However, although values and normal forms are not the same in this language, the set of values is a subset of the set of normal forms. This is important because it shows we did not accidentally define things so that some value could still take a step.

Exercise: 3 stars, standard (value_is_nf)

Exercise: 3 stars, advanced (finish_progress_informal)

Complete the corresponding informal proof: Theorem: If ⊢ t ∈ T, then either t is a value or else t --> t' for some t'. Proof: By induction on a derivation of ⊢ t ∈ T.

• If the last rule in the derivation is T_Test, then t = test t₁ then t₂ else t₃, with ⊢ t₁ ∈ Bool, ⊢ t₂ ∈ T and ⊢ t₃ ∈ T. By the IH, either t₁ is a value or else t₁ can step to some t₁'.
  • If t₁ is a value, then by the canonical forms lemmas and the fact that ⊢ t₁ ∈ Bool we have that t₁ is a bvalue — i.e., it is either tru or fls. If t₁ = tru, then t steps to t₂ by ST_TestTru, while if t₁ = fls, then t steps to t₃ by ST_TestFls. Either way, t can step, which is what we wanted to show.
  • If t₁ itself can take a step, then, by ST_Test, so can t.
• (* FILL IN HERE *)
  

This theorem is more interesting than the strong progress theorem that we saw in the Smallstep chapter, where all normal forms were values. Here a term can be stuck, but only if it is ill typed.

Exercise: 3 stars, advanced (finish_preservation_informal)

Complete the following informal proof: Theorem: If ⊢ t ∈ T and t --> t', then ⊢ t' ∈ T. Proof: By induction on a derivation of ⊢ t ∈ T.

• If the last rule in the derivation is T_Test, then t = test t₁ then t₂ else t₃, with ⊢ t₁ ∈ Bool, ⊢ t₂ ∈ T and ⊢ t₃ ∈ T.
  Inspecting the rules for the small-step reduction relation and remembering that t has the form test ..., we see that the only ones that could have been used to prove t --> t' are ST_TestTru, ST_TestFls, or ST_Test.
  • If the last rule was ST_TestTru, then t' = t₂. But we know that ⊢ t₂ ∈ T, so we are done.
  • If the last rule was ST_TestFls, then t' = t₃. But we know that ⊢ t₃ ∈ T, so we are done.
  • If the last rule was ST_Test, then t' = test t₁' then t₂ else t₃, where t₁ --> t₁'. We know ⊢ t₁ ∈ Bool so, by the IH, ⊢ t₁' ∈ Bool. The T_Test rule then gives us ⊢ test t₁' then t₂ else t₃ ∈ T, as required.
• (* FILL IN HERE *)
  

Exercise: 3 stars, standard (preservation_alternate_proof)

Now prove the same property again by induction on the evaluation derivation instead of on the typing derivation. Begin by carefully reading and thinking about the first few lines of the above proofs to make sure you understand what each one is doing. The set-up for this proof is similar, but not exactly the same.

The preservation theorem is often called subject reduction, because it tells us what happens when the "subject" of the typing relation is reduced. This terminology comes from thinking of typing statements as sentences, where the term is the subject and the type is the predicate.

Exercise: 2 stars, standard (variation1)

Suppose that we add this new rule to the typing relation: Which of the following properties remain true in the presence of this rule? For each one, write either "remains true" or else "becomes false." If a property becomes false, give a counterexample.

• Determinism of step (* FILL IN HERE *)
  
• Progress (* FILL IN HERE *)
  
• Preservation (* FILL IN HERE *)
  

Exercise: 2 stars, standard (variation2)

Suppose, instead, that we add this new rule to the step relation: Which of the above properties become false in the presence of this rule? For each one that does, give a counter-example. (* FILL IN HERE *)

Exercise: 2 stars, standard, optional (variation3)

Suppose instead that we add this rule: Which of the above properties become false in the presence of this rule? For each one that does, give a counter-example. (* FILL IN HERE *)
☐

Exercise: 2 stars, standard, optional (variation4)

Suppose instead that we add this rule: Which of the above properties become false in the presence of this rule? For each one that does, give a counter-example. (* FILL IN HERE *)
☐

Exercise: 2 stars, standard, optional (variation5)

Suppose instead that we add this rule: Which of the above properties become false in the presence of this rule? For each one that does, give a counter-example. (* FILL IN HERE *)
☐

Exercise: 2 stars, standard, optional (variation6)

Suppose instead that we add this rule: Which of the above properties become false in the presence of this rule? For each one that does, give a counter-example. (* FILL IN HERE *)
☐

Exercise: 3 stars, standard, optional (more_variations)

Make up some exercises of your own along the same lines as the ones above. Try to find ways of selectively breaking properties — i.e., ways of changing the definitions that break just one of the properties and leave the others alone.

Exercise: 1 star, standard (remove_prdzro)

The reduction rule ST_PrdZro is a bit counter-intuitive: we might feel that it makes more sense for the predecessor of zro to be undefined, rather than being defined to be zro. Can we achieve this simply by removing the rule from the definition of step? Would doing so create any problems elsewhere? (* FILL IN HERE *)

Exercise: 4 stars, advanced (prog_pres_bigstep)

Suppose our evaluation relation is defined in the big-step style. State appropriate analogs of the progress and preservation properties. (You do not need to prove them.) Can you see any limitations of either of your properties? Do they allow for nonterminating commands? Why might we prefer the small-step semantics for stating preservation and progress? (* FILL IN HERE *)