Written Assignment 

English only 

Written AssignmentThese exercises are to be handed in December 22nd, either during the course or by email to Burak. Most tasks here by courtesy of Fritz Henglein and Andrzej Filinski. For your convenience, there is a page with some figures from TAPL. Arithmetic Expressions
Task 1)
Recall the language of Arithmetic Expressions.
Extend this language by two constructs " Task 2) Consider the extension with plus only. Show that evaluation always terminates, i.e., that any welltyped term reduces in a finite number of steps to a normal form. Hint: consider a variant of the size function that weighs certain subterms heavier than others. Lists and lambda calculus
Task 3)
Consider the pure lambdacalculus, extended by numerical values
For each function give an example application and show how it evaluates to the desired result.
Typing Rules Task 4) Consider the typing rule
If this rule is added to pure simply typed lambda calculus (Fig 91, p.103, like Exercise 3 without let,fix,numbers, booleans, if) then does Lemma 9.3.1 (inversion of the typing relation) hold as stated? What about Theorem 9.3.3 (uniqueness of types)? Is the rule admissible, that is, whenever the premises can be derived, the conclusion can be derived also? Subtyping Task 5) Consider a type system with rules (TVar), (TAbs), (TApp), (SRefl), (STrans), and (TSub) from Fig 151 (p. 186, TAPL), together with some unspecified collection of subtyping axioms at base types (e.g., Int <: Real). Show that, for typability in this system, (TEta) is equivalent to (SArrow). That is, show that a judgment Γ  t : T is derivable in the system extended with (TEta) iff it is derivable in the system exteded with (SArrow). (If you can't show the result in full generality, try for a weaker variant, such as only one direction of the biimplication, or a representative special case. Just be precise about what exactly you are showing.) (Optional tasks give bonus points.) Task 6) [optional] Write out the proof (in detail) of the progress theorem for FJ. Task 7) [optional] Exercise 22.3.9 (pp. 325326 of TAPL) (Write out constraint typing rules where a set of unused fresh variables F is threaded through a derivation). Prove that the system are equivalent, in an appropriate sense. 
