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          • ACL2 as an Interactive Theorem Prover
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          • Pages Written Especially for the Tours
            • A Tiny Warning Sign
            • About the Prompt
            • ACL2 Symbols
            • Common Lisp
            • About Types
            • The Event Summary
            • Tours
            • An Example of ACL2 in Use
            • Corroborating Models
            • The Proof of the Associativity of App
            • Guards in ACL2
            • Numbers in ACL2
            • About the Admission of Recursive Definitions
            • Free Variables in Top-Level Input
            • Common Lisp as a Modeling Language
            • ACL2 as an Interactive Theorem Prover (cont)
            • A Flying Tour of ACL2
            • Analyzing Common Lisp Models
            • A Walking Tour of ACL2
            • About the ACL2 Home Page
            • The Theorem that App is Associative
            • The End of the Walking Tour
            • Proving Theorems about Models
            • Functions for Manipulating these Objects
            • ACL2 Conses or Ordered Pairs
            • The Associativity of App
            • Models of Computer Hardware and Software
            • How To Find Out about ACL2 Functions (cont)
            • A Sketch of How the Rewriter Works
            • ACL2 System Architecture
            • Running Models
            • ACL2 is an Untyped Language
            • How To Find Out about ACL2 Functions
            • How Long Does It Take to Become an Effective User(Q)
            • The Admission of App
            • Guiding the ACL2 Theorem Prover
            • What Is ACL2(Q)
            • Conversion
            • An Example Common Lisp Function Definition
            • ACL2 Characters
            • You Must Think about the Use of a Formula as a Rule
            • The WARNING about the Trivial Consequence
            • Guessing the Type of a Newly Admitted Function
            • ACL2 Strings
            • What is a Mathematical Logic(Q)
            • Revisiting the Admission of App
            • What is a Mechanical Theorem Prover(Q)
            • About Models
            • What is Required of the User(Q)
            • Hey Wait! Is ACL2 Typed or Untyped(Q)
            • The Falling Body Model
            • Rewrite Rules are Generated from DEFTHM Events
            • Other Requirements
            • A Typical State
            • The Time Taken to do the Associativity of App Proof
            • The Simplification of the Induction Conclusion (Step 1)
            • The Rules used in the Associativity of App Proof
            • Subsumption of Induction Candidates in App Example
            • Models in Engineering
            • Symbolic Execution of Models
            • Suggested Inductions in the Associativity of App Example
            • Overview of the Proof of a Trivial Consequence
            • Evaluating App on Sample Input
            • The Induction Step in the App Example
            • Modeling in ACL2
            • The Simplification of the Induction Conclusion (Step 9)
            • The Induction Scheme Selected for the App Example
            • The First Application of the Associativity Rule
            • Overview of the Expansion of ENDP in the Induction Step
            • Flawed Induction Candidates in App Example
            • ACL2 as an Interactive Theorem Prover
            • Using the Associativity of App to Prove a Trivial Consequence
            • The Simplification of the Induction Conclusion (Step 8)
            • The Instantiation of the Induction Scheme
            • The Final Simplification in the Base Case (Step 0)
            • Overview of the Simplification of the Induction Step to T
            • Overview of the Simplification of the Induction Conclusion
            • On the Naming of Subgoals
            • Nontautological Subgoals
            • What is a Mechanical Theorem Prover(Q) (cont)
            • The Simplification of the Induction Conclusion (Step 12)
            • The Simplification of the Induction Conclusion (Step 10)
            • The End of the Flying Tour
            • The Base Case in the App Example
            • The Simplification of the Induction Conclusion (Step 11)
            • The Final Simplification in the Base Case (Step 3)
            • The Expansion of ENDP in the Induction Step (Step 1)
            • The Expansion of ENDP in the Induction Step (Step 0)
            • The End of the Proof of the Associativity of App
            • Overview of the Simplification of the Base Case to T
            • The Summary of the Proof of the Trivial Consequence
            • The Simplification of the Induction Conclusion (Step 7)
            • The Simplification of the Induction Conclusion (Step 6)
            • The Simplification of the Induction Conclusion (Step 5)
            • The Simplification of the Induction Conclusion (Step 4)
            • The Simplification of the Induction Conclusion (Step 2)
            • The Simplification of the Induction Conclusion (Step 0)
            • The Justification of the Induction Scheme
            • The Final Simplification in the Base Case (Step 1)
            • The Expansion of ENDP in the Induction Step (Step 2)
            • Popping out of an Inductive Proof
            • Overview of the Expansion of ENDP in the Base Case
            • The Simplification of the Induction Conclusion (Step 3)
            • The Q.E.D. Message
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            • Name the Formula Above
            • A Trivial Proof
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The Associativity of App

The Associativity of App

ACL2!>(let ((a '(1 2)) 
            (b '(3 4)) 
            (c '(5 6))) 
        (equal (app (app a b) c) 
               (app a (app b c)))) 
T 

Observe that, for the particular a, b, and c above, (app (app a b) c) returns the same thing as (app a (app b c)). Perhaps app is associative. Of course, to be associative means that the above property must hold for all values of a, b, and c, not just the ones tested above.

Wouldn't it be cool if you could type

ACL2!>(equal (app (app a b) c) 
             (app a (app b c))) 
 

and have ACL2 compute the value T? Well, you can't! If you try it, you'll get an error message! The message says we can't evaluate that form because it contains free variables, i.e., variables not given values. Click here to see the message.

We cannot evaluate a form on an infinite number of cases. But we can prove that a form is a theorem and hence know that it will always evaluate to true.