Naming conventions
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These are naming conventions for Twelf code. This article should be considered an appendix to/chapter of the style guide.
Contents
Syntax
The uninhabited type
Defining an uninhabited type is important for reasoning from false - however, the name of that type varies from user to user, with void, false, and absurd perhaps being the most common.
Standard usage is to use void, and to freeze the type immediately after its introduction, like this:
void : type. %freeze void.
Judgments
Factored judgments
Factoring is a technique used to reduce backtracking. If a rule does some computation, it might then sends the result to another rule that continues based on the output of its computation. The style introduced here works well with factoring - if you are creating "helper judgment," just replace the forward slash in the primary judgment with a hyphen to come up with the name.
eval : state -> state -> type. eval-branch-if-nil : list -> state -> state -> type. ... eval-branch-if-nil/nil : eval-branch-if-nil list/nil ... eval-branch-if-nil/cons : eval-branch-if-nil (list/cons _ _) ... eval/branch-if-nil : eval S1 S2 <- do-some-computation-on-state S1 L <- eval-branch-if-nil L S1 S2.
Introduction rules
If a type family is inhabited by exactly one rule, it is recommended to use i (short for introduction) as the case name for the lone inference rule. For example, the identity (syntactic equality) relation for natural numbers would be defined in the following way.
seq-nat : nat -> nat -> type. seq-nat/i : seq-nat N N.
This also applies to pieces of syntax in the object language, but it appears less frequently there.
A note on word order
Here as in the previous section, there is a legitimate case for reversing the word order, calling a list of natural numbers nat-list instead of list-nat on the other, and for calling the three-place relation implementing addition of natural numbers nat-plus instead of plus-nat
The construction that is not used here looks more like declarations would look were a module system is in place (i.e. nat::list, but the first is more readable after combination - for instance list-list-nat can be read out loud as "a list of lists of nats," whereas the equivalent formulation does not compose as well - nat-list-list can best be read as "a natlist list."
Internal consistancy is most important here, but the construction (structure)-(object) for syntax (i.e. list-nat) and (adjective/verb/operation)-(object) (i.e. wellformed-env, sort-list, or plus-nat) is preferred.
Different types of equality
Metatheorems
Metatheorem names
One way to think about a naming strategy for many metatheorems is that many simple theorems act like admissibility arguments for other rules. For instance, imagine an alternate form of plus-nat/z that has the zero in the second place instead of the first. It might be written like this if it were a real definition:
plus-nat/z-alt : plus N z N.
Following that logic, a lemma establishing the admissibility argument might be titled
plus-nat-z-alt : {N} plus-nat N z N -> type.
Beyond that suggestion, the properties that we wish to prove about our inference rules vary to the point where some measure of consistency is the only reasonable guideline.
can- and -unique metatheorems
A judgment behaves like a function (in the meta-logical sense) on particular "input" derivations if it satisfies two properties. The first is that given the "input" derivations, the judgment can be derived, i.e. an effectiveness lemma. The second is that given two derivations of the judgments with the same "input" derivations, the results are actually the same, i.e. a uniqueness lemma. It is common practice to name the metatheorems proving the first property can-judgmentname and the second property judgmentname-unique.
can-plus-nat : {N1:nat} {N2:nat} plus-nat N1 N2 N3 -> type. %mode can-plus-nat +D1 +D2 -D3. ... plus-nat-unique : plus-nat N1 N2 N3 -> plus-nat N1 N2 N3' -> seq-nat N3 N3 -> type. %mode plus-nat-unique +D1 +D2 -D3. ...