoolRef, None, None]:
+    op_mapping: Dict[Operation, "z3.FuncDeclRef"]
+) -> Generator["z3.BoolRef", None, None]:
     """
     Encode a logic rule as SMT constraints.
 
@@ -63,10 +71,10 @@ def logic_rule_to_smt_constraints(
 
 def logic_falsification_rule_to_smt_constraints(
     rule: Rule,
-    IsDesignated: z3.FuncDeclRef,
+    IsDesignated: "z3.FuncDeclRef",
     smt_carrier_set,
-    op_mapping: Dict[Operation, z3.FuncDeclRef]
-) -> z3.BoolRef:
+    op_mapping: Dict[Operation, "z3.FuncDeclRef"]
+) -> "z3.BoolRef":
     """
     Encode a falsification rule as an SMT constraint.
 
@@ -132,7 +140,7 @@ class SMTLogicEncoder:
         self.carrier_sort, self.smt_carrier_set = EnumSort("C", element_names, ctx=self.ctx)
 
         # Create operation functions
-        self.operation_function_map: Dict[Operation, z3.FuncDeclRef] = {}
+        self.operation_function_map: Dict[Operation, "z3.FuncDeclRef"] = {}
         for operation in logic.operations:
             self.operation_function_map[operation] = self.create_function(operation.symbol, operation.arity)
 
@@ -143,10 +151,10 @@ class SMTLogicEncoder:
         self._add_logic_constraints()
         self._add_designation_symmetry_constraints()
 
-    def create_predicate(self, name: str, arity: int) -> z3.FuncDeclRef:
+    def create_predicate(self, name: str, arity: int) -> "z3.FuncDeclRef":
         return Function(name, *(self.carrier_sort for _ in range(arity)), BoolSort(ctx=self.ctx))
 
-    def create_function(self, name: str, arity: int) -> z3.FuncDeclRef:
+    def create_function(self, name: str, arity: int) -> "z3.FuncDeclRef":
         return Function(name, *(self.carrier_sort for _ in range(arity + 1)))
 
     def _add_logic_constraints(self):
@@ -171,9 +179,9 @@ class SMTLogicEncoder:
             )
             self.solver.add(constraint)
 
-    def extract_model(self, smt_model) -> Model:
+    def extract_model(self, smt_model) -> Tuple[Model, Dict[Operation, ModelFunction]]:
         """
-        Extract a Model object from an SMT model.
+        Extract a Model object and interpretation from an SMT model.
         """
         carrier_set = {ModelValue(f"{i}") for i in range(self.size)}
 
@@ -185,7 +193,8 @@ class SMTLogicEncoder:
         designated_values = {ModelValue(str(x)) for x in smt_designated}
 
         # Extract operation functions
-        model_functions = set()
+        model_functions: Set[ModelFunction] = set()
+        interpretation: Dict[Operation, ModelFunction] = dict()
         for (operation, smt_function) in self.operation_function_map.items():
             mapping: Dict[Tuple[ModelValue], ModelValue] = {}
             for smt_inputs in product(self.smt_carrier_set, repeat=operation.arity):
@@ -193,9 +202,12 @@ class SMTLogicEncoder:
                 smt_output = smt_model.evaluate(smt_function(*smt_inputs))
                 model_output = ModelValue(str(smt_output))
                 mapping[model_inputs] = model_output
-            model_functions.add(ModelFunction(operation.arity, mapping, operation.symbol))
+            model_function = ModelFunction(operation.arity, mapping, operation.symbol)
+            model_functions.add(model_function)
+            interpretation[operation] = model_function
 
-        return Model(carrier_set, model_functions, designated_values)
+
+        return Model(carrier_set, model_functions, designated_values), interpretation
 
 
     def _add_designation_symmetry_constraints(self):
@@ -218,7 +230,7 @@ class SMTLogicEncoder:
                 )
             )
 
-    def create_exclusion_constraint(self, model: Model) -> z3.BoolRef:
+    def create_exclusion_constraint(self, model: Model) -> "z3.BoolRef":
         """
         Create a constraint that excludes the given model from future solutions.
         """
@@ -254,7 +266,7 @@ class SMTLogicEncoder:
 
         return Or(constraints)
 
-    def find_model(self) -> Optional[Model]:
+    def find_model(self) -> Optional[Tuple[Model, Dict[Operation, ModelFunction]]]:
         """
         Find a single model satisfying the logic constraints.
 
@@ -274,12 +286,12 @@ class SMTLogicEncoder:
             pass
 
 
-def find_model(logic: Logic, size: int) -> Optional[Model]:
+def find_model(logic: Logic, size: int) -> Optional[Tuple[Model, Dict[Operation, ModelFunction]]]:
     """Find a single model for the given logic and size."""
     encoder = SMTLogicEncoder(logic, size)
     return encoder.find_model()
 
-def find_all_models(logic: Logic, size: int) -> Generator[Model, None, None]:
+def find_all_models(logic: Logic, size: int) -> Generator[Tuple[Model, Dict[Operation, ModelFunction]], None, None]:
     """
     Find all models for the given logic and size.
 
@@ -294,13 +306,14 @@ def find_all_models(logic: Logic, size: int) -> Generator[Model, None, None]:
 
     while True:
         # Try to find a model
-        model = encoder.find_model()
-        if model is None:
+        solution = encoder.find_model()
+        if solution is None:
             break
 
-        yield model
+        yield solution
 
         # Add constraint to exclude this model from future solutions
+        model, _ = solution
         exclusion_constraint = encoder.create_exclusion_constraint(model)
         encoder.solver.add(exclusion_constraint)
 
@@ -346,13 +359,13 @@ class SMTModelEncoder:
             is_designated = model_value in model.designated_values
             self.solver.add(self.is_designated(self.model_value_to_smt[model_value]) == is_designated)
 
-    def create_predicate(self, name: str, arity: int) -> z3.FuncDeclRef:
+    def create_predicate(self, name: str, arity: int) -> "z3.FuncDeclRef":
         return Function(name, *(self.carrier_sort for _ in range(arity)), BoolSort(ctx=self.ctx))
 
-    def create_function(self, name: str, arity: int) -> z3.FuncDeclRef:
+    def create_function(self, name: str, arity: int) -> "z3.FuncDeclRef":
         return Function(name, *(self.carrier_sort for _ in range(arity + 1)))
 
-    def add_function_constraints_from_table(self, smt_fn: z3.FuncDeclRef, model_fn: ModelFunction):
+    def add_function_constraints_from_table(self, smt_fn: "z3.FuncDeclRef", model_fn: ModelFunction):
         for inputs, output in model_fn.mapping.items():
             smt_inputs = tuple(self.model_value_to_smt[inp] for inp in inputs)
             smt_output = self.model_value_to_smt[output]
diff --git a/vsp.py b/vsp.py
index 0de5b10..fe97ccb 100644
--- a/vsp.py
+++ b/vsp.py
@@ -10,12 +10,12 @@ from model import (
     Model, model_closure, ModelFunction, ModelValue
 )
 
-SMT_LOADED = True
+from smt import SMTModelEncoder, SMTLogicEncoder, smt_is_loaded
+
 try:
     from z3 import And, Or, Implies, sat
-    from smt import SMTModelEncoder, SMTLogicEncoder
 except ImportError:
-    SMT_LOADED = False
+    pass
 
 class VSP_Result:
     def __init__(
@@ -139,7 +139,7 @@ def has_vsp_smt(model: Model, impfn: ModelFunction) -> VSP_Result:
     Checks whether a given model satisfies the variable
     sharing property via SMT
     """
-    if not SMT_LOADED:
+    if not smt_is_loaded():
         raise Exception("Z3 is not property installed, cannot check via SMT")
 
     encoder = SMTModelEncoder(model)
@@ -198,7 +198,7 @@ def has_vsp(model: Model, impfunction: ModelFunction,
     if model.is_magical:
         return has_vsp_magical(model, impfunction, negation_defined, conjunction_disjunction_defined)
 
-    return has_vsp_smt(model)
+    return has_vsp_smt(model, impfunction)
 
 
 def logic_has_vsp(logic: Logic, size: int) -> Optional[Tuple[Model, VSP_Result]]:
@@ -254,15 +254,15 @@ def logic_has_vsp(logic: Logic, size: int) -> Optional[Tuple[Model, VSP_Result]]
             )
         )
 
-    model = encoder.find_model()
+    solution = encoder.find_model()
 
     # We failed to find a VSP witness
-    if model is None:
+    if solution is None:
         return None
 
     # Otherwise, a matrix model and correspoding
     # subalgebras exist.
-
+    model, _ = solution
     smt_model = encoder.solver.model()
     K1_smt = [x for x in encoder.smt_carrier_set if smt_model.evaluate(IsInK1(x))]
     K1 = {ModelValue(str(x)) for x in K1_smt}