ImpCEvalFun: Evaluation Function for Imp

Evaluation Function

Here's a first try at an evaluation function for commands, omitting WHILE.

In a traditional functional programming language like ML or Haskell we could write the WHILE case as follows:

    | WHILE b1 DO c1 END => 
        if (beval st b1) 
          then ceval_step1 st (c1;; WHILE b1 DO c1 END)
          else st 

Coq doesn't accept such a definition (Error: Cannot guess decreasing argument of fix) because the function we want to define is not guaranteed to terminate. Indeed, the changed ceval_step1 function applied to the loop program from Imp.v would never terminate. Since Coq is not just a functional programming language, but also a consistent logic, any potentially non-terminating function needs to be rejected. Here is an invalid(!) Coq program showing what would go wrong if Coq allowed non-terminating recursive functions:

     Fixpoint loop_false (n : nat) : False := loop_false n.

That is, propositions like False would become provable (e.g. loop_false 0 would be a proof of False), which would be a disaster for Coq's logical consistency. Thus, because it doesn't terminate on all inputs, the full version of ceval_step1 cannot be written in Coq -- at least not without one additional trick... Second try, using an extra numeric argument as a "step index" to ensure that evaluation always terminates.

Note: It is tempting to think that the index i here is counting the "number of steps of evaluation." But if you look closely you'll see that this is not the case: for example, in the rule for sequencing, the same i is passed to both recursive calls. Understanding the exact way that i is treated will be important in the proof of ceval__ceval_step, which is given as an exercise below. Third try, returning an option state instead of just a state so that we can distinguish between normal and abnormal termination.

We can improve the readability of this definition by introducing a bit of auxiliary notation to hide the "plumbing" involved in repeatedly matching against optional states.

Exercise: 2 stars (pup_to_n)

Write an Imp program that sums the numbers from 1 to X (inclusive: 1 + 2 + ... + X) in the variable Y. Make sure your solution satisfies the test that follows.

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Exercise: 2 stars, optional (peven)

Write a While program that sets Z to 0 if X is even and sets Z to 1 otherwise. Use ceval_test to test your program.

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Equivalence of Relational and Step-Indexed Evaluation

As with arithmetic and boolean expressions, we'd hope that the two alternative definitions of evaluation actually boil down to the same thing. This section shows that this is the case. Make sure you understand the statements of the theorems and can follow the structure of the proofs.

Exercise: 4 stars (ceval_step__ceval_inf)

Write an informal proof of ceval_step__ceval, following the usual template. (The template for case analysis on an inductively defined value should look the same as for induction, except that there is no induction hypothesis.) Make your proof communicate the main ideas to a human reader; do not simply transcribe the steps of the formal proof. ☐

Exercise: 3 stars (ceval__ceval_step)

Finish the following proof. You'll need ceval_step_more in a few places, as well as some basic facts about ≤ and plus.

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Determinism of Evaluation (Simpler Proof)

Here's a slicker proof showing that the evaluation relation is deterministic, using the fact that the relational and step-indexed definition of evaluation are the same.