# User:Boyland

John Boyland, Professor, University of Wisconsin--Milwaukee.

I'm interested in using Twelf and have written a biased and under-informed tutorial of Twelf. I also have defined some library signatures for Twelf.

## Contents

## Desired Features

Here are some features that I would like to see in Twelf. I start with the simplest to implement (in my opinion).

### What free variables?

In Twelf, if you neglect to declare all the free variables in a theorem, you get the error message "Free variables In theorem." It would be nice if the error message included the names of these free variables. Update: of course Control-C Control-L will tell you all the free variables that have names. Of course, the implicit ones don't show. A %block declaration will show you the reconstructed term; it would be nice if %theorem did too.

### An `exists*` keyword

In Twelf currently, a theorem has the form

my-theorem : forall* {K1} {K2} {V1:term K1} {V2: term K2} forall {D:rule V1 V2} exists {K3} {V3:term K3} {DP:rule2 V1 V2 V3} true.

Sometimes the existentially bound variables are as extraneous as the `forall*`
bound variables. One would like to write

my-theorem : forall* {K1} {K2} {V1:term K1} {V2: term K2} forall {D:rule V1 V2} exists* {K3} exists {V3:term K3} {DP:rule2 V1 V2 V3} true.

Currently my workaround is to avoid the theorem syntax and use an explicit mode declaration:

my-theorem : (rule V1 V2) -> ({V3} (rule2 V1 V2 V3)) -> type. %mode +{K1} +{K2} +{V1:term K1} +{V2: term K2} +{D:rule V1 V2} -{K3} -{V3:term K3} -{DP:rule2 V1 V2 V3} (my-theorem D DP).

An `exists*` keyword should be a trivial addition.

### Multiple `%reduces` for a type family

If one declares two `%reduces` for a type family, the first is forgotten.
Since '%reduces` is very important for later reduction/totality checks, it would be
very helpful to collect all that are attached.

### Output coverage checking

It would be desirable to have output coverage checking done at the same time as mode checking, or at least to have the ability to check output coverage before a total declaration. I have often found it frustrating to get an output coverage error late in the process of writing a proof. And the "covers" check doesn't include output coverage

### Totality checker and theorem prover integration

Currently the theorem prover and totality checker live in different worlds, and neither trusts the other. This seems peculiar to a user of Twelf who may wish to use the theorem prover for certain obvious lemmas but the totality checker for more complex theorems. Currently this is impossible except in unsafe mode with the addition of assertions and "trustme" declarations.

I propose

1. The theorem prover should accept proved theorems: If a totality flag is set on a metatheorem, it should accept it as proved.

2. The totality checker should accept metatheorems proved automatically. If necessary, the theorem prover should deliver the proof to the totality checker internally (invisibly) for re-checking.

3. The `%assert` and `%trustme` declarations should have the same effect: in unsafe mode make the metatheorem trusted by both the totality checker and the theorem prover.

### Local Type Inference

After using Twelf for about a year and writing 100K lines of Twelf, the most frustrating and time consuming process is "debugging" coverage errors. Because unification is bidirectional, any mistake in the middle of a theorem will cause the pattern taht one is trying to match to not be exhaustive, even though it appears to be. It would be nice to use a system with a slightly less powerful type inference system that would prevent this long-distance effect of unification. Pierce and Turner have defined "Local Type Inference". It is interesting to wonder what this would mean for Twelf.

## Known Bugs

In decreasing order of imporance.

### Type Reconstruction Nontermination

Type reconstruction sometimes fails to terminate

xxtag : type. xxtag/a : xxtag. xxterm : xxtag -> type. xxa : xxterm xxtag/a. xxoutput : xxtag -> type. xxoutput/base : xxterm K -> xxoutput K. xxoutput/exists : (xxterm K -> xxoutput K') -> xxoutput K'. xxtrans : xxoutput K -> xxoutput K -> type. xxtrans/transitive : xxtrans E1 E2 -> xxtrans E2 E3 -> xxtrans E1 E3. xxtrans/inside : ({v} xxtrans (FE1 v) (FE2 v)) -> xxtrans (xxoutput/exists FE1) (xxoutput/exists FE2). xxtrans/gen-exists : {K1} {K2} {F:xxterm K1 -> xxoutput K2} {T} xxtrans (F T) (xxoutput/exists F). xxtrans-open-exists : (xxtrans (xxoutput/exists FE1) E2) -> ({v:xxterm K} xxtrans (FE1 v) E2) -> type. - : {K1} {K2} {O} xxtrans-open-exists (xxtrans/gen-exists K1 K2 ([v] (xxoutput/exists ([v'] FF v v'))) O) ([v'] (xxtrans/transitive (xxtrans/gen-exists _ _ ([v] (FF v v')) O) (xxtrans/inside ([v] (xxtrans/gen-exists _ _ _ v'))))).

The last term is incorrectly typed because `FF` depends on K1 and K2, but is not declared explicitly.
But in my version of twelf (twelf 1.5R3, with trustme, allegedly from August 2005) doesn't
appear to terminate when attempting to type it.

- Update: Apparently this is a known bug for which a fix was found by Jason Reed. http://en.scientificcommons.org/47856338

- The bug is still is in Twelf 1.7.1+ (r1887M)

### Coverage Checker Nontermination

The coverage checker sometimes fails to terminate.

%%%% coverage checking doesn't always terminate %%%% Definitions void : type. expr : type. unit : expr. add : expr -> expr -> expr. %%% Nested for evaluation expr-in-expr : (expr -> expr) -> type. expr-in-expr/add1 : expr-in-expr ([E] (add E _)). expr-in-expr/add2 : expr-in-expr ([E] (add unit E)). %%% ready for atomic evaluation atomic-expr : expr -> type. atomic-expr/add : atomic-expr (add unit unit). %% this "theorem" is actually false %theorem expr-in-atomic-expr-contradiction : forall* {E} {T} forall {AE:atomic-expr (T E)} {Tok:expr-in-expr T} exists {F:void} true. %worlds () (expr-in-atomic-expr-contradiction _ _ _). %total { } (expr-in-atomic-expr-contradiction _ _ _).

### Broken Invariant in Coverage Checker

Apparently (I learned of this problem from Rob Simmons), the coverage checker sometimes reports a broken invariant when checking code that uses the explicit-context idiom. This problem is not well understood, but manifests itself in several examples from the wiki, including Concrete representation and Tethered modal logic.

### Coverage checker checking with functions of type A -> A when A is not subordinate to itself.

If the coverage check, when expanding cases comes upon a term that might depend on a formal parameter of the same type, it will require a case that handles such a function, even if the type in question in not subordinate to itself. Such a case normally cannot be written because of freezing. (Example to come later).

### Higher-Order Blocks

Again, I learned of this problem from Rob Simmons: If a "block" includes a higher order function, then the coverage checker doesn't work. No example yet. This bug is considered low-priority.

### Not a block

If you use the name of something that is not a block where a block is expected:

%worlds (nat) (foo _ _ _).

then there is an "uncaught exception" rather than a helpful error message.

### Theorem Prover Nontermination

The prover sometimes fails to terminate. See example in my tutorial (Section 2.3) I have copied it here for completeness:

%%% Example of Twelf from "using-twelf" paper term : type. true : term. false : term. if : term -> term -> term -> term. zero : term. ty : type. bool : ty. int : ty. %% relations is_value : term -> type. is_value/true : is_value true. is_value/false : is_value false. is_value/zero : is_value zero. eval : term -> term -> type. eval/if_true : eval (if true X _) X. eval/if_false : eval (if false _ X) X. eval/if : eval (if E E1 E2) (if E' E1 E2) <- eval E E'. not_stuck : term -> type. not_stuck/value : not_stuck X <- is_value X. not_stuck/eval : not_stuck X <- eval X X'. of : term -> ty -> type. of/true : of true bool. of/false : of false bool. of/zero : of zero int. of/if : of X bool -> of Y T -> of Z T -> of (if X Y Z) T. %% theorems progress : (of X T) -> (not_stuck X) -> type. %mode progress +D1 -D2. %prove 2 T (progress T _).

The theorem prover is deprecated in Twelf, and fixing this bug is not required before producing a new release of Twelf.

## Challenges

### Is HOAS countable?

A challenge: define a mapping from HOAS to the natural numbers (or equivalently to a nameless term representation) and then prove that the mapping is isomorphic *using Twelf*. For details and sample mappings, see question on the Twelf Elf page: Talk:Ask_Twelf_Elf#Mapping_HOAS_isomorphically_to_the_natural_numbers.

Update: Rob has posted a solution (Concrete_representation) that defines a mapping between HOAS and nameless terms and proves that it is bijective. (I asked for an "isomorphic" mapping between HOAS and natural numbers, but should have said "bijective mapping.") Thanks Rob!

My personal challenge is to write a general technique for generating bijective mappings (with proof) between many kinds of terms and natural numbers. In general, of course, this would be undecidable. I hope only to handle cases that are used as AST types.

Update: I did it for a simple HOAS before POPL 2008 (see HOAS nat bijection), and in May 2008, finished a more complex example Indexed HOAS nat bijection that handle an indexed term language.

Success: I have implemented a bijection generator. See http://www.cs.uwm.edu/~boyland/papers/map-natural.html